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By John Marshai,!.. f>!;, A Guide to Greek Tragedy. For English Readers. By L. Campbell. 7^. 6ci. The School Euclid. An Edition of Euclid, Books I.-VI. With Notes and Exercises. By Daniel Bkent. Books I. and II. separately. D..U,— - ... - ^^^ orms /ELL. 1924 012 324 970 Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924012324970 PEACTICAL PHYSICS AN IKTRODUCTOEY HANDBOOK FOE THE PHYSICAL LABOEATOEY 1 / BY W. F. BAREETT, F.E.S.E,, M.E.I.A, Etc. PROFEiSSOR OF EXPERIMENTAL PHYSICS IN THE ROYAL COLLEGE OF SCIENCE FOR IRELAND AND W. BEOWN DEMONSTRATOR OF PHYSICS IN THE SAME gOLLEGE PART I PHYSICAL PROCESSES AND MEASUREMENTS THE PROPERTIES OE MATTER fLontian peecival and CO. 1892 hrr^f+M- Practical Physics. An Introductory Handbook for the Physical Laboratory. In Three Parts. Part I. — Physical Processes and Measurements — The Properties of Matter. Part II. — Heat, Sound, and Light. Part III. — Electricity and Magnetism — Electrical Measure- ments. LONDON: PERCIVAL AND CO. PREFACE It is needless at the present day to insist on the fact that the study of Experimental Physics is incomplete without a course of laboratory practice ; Professor Huxley wrote upwards of thirty years ago " Mere book-learning in physical science is a sham and a delusion, real knowledge in science means persojial acquaintance with the facts, be they few or many." Moreover, as a means of education the value of Practical Physics is obvious, for it affords a training in careful observation, reasoning power, and alertness of mind, probably unsurpassed by other kinds of laboratory work ; like all practical knowledge, it is a valuable aid in the cultivation of comynon sense, by which is meant a just estimate of the relative value and due proportion of different events. Nevertheless, it is only in recent years that Practical Physics has been recognised as an integral part of scientific education ;* this, doubtless, has been due to various causes : the equipment of a physical laboratory is expensive, but, on the other hand, its working expenses are less than many other kinds of laboratory work ; the chief cause, * The enlightened policy of the Director of the Science and Art Depart- ment enabled certificated science teachers, so long ago as 1871, to obtain practical instruction in Physics, Cliemistry, and Biology, in the labora- tories £rt South Kensington. At that time systematic instruction in Prac- tical Phy^fci was so much the exception that no English text-hook on this subject eSRed to aid those of us who then took part in giving the course. IV PRACTICAL PHYSICS however, of the recognition and the growing importance of Practical Physics is doubtless the rapid progress which the last decade has witnessed in electrical dis- covery and invention, and the conseqixent development of new and important industries. The number of exist- ing text-books in Practical Physics is still, however, very small compared with the many admirable handbooks on Practical Chemistry. Moreover, less uniformity exists in the mode of teaching the former than of the latter ; each laboratory has its own methods, this, whilst not undesirable, throws greater labour on the teacher, who usually prepares for his students MS. notes of each ex- periment with such brief instructions as are necessary. Though there is a freedom and flexibility in this system of descriptive sheets of separate experiments, it has the serious disadvantage of piecemeal instruction : the student does not see beyond the experiment immediately before him, and has only his laboratory note-book to fall back upon for reference. The present little volume was undertaken to meet the needs of students in my own laboratory, but it is hoped that it may be found useful to others who are beginning the study of Practical Physics, in a fairly furnished physical laboratory. Whilst the student is supposed to have the use of ordinary instruments of precision, he should be encouraged as far as possible to make for himself the simpler forms of apparatus. The time devoted to Practical Physics by the general student is usually insufficient to allow of this being done, or of much practice in manipu- lation beyond what is afforded by a systematic course of physical measurement. This is a defect the teacher should recognise : occasional exercises, such as Experi- ments 7, 18, etc., are therefore introduced to give facility in the construction of physical apparatus, though it PREFACE V is impossible to include within the limits of this small volume much information in this direction. The arrange- ment and scope of this book will be seen to differ in some respects from those of the two or three excellent text-books which now exist in England, and the value of which I desire cordially to acknowledge. Only General Physics is treated in the present Part, but under this head, as will be seen by the Table of Contents, a somewhat wide range of subjects is covered, for, as in other colleges, our course on Practical Physics requires to be adapted to various classes of students. It may therefore be convenient to point out that Chapters I. to V. are intended for general students ; Chapters VI. to IX. are adapted chiefly for those who intend to take up Engineering, whilst Chapter X. is intended for students who are taking up more fully Physics or Chemistry. In the Introductory Chapter the maiu object aimed at is to awaken thought in the mind of the student : if any ob- ject to certain parts of this chapter as going beyond the proper boundary of Physics, they can instruct their class to begin at once with the practical work of Chapter II. As far as possible a logical sequence is followed in the order of subjects, and the experiments lead up from sim- pler to more difficult ones. After stating the nature of each experiment and the instruments needed to perform it, the student is told as briefly as possible what to do, and an illustrative example is worked out for his guid- ance; a corresponding exercise is then given for the student to perform. The object of each lesson is to confine the student's attention to the particular experi- mental work in hand, and to avoid distracting his attention and encumbering the page with theoretical considerations. The student who uses this book ought therefore to have already attended a course of lectures on Experi- VI PEACTICAL PHYSICS mental Physics, and if unacquainted with the theoretical explanation of the experiment he is pursuing, he should consult at his leisure the references given in the footnotes. In some cases, however, where the ordinary text-books give less help in this direction, my collaborateur has added in the form of appendices the necessary mathematical demonstrations. As the labour of the preparation of this volume, with the exception of Chapters I. and X., has been shared by my colleague, Mr. W. Brown, his name is in- cluded on the title-page. In conclusion, I have gratefully to acknowledge the kind help given by many friends in the revision of the proof-sheets ; at the same time it will of course be under- stood that those who have been good enough to add to the accuracy of this little volume are in no wise responsible for its shortcomings. To Principal Garnett, M.A., D.C.L., and to Professor Fitzgerald, F.T.C.D., F.E.S., thanks are especially due for their valuable revisions and suggestions, and to Mr. F. J. Trouton, M.A., D.Sc, for his careful reading of the whole of the proofs. I have also warmly to thank Professor Eticker, M.A., F.RS., and the Eev. M. H. Close, M.A., Treasurer of the Eoyal Irish Academy, for their kind and important revision of the parts sub- mitted to them. For several of the woodcuts I am indebted to the kindness of Mr. Hicks of Hatton Garden, London, and Mr. Yeates of Dublin ; these or other well- known instrument-makers can supply all the instruments required in this book. W. F. BAEEETT. Royal College of Science for Ireland, September 1892.- CONTENTS CHAPTER I INTRODUCTION AND DEFINITIONS CHAPTER n MEASUREMENT OF SPACE Section I. — Length Measurement EXPERIMENT PAGE 1. Compass and Beam Compass . . , .11 2. Verniers — Baeombtek Reading . . . .13 3. Calipees — Mbasueement of Solids . . .15 4. Miceombtee Screw ... .16 5. Spherometee . . . . . .18 6. Cathetometee .... .20 7. Construction of mm. Scale on Glass . . .24 8. Dividing Engine . . . . . .26 Section II. — Angular Measurement 9. Methods of Measuring Angles . . .27 10. Optical Levee ..... 29 11. The Sextant — Measurement of Heights . . 31 12. Simple Methods of Measuring Heights . . 36 Section III. — Area Measurement 13. Methods of Measuring Aeeas . . . .37 14. Planimetbe . . . . . .39 15. Geaphic Representation of Results . . 40 mi PKACTICAL PHYSICS Section IV. — Volwme Measurement experiment pace 16. Methods of Measuring Volumes . • .43 17. Calibeation and Area of Tubes . . .45 18. CONSTEUOTION OF BuKET'l'E . . • .47 CHAPTER III MEASUREMENT OF TIME 19. Method op Passages . . . • .49 20. Method of Coincidences . ... 51 21. Chronoscopbs . • 63 CHAPTER IV MEASUREMENT OF MASS 22. Use of the Balance ... .57 Specific Gravities of — 23. An Insoluble Body Heavier than Water . . 63 24. An Insoluble Body Lighter than Water . . 64 25. A Soluble Body Heavier than Water . . 65 26. A Liquid by Immersion of a Solid . . .65 27. A Liquid by Mohe's Balance . . . .66 28. A Liquid by U-Tubb, Simple and Double . . 68 29. A Liquid by the Specific Gravity Beads . . 70 30. A Liquid by Various Hydrometers — (i.) Ordinary Hydrometer, (ii.) Jwaddell's Hydrometer, (iii.) Sikb's Hydrometer, (iv. ) Nicholson's Hydrometer 73 31. A Liquid by the Specific Gravity Bottle . . 77 32. A Powder by the Specific Gravity Bottle . 79 33. Specific Gravity by the Stekeometeb . . 83 34. Specific Gravity of a Mechanical Mixture of Solids 86 35. ,, ,, A Mechanical Mixture op Liquids 87 36. Weight op a Body in Vacuo . . . .88 37. Density of Gases ... . . 89 CONTENTS IX CHAPTEE V MEASUREMENT OF FLUID PRESSURE EXPERIMENT PAGE 38. Peessuee proportional to Depth . ■ 93 39. The Barometeb ... 95 40. Atmospheric Pressure bt simple means 99 41. Measurement of Heights by Barometer 101 42. Boyle's Law, above Atmospheric Pressure 103 43. Boyle's Law, below Atmospheric Pressure . . 104 44. M'Leod Gauge . . .106 CHAPTER VI MEASUREMENT OF FORCE 45. Dynamometer . 110 46. Whirling Table . . Ill 47. Brake Horse-Power . . 112 48. Atwood's Machine . 115 49. Laws of Falling Bodies . . . 117 50. Isoohronism of a Pendulum 120 51. Simple Pendulum, Value of ' = MY; unit density = [M] / \l}\ Telocity (t) is the time rate of change of position ; unit velocity is the velocity of 1 cm. per sec. = [L] / [T]. AccELEEATiox (ffl) is the time rate of change of velocity; unit acceleration is the gain of unit velocity {i.e. 1 cm. per sec.) every second = [L] / [T*]. Momentum (w) is the quantity of motion in a moving body; and is the product of the velocity of a body iuto its mass, r/i = Mr ; unit momentum = [M] [L] / [T]. FoECE (F) is the time rate of change of momentum; and is the product of mass into acceleration, F = Ma ; unit force = [M] [L] / [T*]. The unit of force is the dyne, and is that * The English equivalent of "per" Is " divided bv," and is indicated by the sloping line /, or h j the negative sign to the index ; thus velocity is L/r or LT~S and so on. 10 PRACTICAL PHYSICS force ■which, acting on a gramme of matter for 1 second, generates in it a velocity of one centimetre per second. The moment of a force about a point or axis is the pro- duct of the magnitude or component of the force in a plane perpendicular to the axis, into the distance from the axis of the line of action of the component. If a force of magnitude F acts at a distance p from and per- pendicularly to the given axis, Fp is the magnitude of its moment about the axis. Work (W) is done by a force on a body when the body is displaced in the direction of the force, and is measured by the product of the force into the dis- placement (s), W = Fs. The unit of work is the erg, and is the force of a dyne moving its point of application through the length of a centimetre, therefore = [M] [L"] / [T^]. ENERav (E) is the capacity for doing work, and is measured by the quantity of work it can do ; it has there- fore the same dimensions as work : [M] [L-] / [T^]. If the energy given out or gained by a system (1) results in motion, the system is said to lose or gain kinetic encrgi/ ; if, on the other hand, (2) the result be a change of con- figuration in its parts, the system is said to gain or lose potential energy. PoAVER or activity (P) is the mean rate at which a force does work in a given time, and is measured by the quotient of the work done in the time by the time taken : P = W/T. The unit is one erg per second = [M] [L-'] / [T']. CHAPTER II MEASUEEIIEXT OF SPACE Section I. — Length Measurcnunt Experiment 1. — To find the distance between two marks on a sheet of paper. Instruments required. — An ordinary or "hair" compass, a beam compass, and a millimetre scale. By adjusting the points of the compass to the extremities of the length to be measured, and then applying the compass to the scale, the number of units and fractional parts of a unit of length in the given space can be determined. To enable the adjustment of the compasses to be made with accuracy, a fine motion can be given to one leg of the "hair compass" (Fig. 1) by means of a screiv, or an eccentric, A. For lengths greater than the space of an ordinary compass the " beam compass" (Fig. 2) is employed. This consists of a long brass rod A, vdth tvro steel points B and C, the latter capable of sliding along the rod A, and of being clamped 12 PRACTICAL PHYSICS in any required position by the screw F. The point B has also a fine adjustment by means of the screw D. To make a measurement, apply the compasses to the two points whose distance apart is required, adjusting Fig. 2. accurately by means of the screw J), then apply the points to the scale, estimating by eye the decimal parts of a scale division. Eepeat the measurement three or four times, using different parts of the scale. The brass rod A is itself graduated in some instruments, so that the required length can be read directly. Example. — Find on a millimetre scale the length that corresponds to 5 inches. Enter your results thus : — 1st trial . .12-7 mm. 2nd „ . . . 12-6o „ 3rd „ . . . 12-75 „ Mean =12-7 mm. Exercises. 1. Find the distance between two scratches on a plate of glass. MEASUEEMENT OF LENGTH 13 2. Test the accuracy of the ruling of the piece of millimetre or curve paper supplied to you.* 3. Compare three centimetre scales, engraved re- spectively on glass, steel, and paper. c - 24 0'048 of an inch. 3+ Experiment 2. — How to use a verniep and to find the " least count " of a vernier. Instruments required. — A scale with its accompanying vernier. In Fig. 3 is shown a scale and vernier wherein 25 divisions on the vernier CD correspond to 24 divisions on the scale AB. Each scale division is equal to ^^ or '05 of au inch, therefore each vernier division will be m B Fig. Hence each scale division exceeds each vernier division by ('OS - -048) = -002 of an inch, which is the " least count " of this particular vernier. In order to use the vernier after getting the " least count " (d), first read off the graduation (E) immediately below the zero of the vernier; then count wp the vernier from its zero to the division that exactly coincides with a scale division, say the nth vernier division from its zero ; the total reading will then he (E + nd). * Millimetre paper is paper ruled into small squares, such as is shown in Fig. 19 ; it is essential for work in the physical laboratory, and can be obtained from Messrs. Williams & Norgate, of London, or Messrs. DoUard, 14 PRACTICAL PHYSICS The general theory of the vernier is given in Ap- pendix, § 1. Example. — The vernier zero stands between 30 '3 5 and 30 '4 inches, and coincidence occurs at the 18th division from zero, therefore the total reading is 30-35 + (18 X -002) =30-386 inches. In practice proceed as follows : (i.) Eead first the inches and nearest fraction of an inch on the scale, e.g. (Fig. 3) 29-65. (ii.) Then note the point of coincidence between the divisions on the scale and vernier, and read the nearest whole number on the vernier at or below the point of coincidence. Let this be 2 ; add this to the scale reading in the second place of decimals, e.g. 29-67. (iii.) Then observe how many (if any) of the divisions on the vernier are between the point of coincidence and the figure 2 ; let this be 3, as the least count is = '002 of an inch, 3 X '002 = '006; add this to the previous reading, then 29-676 is the final reading. This is the usual form of scale and vernier used in standard baro- meters in the United Kingdom. Note. — As each figure marked on the vernier includes 5 divisions, there will be 15 vernier divisions at the figure 3; hence 15 x '002 = -03 of an inch, and so on with the other figures on the vernier, which therefore indicate the number of hundredths of an inch to be added of Dublin, and other stationers. It is to be had ruled into squares whose sides are fractions of an inch or of a centimetre, the most convenient paper being that which has every fifth or tenth division ruled in a thicker or a different-coloured line. MEASTJREMEXT OF LEXGTH 15 to the reading of the scale. For ordinary barometric pur- poses this reading is sufficient, and indeed the construction of the barometer in general makes any finer reading only a misleading attempt at accuracy. Excirisc. Find the " least count " of the verniers attached to the cathetometer (Fig. 8), barometer (Fig. 35), calipers (Fig. 4), spectrometer, and sextant. Eead the barometer each day for a week and enter your readings. E-qxrimint 3. — To find the length of a cylindrical bar. Instrument required. — The slide calipers. The slide calipers (Fig. 4) consist of a rectangular bar of metal A, with one jaw B fixed at right angles to it at Fig. 4. one end, and another jaw C fixed to the sliding piece D, which is kept in a half T-groove by means of the spring E. The jaw C moves parallel to B; A has a scale in centimetres engraved upon it, and D has a vernier also engraved on it, which enables the scale to be read to tenths of a millimetre. To make a measurement, open the jaws and insert the 16 PEACTICAL PHYSICS bar, then close C till the two jaws just touch the ends of the bar ; now read off the scale and vernier. Example. — The scale in another instrument is divided into twentieths of an inch, and the " least count " of the ver- nier is -001 inch. The zero of the vernier stands between 1-55 and 1-6 inch, and coincidence occurs at the 14th division from zero. Therefore the length of the bar is l-55 + (14x -001)= 1-564 inches. N'ote. — In order to measure an inside length — such, for example, as the internal diameter of a glass tube — the width of the two jaws must be added on to the scale reading. This is called the " outside correction." Exercises. 1. Find the mean diameter of a bar, and calculate its cross sectional area and volume. 2. Find the mean diameter of a sphere, and calculate its surface and volume. 3. rind the mean thickness of the glass at the mouth of a jar by direct measurement and by taking the external and internal diameter (see Table III., 6). Experiment 4. — To measure the diameter of a wire. Instrujnents required. — The wire-gauge or micrometer screw, and a millimetre scale. The micrometer screw (Fig, 5) consists of a piece of metal turned twice at right angles ; a steel plug ^'8- ^- A with plane end is fixed in one bend or arm, and in the other arm a fine screw B works MEASUREMENT OF LENGTH 17 smoothly ; its face is also plane and parallel to that of A. Along a fixed arm a scale is engraved, and the cap D, which is a part of the screw B, is also divided into a certain number of equal divisions. One complete turn of the cap moves the faces A and B nearer to or farther from each other by an amount equal to the distance of one scale division of the fixed arm. To make a measurement, screw back B, insert the wire, screw up B till the wire is just caught between A and B, then read off the position indicated on the fixed arm and the division on the cap. Example. — (Each division on the fixed arm is = -^ mm., and there are 20 divisions on the cap D, therefore each division on the cap measures to -^ X -^ = :^^ mm.) When A and B are just touching, the reading on D must be taken. It ought to be ; if not, the difference must be noted and allowed for.* Let the thickness of a piece of wire give a reading of 3 on the fixed scale and 18 on the cap, therefore the diameter of the wire is (3x^)+l§=l-95 mm. Another and superior form of this instrument, first de- vised by Mr. Yeates of Dub- lin, is shown in Fig. 6. Pig ,. * Care must be used that similar pressure, as indicated by the sense of touch, is given in each case. PART I 18 PRACTICAL PHYSICS Here greater accuracy is obtained by enlarging t e engraved circle attached to the micrometer screw, which is turned by the milled head C. The zero of the instrument can also be adjusted by loosening the clamping screw which fastens the soUd plug A. A scale engraved on D shows the number of complete revolutions made hy the screw ; if the pitch of the screw be 100 to the inch, and the engraved circle be divided into 100 parts, each division will correspond to ^^ of ^-^ of an inch or 10000 of a° "ich. Exercises. 1. Find the diameter of a wire ; measure at several points and take the mean. 2. Find the average thickness of a strip of paper. 3. Find the thickness of a piece of microscopic cover- ing glass. Experiment 5. — To measure a thin plate and the radius of curvature of a curved surface. Instruments required. — A sphero- meter, a plane glass surface, and a millimetre scale. The spherometer (Fig. 7) consists of a triangular frame H supported on three feet, whose extremities form an equilateral triangle, and in the middle of the frame is a fourth foot capable of moving up and down perpendicularly by means of the fine screw chased upon it. The disc D, which is graduated round the circumference SIEASrEKMEvT OF LENGTH 19 turns with the central foot ; the edge of this disc serves as an index to read the scale S which is attached to the frame. One complete turn of the disc D rai;es or lowers the centre foot through one division of the scale S, and there- fore gives the perpendicular distance between the central foot and the plane of the other three feet. To use the spherometer, first test the zero by placing the instrument on the glass plane and screw D up or down tUl the whole four feet rest equally on the surface. This is best ascertained by the sense of touch, i.e. when the rocking of the instrument on the plane just ceases. To find the thickness of a thin plate, place the central foot over the thin plate, the other three feet resting on the plane surface ; then raise or lower the screw until the four feet bear equally ; when the rocking ceases read the distance through which the screw has been raised. The difference between this reading and the zero is equal to the thickness required. To find the radius of curvature of a surface, such as a lens or mirror, place the whole instrument on the curved body, and raise or lower the central foot until all four bear equally, and no rocking is felt. If E = the radius of curvature of the surface to be measured, I = the distance AB, a = the distance of D above or below the plane of A, B, C,as measured by the scale E and disc F, P a then E = — + - (see Appendix, § 2). 20 PRACTICAL PHYSICS Example.— {Vae scale S is divided into I mm. and there are 100 divisions on the disc D, therefore one division on D reads to ^y-rhv^^ '^^™-)- ^"^^ reading on the scale S stands between 4 and 5, and the disc reads 70, a convex lens being used. a = (4 X i) + ^Vu =1-175 mm., and I = 42 mm., 42^ 1-175 ^^^„ . T>^ L — -—=250-8 mm. ••7-05^ 2 Exercises. 1. Find the thickness of a piece of microscopic cover- ing glass. 2. Determine the radius of curvature of a concave mirror and also that of a convex lens. Experiment 6. — To measure the difference in height between two marks by means of the cathetometer. Instruments required. — A cathetometer, a centimetre scale, and a plumb Hne. The cathetometer (Fig 8) consists of a strong vertical support AB, with a scale engraved upon it. This rod is capable of moving round a vertical axis by means of a journal at its lower end turning in the tripod stand. This tripod stand contains three levelling screws, which enable the rod AB to be made perpendicular, as indicated by the two spirit-levels placed at right angles to each MEASUREMENT OF LENGTH 21 other. This is the iirst adjustment to be made in the use of the cathetometer. A carriage C, support- ing the telescope T, slides up and down the vertical rod, and can be clamped at auy position. This car- riage has a vernier attached to it (see enlarged drawing, Fig. 9), which, together with the scale, enables us to read off the vertical height of the telescope. D is a screw with fine motion to give a small final adjustment to the telescope when making an observation. Before taking each ob- servation the telescope is carefully levelled by means of the screw E and the graduated spirit-level L on T. The eye end of the telescope contains fine cross hairs, which must be focused by adjusting the eye-piece. To avoid paral- j,._ ^ lax * the telescope should * By parallax is meant the displacement of the image oT an object due 22 PKACTICAL PHYSICS first be turned to a distant object and the cross hairs focused for parallel rays, then focus the object glass on the mark to be read, by means of the rack and pinion ; now carefully adjust the eye-piece till both mark and cross hairs are seen distinctly, and no altera- tion of their relative positions is made by moving the eye slightly out of the axis of the telescope. In using the catheto- meter it is essential to remember that the axis of the telescope must always remain at the same angle to the axis of the vertical rod carry- ing the scale. It is most convenient to make this angle a right angle, that is to say, to make the rod vertical and the axis of the telescope to the fact that the image does not coincide with the plane of the cross hairs, or of the scale against which it is to be read. MEASUREMENT OF LENGTH 23 horizontal* This is easy enough to adjust for one position; the difficulty is to preserve this exact relation when the telescope is moved vertically up or down, for it is seldom that the vertical support is absolutely true throughout its length ; it is therefore necessary accurately to adjust the level of the telescope before each observation by means of the screw E and graduated level on C. Any error in the levelling of the telescope will obviously be more serious the farther the instrument is distant from the object measured ; hence this distance should be made as small as possible, t In making the experiment with the cathetometer the telescope is focused on one mark, and the fine adjust- ment made by the screw D ; the scale and vernier are then read off. The carriage with the telescope is now slid up or down the vertical, and the telescope similarly focused on the other mark, and the reading again taken. The difference between these two readings is the vertical distance between the two marks in scale divisions of the cathetometer. Exavvple. — Find by means of the cathetometer the distance between the 20th, 30th, and iOth divisions on the centimetre scale. The cathetometer has a millimetre scale on the stem, and the "least count" of the vernier is "05 mm. (Pig. 9). Enter results thus : — * It is essential to do this when the two marks observed are not at the same distance from the instrument. t See description of cathetometer microscope and reading telescope in Appendix, § 3. 24 PRACTICAL PHYSICS Reading on the Scale in cms. Heading on the Cathetoinetor in cms. Difference. 20-00 30-00 40-00 40-43 30-41 20-42 •02 •01 Exercise. Compare the millimetre scale of the cathetometer with a scale of inches, and find the factor for converting inches to cms., and vice versd. Experiment 7. — To construct and etch a millimetre scale on glass. Instruments required. — Strip of plate-glass, a standard scale, and beam compass; also hydrofluoric acid. Coat a warm, dry glass strip with bees'- wax, in which a little turpentine has been mixed, fasten it to the bench with soft wax, the coated side uppermost. End to end, about 10 cms. off, similarly fasten the standard scale, which may be a steel rule 20 to 30 cms. long, divided into centimetres and millimetres. Place one point of the beam compass on any division of the standard scale, and with the other point draw a line on the coated glass surface, cutting through the wax ; then lift the points, and repeat the operation at the next division of the scale, and so on. To ensure regularity in the length of the divisions, fix a thin strip of brass or stout tinfoil over the coated glass strip, which will serve as a stop or guide to the marking point ; for the larger marks at the fifth and tenth divisions MEASUREMENT OF LENGTH 25 corresponding niches can be made in the guide strip, or these divisions may be omitted in the first instance, and put in after the metal strip has been set farther back. Neatly write the numbers on the scale with a fine dry steel pen. Carefully inspect and, if necessary, test by compasses the scale when made ; if any mistakes have been made re-melt the wax on the bad spot by means of a hot wire and mark it again ; if found correct then etch in the divisions as follows : Lay a strip of blotting paper on the whole length of the glass, and pour on it a little dilute hydrofluoric acid, which can be obtained in solution, in small gutta-percha bottles ; or tie a tuft of cotton wool to a stick, moisten it with hydrofluoric acid, and gently dab on the waxed surface. Breathe on the scale before applying the acid, or the latter may not bite. The fumes of hydrofluoric acid gas, liberated by pouring strong sulphuric acid on some powdered fluorspar con- tained in a lead trough, which is gently warmed, produce opaque and easily-read markings. After exposure to the acid fumes or to the liquid acid for a few minutes, wash carefully and clean off the wax. To render the markings more visible, it may be necessary to rub over the whole with cotton wool moistened with Brunswick black, or coloured paint, and then wipe off with glazed notepaper or a smooth stick. The colour remains in the markings, and when dry is fairly durable. Another plan, adopted by scale-makers, is to use a ruler of the shape of a right-angled triangle, and lay it over the standard scale and the scale to be made, which 26 I'ltAOTICAr, I'KVBIIIH must, ill Uu'h case, l)c fixod Hides by Midn. Ono udgo oi' tho ruler lias a liidj^o wliiuli Ih \)Vi:m<:(\ iit^aiiiHl, iln) hjiIo of Llio HLaridiird scalu, ho that Uu; iiiiukiii|,'H can )m iiiado pandlol to each uiUf.r. A. HiiiiiJl poinl- in iixed lo and proJcclH hclow Uie riKliL-anglcd inlcr; Uiis !« allowed l,o drop into Llie divisions of Uie HUuidurd nride aJid UuiH ilcLh as a guid be longer, and nvciy U:nih longer still ; or, by another notched wluiul t;vi;iy «ecoiuI and fouiih divi«i<^n Uj be longer, according to tlie »cal« we MEASUREMENT OF ANGLES 27 require to construct. To divide a space between two marks into a given number of equal divisions, such as 100 or 180, as in tlic construction of a thermometer scale, bring the carriage and marking point to the first mark, and read the position of the screw, then turn the handle till the second mark is reached and again read the number of turns and parts of a turn of the screw included between the two marks. Divide this number by 100 or 180, as the case may be, and set the instrument to turn through this range. Now lift the carriage and bring it back to the iirst mark, adjust accurately by small motions of the handle of the screw until the marking point exactly coincides with the first mark, and proceed to divide the waxed surface, etching in afterwards as ex- plained in the previous experiment. Section II. — Angular Measurement. E,q)er'iinent 9. — The measurement of angles. Tndrumc'fUs recpiiral. — A pair of compasses, a pro- tractor, a millimetre scale, and a scale of chords. ( 1 ) By the Protractor. In Fig. 1 0, if we want to measure the angle A, we put the centre point of the protractor on the point A and its base along AB, then read off the division on the protractor which coincides with the 28 PRACTICAL PHYSIOS direction of the line AC, which will he the angle rcciuired in degrees. (2) By the Scale of Chords. Take off on a pair of compasses tho distance from to 60 on a scale of chords, and with A as centre describe the arc of a circle mn, then take off the diHtanoe 7im in the compasses and apply it to the scale of chords, one point of the compasses hcing at zero, when tliy reading of the other point will be the required angle in degrees. (3) By Trigonometrical Katios. At any point B in AB draw a perpendicular, meeting AC at C, and measure off BC and AB with the compasses and a millimetre scale, then tan BAC = — ^ (Fig. 1 0),and by consulting a table of natural tangents we get the required angle. This metliod of measuring angles is of frequent occurrence in magnetic and electric rneasurejnenta ; thus in a mirror galvanometer, since the angle wliich the re- flected beam is turned through is twice that of the mirror itself, then d the deflection on the scale, divided bylJ the distance from the scale to the galvanometer mirror, i«the tangent of twice tlie angle of the mirror. That is ,,„ 2 tan d tan 20 = ^ „ „ = - , .•.tan^= "^ .* 21) * The «ign '—. ) by trigonometrical ratios. 2. Determine the angle through which the mirror of a reflecting galvanometer moves when it has turned tlie spot of light through 100 scale divisions, the scale being 100 cms. from the mirror. E.eperiincnt 10. — To detepmine a small thickness by means of the optical lever. Instruments required. — An optical lever, a lamp, and scale. .s:::::l - 8 \ C L^ i ^^^_^ » c a 1 _ 4 Fig. 12. Fig. 11. A modified form of Cornu's optical lever (Fig. 11) consists of a small rectangular mirror A fixed at right angles to a base piece. On the under side of this base are three small conical toes, shown in section in Fig. 11 and in plan at a, &, c in Fig. 12. 30 PRACTICAL PHYSICS B and C are two rigid supports, with plane level surfaces, on which the optical lever rests during an experiment. A shallow V-groove is cut along B, in which the two toes a and h (Fig. 12) rest ; the other toe c rests on the plane surface C. DD' is a lamp and scale, or telescope and scale arrange- ment for measuring the deviation of the mirror during an experiment. Three measurements are required in the experiment : (i.) the line cd (Fig. 12); (ii.) the distance in corresponding units of AE ; and (iii.) in similar units the scale reading DE. An easy way of obtaining the first is to gently press the toes of the optical lever on to a piece of paper or card, and measure by means of the impressed pricks. To measure a thin parallel plate it is put on C under- neath the conical toe, and the beam of light which was re- flected (back along its own path) from the mirror to E on the scale is now reflected to D, and the deflection DE is read off. To measure a small increment of length, such as the expansion of a given bar C, a micrometer screw (not shown in the figure) is used to bring the mirror to the zero of the scale, and the experiment then made. Let o: = thickness or increment of length required, D = the distance from the mirror to scale, d = half the deflection of the reflected beam, I = the distance cd in Fig. 1 2 between one foot C and the line joining ah, then, since D = AE, d = DF, x = mn, by similar triangles we get — X d Id MEASUREMENT OF ANGLES 31 Example. — Find the thickness of a piece of micro- scopic glass by means of the optical lever. D = 320 cms., (?=10 cms., ^=1 cm. 10x1 320 = •0312 cms. By measurement with the screw -gauge the thickness of glass was found to be '031 cms.* Exercises. 1. Find the thickness of a plate of mica. 2. By means of the micrometer screw at the lower end of the support C find what number of scale divisions correspond to a tenth of a milKmetre when the scale is 5 metres from the mirror. Experiment 11. — Measurement of vertical heights by the sextant. Instruments required. — Sex- tant and a measuring tape. The sextant (Fig. 13) consists of a graduated circular arc AB of about 60°, joined to the centre D by the two rigid arms BD and AD. DC is a third arm, movable round the centre D, and having at one end a vernier with tangent screw attached, and at the other a mirror fixed at right angles to the plane of the scale. E is the horizon glass, having * The "horizontal pendulum" affords another delicate method of measuring minute displacements, Appendix, § 4. Fig. 13. 32 PRACTICAL PHYSICS one half silvered and the other half unsilvered, so that the eye on looking through a small hole in the disc F can see an object directly through the unsilvered part and simultaneously the reflection of another object from the silvered part. Since in a rotating mirror the angle through which the reflected beam of light is turned is twice that of the mirror itself, the graduations on the scale are purposely marked double of what they really are, so as to enable the scale to be read off directly in an observation. In using the sextant to measure a vertical height, we first measure off a vertical distance equal to the height of the eye of the observer, and marking this point make it our horizon. Then look through the hole in the disc F directly at the horizon mark (some instruments are fur- nished with a telescope and cross hairs instead of a disc at F) and at the same time move the arm DC till the image of the summit whose height we are measuring, after being reflected from D and E, coincides with the horizon mark as seen directly. Then the reading on the scale and vernier gives the angle subtended by the object at the eye of the observer. Knowing this angle and measuring by a tape the distance from the object to the observer, we can calculate the heiglit required, and by adding to this the height of the horizon mark, we get the total height. The zero of the sextant should always be tested before making an observation, and this is done by clamping the vernier at zero, and looking at a distant object, such as a horizontal window bar, when, if the zero be correct the MEASUREMENT OF HEIGHTS 33 part of the bar as seen directly will be continuous with the part seen by reflection. If they do not coincide a small motion of the horizon glass round its axis makes this adjustment complete ; that is to say, the horizon glass and the mirror D are now parallel, and both are perpendicular to the plane of the scale, which latter condition is necessary for the reflected ray to have an angular ^ motion twice that of the mirror. Example 1. — To find the height of a building, whose lose is accessible, by means of ' •' Fig. 14. the sextant. In Kg. 14 the height of the observer's eye AB = ED = 5 feet, which is marked off. The base BE = AD = 1 1 feet. The angle CAD = 19° 15'; .-. CD = AD X tan CAD = 110x ■3492 = 38-41 ft.; .-. CE = 38-41 + 5 = 43-41 ft. The height by direct measurement was found to be 43'5 feet, hence the error in the calculated height is about 1 inch, or 0'2 per cent. Example 2. — To find the height of a building with iTiaccessible hose (in this experiment a sunk area). In Fig. 15 measure the distance BG, and observe the angles DCE, DAC, DCF. Thus the base — PART I D 34 PRACTICAL PHYSICS EG = 30 ft., DOE =28° 45', DAC = 16°5', DCr = 45°55'. Then the angle ADC = 28° 45' -16° 5' = 12° 40'; 30 X sin 16° 5' _ 30 X ■277 •■■ ^^ " sin 12° 40' ~ -2192 ' and DE = DC sin 28° 45' =D0 x -481 ; ^^ 30 X ■277 X -481 ,.„, „ , •■•^^= ^2l92 = 18-24 feet. Fig. 15. Then the width of the area (required to find EF) 18-24 18-24 CE = 33-25 feet. and but tan 28° 45' ~ -5486 EF = CE X tan ECF, ECF = DCF - DCE = 45°55'-28°45'=l7° 10'; .-. EF= 33-25 X -3067 = 10-2 ft. Therefore the total height DF is 18-24 + 10-2 = 28-44 feet. The height by direct measurement was found to be 28-3 feet, hence the error in the calculated height is 1-68 inch, or 0-5 per cent. SIEASUBEMEVr OF HEIGHTS 35 1. Find the height of the gutter of the laboratory roof from the ground. 2. Find the height of a house with a street interven- ing when the traffic on the street prevents direct measure- ment of its width ; calctilate its width. Ei-ptTivient 12. — Approximate measurement of heights by simple methods. (1) Cur an inch square hole throtigh a piece of wood an inch thick. Fig. 16 shows a section of the hole, AB is the height to be measured. a The eye being placed at C, walk backwards or forwards till the rav of light CA from the top of the object is seen to graze the upper edge D, the bottom B of the object being also seen along the lower side of the hole CE. Then the distance from the observer to the object is the height required, because the angle DCF = 45\ (2) The same result may be obtained by using a card or piece of wcc^d cut in the shape of a right-angled triangle with equal sides. The eye is placed at one of the acute angles, the bottom of the object being viewed along the base and the top along the hypothenuse : then proceed as abova f^ i Tiz. IS. 36 PRACTICAL PHYSICS (3) In sunlight put up a rod of known length and measure its shadow, then measure the length of the shadow of the object whose height is required. If I = length of the rod, s = length of its shadow, L = height required, S = length of shadow of height required, then, by similar triangles, s (4) In Fig. 17 AB is the height to be measured. On H^ /w' F , .. g' c Fig. 17. the level ground BE measure off a convenient distance DC, and place rods CH' and DH at each extremity of the length CD. CGr' = EG, the height of the eye of the observer. Let F be the position of the eye when the top of the object A is just seen over the top of the rod CH', aud measure EG'. Then find another position of the eye G where the same point A is just seen over the top of the second rod DH, and measure DE. MEASUREMENT OF AEEAS 37 Now let us call CD = HH' = d, FG' = a, DE = h, CG' = c, DH = h, and AB = H, then H = ^fcl) + A. — a This formula is proved by aid of the similar triangles AGF and AHH', thus d B.-JI b + {d-a) H - c H(5 -a) = d{h -c) + h{b - a), o — a Section III. — Area Measurement. Experiment 13. — Measurement of the area of a plane surface. I^istmments required. — Millimetre paper, millimetre scale, compasses, scissors, and ruler. First method. — From geometrical dimensions (see Table III.) Second method. — Transfer the figure whose area is required to curve paper, and count all the included areas, estimating by eye the decimal parts of the small areas just round the boundary. ZH vhA(m<;Ai. )'iivi',(';(( TMrd mdJhod. — '\'miw,\'';r i\\<; \'ni}m', Ui nUmi \M\)",r nr tinf'oi] of unifro-Ki Uiick/ico*, i\\';n <;iti i\i'i /if/uro out, nml w() tJx; ratio i)i' i\ii; two ur'tiirt, Mxd/m'iiU.' — Y'm'\ l/y Uh; U/r'}*) ;;/';1,)/0'l« ilxj (t/''!«i H-J^(f; f ;^ + ) Jj» J-"/; M«t;}M<]. riiir'i . hro.n,, %, '-Am. '^A MKti \m>i^ l> 'AAmu'-A i}jui<, /. 2(yl2'.'if)();'.h-2t/ ',:/;, in imttd Uj \Mt ) -22, MHL\SUKEMENT OF AKE/\S 39 ill ■(■ ;•(••(>•(>■. 1. Find by tho tluoo methods named the area of a (.'uvlo 10 coutimotivs in diameter. -. Find bv the throe methods the area of the reuuiiuder obtained by euttiiig the inscribed circle out of a square of 8 cms. in the side. Mv^vriment 14. — MeasuFement of areas by the planimeter. Iiuiu-nt mpiirtd. — Amshn-'s planimeter. lu the pkuimeter. of which the form devised by t\s:. IS. Amslor is shown in Fig-. 18, a point is made to traverse the boundary of the surface the area of which has to be meastuwl. and the required area is given dire^'tly by reading off the graduated rim of a wheel. As usually constructed, the instrument reads to square inches and hundiwlths of a square inch, or square centimetres and hundredths of a square centimetre. 40 PEACTICAL PHYSICS To find the area of any surface, place the instrument on the drawing or the irregular surface to be measured, with the tracing point F at a mark on the curve. Press the needle point E slightly into the paper anywhere outside the curve to be measured. Bead the roller above the letter D and the disc G; say the reading is 2-368, i.e. 2 from the disc G, 3 6 from the roller, and 8 from the vernier ; note this down. Now carefully move the tracing point all round the area in the direction of the hands of a watch, and when the starting-point is reached, take the reading again ; subtract the first reading from the second, and the difference multiplied by 10 will be the exact area in square inches and hundredths of a square inch. A weight W keeps the fixed point E steady ; this point is fixed in such a position that the point F in a preliminary trial can travel completely round the area to be measured.* Exercises. 1, Eepeat the measurement given in first exercise of Experiment 13 by means of the planimeter. 2. Find the area of Phoenix Park from the Ordnance Map of Dublin. Experiment 15. — The graphical representation of experimental results. Instruments required. — Millimetre paper and a flexible strip of wood for drawing the curves. * The theory of the planimeter is given in Williamson's Integral Calculus ; or in Professor Hele Shaw's paper on "Mechanical Integrators " Proc. Inst. Civil Engineers, 1885. MEASUREMENT OF AREAS 41 The object of a graphic representation is to show to the eye the relation between any two quantities x and y connected in such a way that a change in the one alters the other. To plot a curve on the millimetre paper take the intersection of a horizontal and vertical line as the origin, and lay off the determined values of x along the hori- zontal line, which is called the axis of abscisste, and the corresponding values of y along the vertical, called the axis of ordinates ; then the smooth curve drawn through these various points will represent graphically the relation between the two quantities x and y. In plotting the results of an experiment it will be found that a smooth curve will not always pass through the marked points ; this is usually due to accidental errors in the experiment, but in every case the judgment must be used in drawing the smooth curve so as to repre- sent as nearly as possible the actual results of the experiment. It is not necessary that the same scale should be used for both the horizontal and vertical distances ; this should depend on the size of the paper. By making the curve as large as possible small errors in observation are more easily indicated. This graphic method is of great importance in all physical work, and will be much used throughout this book ; we therefore append some examples for the student to practise. Hxample 1. — Plot the curve whose equation is a;?/ =128. This is an equilateral hyperbola, and is approximately 42 PEACTICAL PHYSICS the curve obtained when volumes of air are noted under varying pressures and the results plotted. Give various values to x, and find by calculation the corresponding values of y ; tabulate the results thus : — X y X y 1 128-00 8 16-00 2 64-00 10 12-80 4 32-00 12 10-66 6 21-33 14 9-14 Now lay off on the horizontal axis the values of x, and on J-* Y ----- - - -- -- _5: _ ip : :: : :± : _ ._ u % - - — - - - _:^!: _ ^ ~- ~r ------ - ------ iii'ii " ":::_- :: s : : :_ _" __:" ^ : : : ^s^ ::: :: :_ : " ±'::: lllllllllllllllllllll^ilttllilti''''tlti [ll^^ -i---:2L:::^::::$:ii:::*>:::ffi:::5»::^^ ±- X it i-xi X-- XX Fig. 19. the vertical the corresponding values of y, divided by 10 for convenience, as shown in the curve, Fig. 19. MEASUEEMENT OF VOLUME 43 ExMmple 2. — Plot the curve whose equation is y = x? + 4:x-2. Give various values to x and calculate y as before, thus : — X y X 2/ - 2 -3 - 5 1 + 3 -4 - 2 2 + 10 -5 + 3 3 + 19 -6 + 10 -1 - 5 -7 + 19 -2 - 6 -8 + 30 On plotting these values the curve will be seen to be a vertical parabola with vertex downwards, the co- ordinates of the vertex being ( — 2, — 6). Exercises. Plot the curves whose equations are — (1) y =l.;+2. (2) 3/2= 8a;. (3) y =2±(25-a^-a;2)i. Section IV. — Volume Measurement. Experiment 16. — Determination of volume. Imtruments required. — Calipers, a vessel with overflow, a beaker, clean mercury, and a balance. First method. — If a solid body is of regular geometrical shape, take its dimensions with the calipers and calculate its volume (see Table III.) 44 PEACTICAl PHYSICS Second method.— li the body has an irregular shape, take a vessel with a lip or overflow and fiU it with water just up to the overflow ; then immerse the body whose volume is required in the vessel and it will displace its own volume of water, which can be caught in a beaker. Now weigh the displaced water in grammes, and the number of grammes weight will be the number of cuMc centimetres in the body, which can be corrected for tem- perature if necessary (see Table VIII.) Example. — Find the volume of an ivory ball 4 cms. in diameter by both methods. ,,, ,, 4 „ 4x3-1416x23 „„-,„, (1) V = -7rr3 = =33-5104 c.c. o o (2) The weight of the water at 15° C. displaced by the ball = 33-483 grammes. Volume = 33-483 c.c. Method. Volume. Per Cent Error. (Page 38.) Calculation . . Displacement . . 33-5104 33-483 -08 JVofe. — The weight of the body in air divided by the volume of the body thus found is a simple way of deter- mining the specific gravity of an insoluble body (p. 62 et seq.) Third method. — To find the internal capacity of a flask up to a certain mark on the neck. If the flask is MEASUREMENT OF VOLUME 46 small use pure mercury, and if large use distilled water. Then if W = weight in grams, of the flask when clean and dry, W = weight of the flask when filled with liquid up to the mark, p = density of the liquid at t° C. Then W — W = the weight of the liquid ; W-W . ■ . Volume = * Eocercises. 1. Find the volume of a ball and of a cube by calcu- lation and displacement. 2. Find the internal capacity of a specific gravity flask. Expemnent 17. — Volume, padius, and calibration of a narrow tube. Instruments required. — Pure mercury, a watch-glass, a millimetre scale, and a balance. Introduce a short thread of mercury into the tube, and measure its length when it is at various positions in the tube, which will indicate the uniformity or otherwise of the bore of the tube. Let the mercury run out into a watch-glass and weigh it. Then if * The volume of a body can also be found by the hydrostatic balance or by the Stereometer (see Experiments 23 and 33). 46 PRACTICAL PHYSICS M = weight in grammes of the mercury thread, I — mean length of the thread, r = mean radius of the tube, p = density of mercury at t° C. (see Table IX.), Volume = irrH, Mass M = irrHp. . • . Sectional area = Trr = — . _r and '■= V -7-- ^ Trip Example. — The mean length of the mercury thread in a tube =5 '41 cms. and the weight = '743 gramme, the temperature 15° C. •743 Sectional area = 7rr^ = — ; = '01 sq. cm. 5-41 X 13-56 Eocercises. 1. Ascertain whether the bore of the capillary tube given you be uniform. 2. Find the mean radius of a capillary tube. Note. — The reading microscope also affords a rapid and accurate way of directly measuring the bore at the end of a capillary tube and the diameter of a fine wire, when the cross-section of each is circular. A description of this instrument is given in Appendix, § 3, which the student should read. In using this method care must be taken to ascertain the uniformity of the section of the tube by a mercury thread as above. MEASUREMENT OF SPACE 47 Experiment 18. — To make and graduate a Burette. Instruvients required. — About 3 feet of stout lead or soda glass tubing about 1 or 1'5 cm. internal diameter, and a blow-pipe flame. A burette is a graduated tube for the delivery of known quantities of a liquid, and its construction affords useful practice in simple blow-pipe work.* To make a burette, heat the tube uniformly in the blow-pipe flame by turning it round all the time ; when the glass is soft remove the tube from the flame and draw it steadily apart. After the heated part has cooled slowly, with a triangular file scratch the glass at the contracted portion, and it will easily snap across.! Fuse the edges of the tube at each end, and when the tube is cold slip on the contracted end a short piece of rubber tubing, nipped by a pinch tap or plugged by a bit of glass rod. Coat the tube with melted bees'-wax, and having fixed it in a clip, pour into it 1 grammes of water, mark the level of the water in the tube, now add another 10 grammes, and make another mark, and so on till the tube is nearly full. Now fasten the tube to the dividing engine,^ and set the instrument so that 10 divisions can be made in the first marked space ; if the marks are equidistant, * The student who is unfamiliar with glass-hlowing should hegin by bending tubes in an ordinary flat luminous gas flame, then proceed to practise drawing out tubes, then sealing platinum wire into tubes, and afterwards blowing bulbs and making glass T-pieces. Full and excellent instructions are given in Mr. Shenstone's Methods of Glass Blowing (pub- lished by Eivingtons), which the student will do well to procure. + A knife, made glass-hard by plunging in cold water when bright red-hot, is better than a file for cutting glass. X If the student has not access to a dividing engine, the method de- scribed in Experiment 7 can be followed for marking in the divisions. 48 PRACTICAL PHYSICS continue the division along the whole tube. Probably it will be found that the length between the marks is not quite the same ; in this case the dividing engine must be set for each pair of marks* Now write the correspond- ing number on the wax, beginning at at the upper- most division near the mouth of the tube, and putting a iigure at every 5 th or 1 th division. Etch in the divisions and figures as already explained (Experiment 7), and remove the wax. If all has gone well, proceed to cali- brate tlie tube by filling it with distilled water, drawing off, say, 5 c.c. at a time and weighing the quantity. After repetition enter the mean weights found in a table, thus : — — 5 c.c. = 5'10 grammes. 5-10 c.c. = 5-00 10-15 c.c. = 4-92 „ etc. Use this table to estimate the true volume of the liquid employed, when say 20 c.c. are drawn off between the divisions 20 and 40. For accurate purposes a correction must be made for the temperature of the water (Table VIII), since at 4° 0. 1 gramme of water has a volume of 1 c.c. In reading the level of the water in the burette, hold a piece of white paper behind and read from the lower edge of the meniscus — a black line will there be seen. * The sub-division of a given length can also be done by means of a "diagonal-scale." A very simple form of Line-divider is described by Miss Marks in Proc. Physical Soc, February 1885. CHAPTEE III MEASUKEMENT OF TIME The unit of time has been already defined as tlae second or the 1/86400 part of the mean solar day. The length of a simple pendulum which beats seconds at every single oscillation is 9 9 '4 3 cms. or 39'14 inches in Dublin. In the accurate estimation of time various methods may be followed, according to the nature of the experiment and the degree of accuracy required. Experiment 19. — Find the time of oscillation (half-vibration period) of a simple pendulum by the ' method of passages.' Instrument required. — A heavy bullet suspended by a fine silk thread. The method of passages consists in finding, at first approximately, the time of oscillation by noting the exact moment when the middle of each swing occurs during, say, 5 consecutive oscillations ; then determining the exact number of oscillations in a longer interval by dividing this interval by the approximate oscillation period previ- ously obtained, and selecting the nearest integer. This gives a closer approximation. A still longer interval PART I e 50 PEACTICAL PHYSICS may now be selected and divided by the last approxima- tion, the nearest integer being selected as before. The whole time being divided by the integer so found gives the true time of a single oscillation. Example. — A heavy bullet was suspended by a fine silk thread and allowed to vibrate through a small arc. The transits of the thread across the wire of a tele- scope were noted and the time taken by listening to and counting aloud the beats of the seconds pendulum of a standard clock. The following results were obtained : — Transit. Time. Transit. Time. Ditference. Time of a Single Transit. Min. Sec. Min. Sec. Sees. 15 5-0 25th 15 28-0 23 0-92 5tli 15 9'5 30 „ 15 32-5 23 0-92 10 „ 15 14-0 35 „ 15 37-0 23 0-92 16 „ 15 18-5 40 „ 15 42-0 23-5 0-94 20 „ 15 23-5 45 „ 15 46'5 23 0-92 Mean 0-924 Subtracting the time at from that at the 25th transit we get the time required for 25 transits, and so on for each pair, as given in the column of differences. By dividing each difference by 2 5 we get the time of a single transit, and the mean gives 0'924 of a second, which is the first approximation. From this the time taken for 100 oscillations should be 100 X '924 = 92-4 seconds. Now observe the seconds hand of the clock as it comes to 6 0, and count aloud from this point, watching the transits of the pendulum ; the MEASUEEMENT OF TIME 51 100th transit occurred at 93-0 sees., and dividing by 100 we get 0'93 of a second as the second approximation. Again estimate from this the time of 500 oscillations, this comes to 7 minutes 45 seconds. Leave the pendulum swinging and return to the observation at 7 minutes 40 seconds ; counting the beats of the clock from this instant, the transit occurred at 7 minutes 45 seconds very approximately. Dividing this by 500 we get '9 30 of a second as the true time of an oscillation. Exercise. Make an experiment similar to the above. Experiment 20. — Determination of tlie period of oscillation by the ' method of coincidences.' Instrument required. — A clock with seconds pendulum and a simple pendulum hung in front of the clock. Suspend the simple pendulum used in the last experi- ment in front of the seconds pendulum of the standard clock, making the length of the suspending thread slightly greater than before, so that its period shall coincide as nearly as possible with that of the clock pendulum. It is best to start the pendulum by pulling it aside with a thread, which is then burnt; this avoids giving any rotatory motion. If the period of the simple pendulum be now slightly greater than a second, a moment will occur when the beat of both pendulums will precisely coincide ; the clock will then gain on the experimental pendulum, and 52 PRACTICAL PHYSICS after a certain interval it will gain a complete oscilla- tion, when coincidences wUl again occur. The interval of tune between the two moments of coincidence is to be accurately noted. The same thing will occur if the ex- perimental pendulum be slightly quicker than the clock. In this case, if n be the interval between two successive coincidences, the experimental pendulum will in n seconds have made n+ 1 oscillations, and the time of each oscilla- tion will therefore be seconds. In the former case 7i,+ l the experimental pendulum will have made n—1 swings n in n seconds, and the time of its oscOlation will be n—1 seconds. This method, it is obvious, can only be used when the time of vibration is very nearly alike in the two cases. Eocample. — Find the time of oscillation of a simple pendulum by the foregoing method. The experimental pendulum was slightly quicker than the clock pendulum. The mean of five coincidences gave w = 21 seconds. Hence the time of oscillation was 0"954 of a second. Hxercise. Eepeat the above experiment, making the experi- mental pendulum first slightly longer then slightly shorter than the clock pendulum. MEASUREMENT OF TIME 53 Experiment 21. — Determination of small intervals of time. Instruments required. — A stop-watch, an ordinary watch, a clepsydra, a chronoscope, and a tuning-fork chronograph. (1) The seconds hand of a stop-watch traverses the whole dial in one minute and is made to indicate fifths of a second. There is a slight loss of time in starting the watch, but a similar one in stopping it, so that the error is extremely small. For most experiments in timing oscillations as on the value of gravity, a stop-watch is the most convenient instrument to use. (2) An ordinary watch generally gives five ticks to a second, with a little practice fractions of a second may be accurately estimated by this means. Proceed as follows : make a series of twenty-five strokes on paper, put an ordinary watch to your ear and listen till you get accustomed to the sound of every second tick ; now hold- ing a pencil in your hand ask a second observer who is holding a stop-watch to start it directly you point to the first stroke, and stop it when you reach the last, you mean- while pointing to the successive strokes at every second tick, or two-fifths of a second. In this way you will, after one or two trials, find 1 seconds accurately registered in the interval, indicated by the twenty-five strokes. Now try in the same way 30 seconds. By this preliminary practice with a stop-watch you will thus obtain confidence in your ability to count the ticks, which at first seems a hopeless undertaking. Por a short interval of time, such as that taken by a falling body, an ordinary watch can be used to replace a stop-watch. 54 PRACTICAL PHYSICS (3) The flow of water through an orifice, under a con- stant head, also gives a means of estimating time. For this purpose the water is caught in a vessel at the com- mencement of the period, and the vessel promptly with- drawn at the end, the quantity of water in the vessel is then weighed. The time is now taken which corresponds to the flow of any given quantity of water, and the in- terval of the time occupied by the experiment is thus estimated, the time being proportional to the quantity of water, all other conditions remaining the same. (4) A falling weight which actuates clockwork can also be used to measure small intervals of time up to 1/3 00th of a second. This form of chronoscope is most easily started and stopped by an electro-magnetic arrange- ment. A preliminary experiment is made in timing the instrument, the rate of which can be varied by adding shots to the bucket which forms the falling weight. A complete revolution of one hand on the chronoscope being made to correspond to one second, each revolution of the other and smaller hand corresponds to 1/3 0th of a second, and as this dial is divided into ten parts, the 1/3 00th of a second can easily be estimated. (5) The most accurate method of registering small intervals of time is by means of a tuning-fork chrono- graph. This method will be described in Part II in the section on Sound. The fork, which is kept vibrating by an electro-magnet, has a style {a) attached to one prong, which writes upon a rotating blackened cylinder a sinuous curve resulting from the vibration of the fork. Its exact period of vibration is determined by a current from a MEASUREMENT OF TIME 55 seconds pendulum electro-magnetically moving another style, which makes a record on the blackened cylinder. When an experiment is to be made, such, for example, as the time taken by a freely falling body, the beginning and the end of the fall are indicated on the rotating cylin- der — simultaneously with the record made by the fork — by the falling body making and breaking an electric circuit, which actuates another style (b) ; or an induction coil may be used, the primary circuit being made and broken, and the spark from the secondary marking the blackened cylinder. This has some advantages over the use of a style. This last experiment is well suited for lecture demonstration, as the moving styles and their record can be easily projected on the screen, the tuning-fork being mounted vertically — a vertical strip of blackened glass sliding in a grooved upright being used instead of the rotating cylinder. The following is the result of an actual lecture experiment : — JSxample. — A small brass weight was fixed at the height of 8 feet 7 inches above a hinged shelf forming a contact breaker. The electric circuit comprised the movable jaw which clipped the weight, the electro-magnet actuating the style h and the contact breaker below. The tuning- fork gave 30 vibrations per second recorded on the strip of blackened glass by the style a. The Pier. 20. glass was raised by hand, the styles a and h lightly pressing upon it. On raising the glass an 56 PRACTICAL PHYSICS assistant released the weight by pulling a string which opened the jaws of the clip and simultaneously made contact; the style b thereupon recorded this instant. On the weight striking the hinged lid contact was again broken and the style b moved. The sinuosities made on the blackened glass by the style a are shown at F, and the marks by the style b are shown at SS', on Fig. 20. The number of sinuosities between S and S' was 22 =§^ of a second. From this the height of fall h can be calculated (see Experiment 49). = 16-1 X my (in feet) = 8 ft. 6-7 in,, the actual height being 8 ft. 7 in. CHAPTER IV MBASUEEMENT OF MASS SPECIFIC GRAVITY Experiment 22. — The comparison of masses by means of the chemical balance. Instruments required. — A balance and box of gramme weights. The chemical balance (Fig. 21) consists essentially of a metal beam A supported at the middle on a knife-edge so as to be free to move in a vertical plane round an axis perpendicular to its length. At the end of this beam there are knife-edges BB' for supporting the scale pans SS', which can turn freely round axes parallel to the axis of rotation of the beam. These axes are usually formed of agate knife-edges resting on plates of agate. To prevent injury when the balance is out of use, or when the weights are being changed, an " arrestment," actuated by a handle D outside the balance case, lifts the knife-edges off the agate planes. A long pointer is fixed to the beam of the balance with its length perpendicular to the line joining the extreme knife-edges, and serves to define the position of the beam as indicated in the short scale fixed to the pillar of the balance. 58 PRACTICAL PHYSICS When the balance is properly levelled and adjusted the following conditions must be observed : — (1) The pointer should be opposite to the middle division of the scale, the beam at the same time being perfectly horizontal. Fig. 21. (2) The arms of the balance must be symmetrical, uniform, and of equal length, that is to say, the knife- edge on which the beam turns must be exactly midway between the knife-edges on which the scale pans hang. (3) The scale pans must be of equal weight. MEASUREMENT OF MASS 59 (4) The centre of gravity of the beam must be verti- cally heloiv the axis of rotation when the beam is hori- zontal, as indicated by the pointer and scale. The practical manipulation necessary to obtain the above conditions will be better acquired by the student with a balance before him and a few words of instruction from his teacher than from a long description accompanied by figures or cuts. For a general theory of the balance with straight beam, see Appendix, § 5. The smaller weights accom- panying the balance are usually marked thus : — (1) 0'5 to O'l of a gramme or decigrammes. (2) 0-05 toO'Ol „ centigrammes. (3) 0-005 to 0-001 „ milligrammes. For fine weighing a wire rider 1 milligramme in weight can be placed on the balance beam, the arm of which is divided into ten equal parts. By putting the rider, using the rod E and hook C, on the first division from the centre of the beam it will counterpoise one-tenth of a milligramme on the opposite scale pan. In order to weigh a body the position of the pointer when at rest must first be determined. If it does not rest at the zero of the scale, note its position and use that as the working zero. If the pointer be not quite at rest, note the numbers at which it turns on each side of the middle of the scale, and take the mean as the zero ; for greater exactness another pair of observations may be taken as the swings grow less. Be careful always to support the pans by the rests, when the body or weights 60 PEACTICAL PHYSICS are put in or taken from the pans ; and never exceed the limited load which the balance was made to weigh. By trial and error find the weights which equipoise the body, i.e. when the pointer rests at its worMng zero. As the pointer will take some time in coming to rest, note the turning-point on each side of the working zero, and add a small weight (or use the rider) to the side at which the swing is greatest, until the pointer vibrates equally on each side of its zero. The true weight can also be found, without the last adjustment, by observing the oscillation of the pointer, as follows : — Note the limits of its swing and estimate the point of rest {a), then add a milligramme or two so that the pointer swings on the other side of its zero ; again note the limits of the swing and estimate the new point of rest (6). The difference of the two points of rest gives the deviation (c) for the small weight added. The total weight in the pan plus the small weight added was therefore too heavy by the difference between (6) and the true zero divided by the deviation (c). Example. — The scale reads from to 50, division 25 being the true zero. If point («) = 29, and with one milligramme the point (&) = 18, .■. c=29 — 18 = 11 divisions, OK 1 Q fr therefore — ^1 = r— of a milligramme to be taken from the weight in the pan. The number of scale divisions between the two estimated points of rest (a) and (J) enables us to MEASUREMENT OF MASS 61 determine the sensitiveness of the balance. Let the small weight added be "W, and the points of rest be a and b, then — -— will be the sensitiveness of the balance w for the particular load on the pans. Example. — If the pointer stands at 25 when 100 grammes are on each pan, and at 1 6 when the difference is ■001 of a gramme, or one milligramme, therefore the sen- . 25-16 „ ,. . . sitiveness is = — = 9 divisions per milligramme for a load of 100 grammes.* If the arms of the balance are not quite of equal length, we may determine the correct weight of a body- by double weighing, that is by weighing with the body in one pan, then changing the body into the other pan and weighing again. Let a and b be the lengths of the left and right arms of the balance respectively, W = true weight of the body, Wj = apparent weight when the body is in the right- hand pan, Wg = apparent weight when in the left-hand pan. Then (1) Wj« = W& in the first weighing (2) ^a = yfj) „ second „ * Theoretically the sensitiveness is the same whatever the load, but owing to the bending of the beam and other causes the sensitiveness varies slightly with the load. 62 PRACTICAL PHYSICS Dividing equation (1) by (2) Wi_ W .■.w= Vw.w^. Example. — The apparent weights of a body when weighed in the left and right-hand pans are 2 6 '2 4 5 and 2 6 "2 17 grammes respectively, therefore the true weight is W= V26-245X 26-217 = 26-231 grammes. Exercises. 1. Find the true weight of a body in a balance by the method of double weighing. 2. Find the sensitiveness of the balance for loads of different amounts, and represent the results graphically with loads as abscissae, and sensitiveness as ordinates. Specific Gravity. The specific gravity of a body is the ratio of its density (p. 9) to that of a standard substance, usually water at 4° C. It is therefore a numerical quantity, and has the same value whatever units are employed. Density and specific gravity coincide in numerical value when the density of the standard substance is taken as unity. The specific gravity of a substance being its weight divided by the weight of an equal volume of water at 4° C, the latter is conveniently found by noting the loss of weight the body undergoes when weighed in water ; as the tem- perature of a room is about 15° C, a correction becomes SPECIFIC GRAVITY 63 necessary for the slight difference in the density of water, and also for the buoyancy of the air (see p. 81). Experiment 23. — To find the speciflc gravity of an insoluble body heavier than water. Instruments required. — A balance, a box of gramme weights, a beaker of distilled water, and a thermometer. When the chemical balance is used for the determi- nation of specific gravities, there is usually a hook fixed immediately below the knife-edge, at the end of the arm, to which the body is hung by a fine thread when it is being weighed in a liquid. There is also a movable platform, which is made to rest on the floor of the balance case, and overspanning the scale pan, thus supporting the beaker of liquid in which the body is immersed. If, owing to the mode of suspension or the smallness of the scale pan, a platform of this kind cannot be used, the pan is taken off the balance. and a counterpoise with suitable hook put in its place. To make the experiment, first weigh the body in air in the usual way, then suspend it by a fine thread from the hook, having the body completely immersed in the water and free from all air bubbles, which latter may be detached by the feather end of a quill (see also p. 82). Then if W = weight in grammes of the body in air, Wj = „ „ „ water, p = the specific gravity of the body, W P - W - Wj' 64 PEACTICAL PHYSICS Example. — Find the specific gravity of a piece of brass. ^ 15-9-14 Exercises. 1. Find the specific gravity of a glass stopper. 2. Find the specific gravity of a piece of basalt. Experiment 24. — To determine the specific gravity of an insoluble body lighter than water. Instruments required. — The same as before, together with a suitable sinker, wliich may be made of a piece of brass vs^ith a hook for suspension on one side and a sharp spike to thrust ia the light body on the other ; this saves tying on the sinker to the light body. Proceed with the weighing as iu the last experiment. Then if W = weight of the solid in air, Wj = weight of the sinker in water* Wg = weight of the solid and sinker together in water, p = the specific gravity of the body, W Example. — ^Find the specific gravity of a piece of paraffin wax. i-.q ^ = 7-9 + 14-05- 12-9 =^'^^- Exercise. Find the specific gravity of a piece of cork. * It isi not necessary to weigh the sinker in air, as this weight cancels out SPECIFIC GRAVITY 65 Experiment 25. — To detepmine the specific gravity of a soluble body heavier than water. Instricments required. — Same as in Experiment 23, and a liquid lighter than the body, and of density p^ in which the body is not soluble. Proceed as before, then if W = weight of the body in air, Wj = weight of the body in the liquid, Pi = specific gravity of the body compared with the liquid, p = specific gravity of the body compared with water, W and p = PiX Pr Example. — Find the specific gravity of a piece of lump - sugar, the liquid used being petroleum with |02= 0-83. 6-6 ^1 " 6-6- 3-19 "■^■^^' p = 1-94 X -83 = 1-61. Exercise. Find the specific gravity of a piece of rock-salt. Experiment 2 6. — To determine the specific gravity of a liquid : method i. Instruments required. — Same as in Experiment 23, also a body for displacement of the liquid, which body PART I F 66 PKACTICAL PHYSICS must be heavier than, and not acted on chemically by, either the liquid or water. Proceed as before, then if W = weight of the body in air, "Wj = „ „ „ water at t° C, "W2= „ „ „ the liquid also at <° C, p = the specific gra\dty of the liquid, W-W, Z' - W - Wi' Hxample. — Find the specific gravity of petroleum at 1 5° C, using a glass stopper as the body for displacement. 15-9-14-37 „„„^ Exercise. Find by this method the specific gravity of a solution of zinc sulphate. Experiment 27. — Determination of the specific gravity of a liquid : method ii. Instrument required. — A Mohr's balance. Mohr's balance (Fig. 22) consists of a beam ABC pivoted at B, and having on the longer arm BC nine notches in which riders can be placed. A float F is hung from a hook C by means of a fine platinum wire. This float is conveniently made of a small thermometer, which at the same time gives the temperature of the liquid. There are three riders, having a weight ratio to each SPECIFIC GRAVITY 67 other of 1, -jlj, y^, accompanying the instrument, and when the rider 1 is hung on the hook C,* the float being _A B Pig. 22. wholly immersed in distilled water at 15° C, the instru- ment is in equilibrium, indicated by the two pointers at A being in a line. The liquid the density of which has to be determined is contained in the vessel M with the float immersed in it ; riders are added to the beam tUl a balance is ob- tained. JExampk 1. — Find the density of spirits of wine. First rider was at 8. Second rider was at 2. Third rider (hanging on the first) was at 8. .•. density =0-828. Example 2. — Find the density of a solution of zinc sulphate. As this liquid is heavier than water, a fourth rider * The hook at C is under the 10th notch, and not as shown in the figure. 68 PRA.CTICAL PHYSICS equal in weight to the first was put on the hook 0, and the other three riders were at the positions 1, 2, 8. .-. the density =1-128. Both examples require correcting for temperature (p. 81). • Exercise. Find by Mohr's balance the specific gravity of 1. Pure alcohol. 2. Fusel oil. 3. Copper sulphate solution. Experiment 28. — To detep- mine the specific gravity of a liquid : method iii. Instruments required. — A U-tube and a cathetometer or a scale. (1) Take a tube (Fig. 23) with a stop-cock C at the bend, and the ends dipping into two vessels A and B containing water in the one, and the liquid, whose specific gravity is required, in the other. When the tube is placed in position and some air drawn out by means of the stop-cock C,the liquids rise in their respective tubes, and the heights above the level of the liquids in the vessels wUl be in the inverse ratio of the specific gravities of the liquids.* If * The scale shown needs shifting till its zero is level with the liq^uids in A B ; if one scale be used the liquid surfaces must be at one level. SPECIFIC GEAVITY 69 h and p = the height and specific gravity of one liquid, K and p' = the height and specific gravity of the other liquid, then ph = p'K. (2) A more accurate way is to take a double U-tube (Fig. 24) with a stop-cock C at the upper bend.* Fig. 24. Fig. 26. Pour the liquids whose specific gravities are to be compared, one into tube A and the other into tube B, the cock C being open. Then draw out some air and close C, when a difference of level wiU be established in each tube ; finally measure this difference of level by means of the cathetometer as in the first case, and calculate as above. Let d = difference of levels in the one tube and d' = difference of levels in the other tube, then pd = p'd'. * A much simpler apparatus (which has the advantage of avoiding any leak in the stop-cock or piueh-tap) can be made by the student for him- self by bending a length of say four feet of glass tubing about half-inch diameter into the shape given in Fig. 25. By pouring the liquids alter- nately, a little at a time, each into its respective tube, the air entrapped in the bend will be compressed, and a difference of level in the liquids established as before. Wide tubing is used to avoid capillarity. 70 PRACTICAL PHYSICS Example 1. — Find the specific gravity of petroleum, having water in the other limb of the U-tuhe. By measurement with the cathetometer we have ^ = 12'6 cms., 71'= 152 cms. ... .■.i^ = 0.8.S. Example 2. — Eepeat Example 1 with the double U- tube. By measurement with the cathetometer the differ- ence of levels of the water = 6'45 cms., and the difference of the petroleum = 7'8 cms. 1 X 6-45 :0'827. Exercises. 1. Find the specific gravity of methylated spirit. 2. Find the specific gravity of glycerine. Experiment 29. — To determine the specific gravity of a liquid by the specific gravity beads : method iv. Instruments required. — A vessel to contain the liquid under experiment, and a set of specific gravity beads. Specific gravity beads are small hollow balls of glass . (Fig. 26), each one marked with the specific gravity of the liquid in which it will just float. Fig. 26. SPECiriC GRAVITY 71 To use the beads in determining the specific gravity of a liquid, take an assortment of beads having different values, and drop them one by one into the liquid tiU one of them just floats in the body of the liquid, i.e. has no tendency to either rise or sink, then the number marked on that bead is the specific gravity of the liquid. An ingenious modification of the specific gravity beads is shown in Fig. 27 ; the tube can be dipped in a liquid whose specific gravity is required, and the bead which floats gives the required density. The beads or discs are coloured differently, so that they can be readily distinguished.* Example. — Find the specific gravity of dilute sulphuric acid by the heads. By trial we find that the bead marked 1'015 sinks, and that the one marked 1'003 floats, whilst that marked 1'012 just floats, therefore the specific gravity of the liquid is=1012. Exercise. Find the specific gravity of oUve oU. Pig 27 A modification of the foregoing method t enables a rapid * This arrangement is due to, and different modifications of it are made by, the well-known instrument maker, Mr. Hicks, of Hatton Garden, London. t Professor SoUas in Nature, 26th February 1891. 72 PRACTICAL PHYSICS determination to be made of the specific gravity of small fragments of a mineral, or of any minute insoluble solid object, up to a density of 3-45. In the case of bodies heavier than water, a heavy liquid such as methylene iodide is diluted with benzol, or potassium mercuric iodide is diluted with water, until a density is reached approximate to that of the body to be tested. It is best partly to fiU a test-tube with the heavy liquid, pour on it the diluent, and leave it undis- turbed till next day. A column of Liquid of regularly increasing density from the top downwards will thus be obtained. Small specific gravity beads, of slightly different densities, are now thrown in, and when three or four sink to different levels their distance apart will he found to be directly proportioned to their respective densities. The distances and densities are now plotted on millimetre paper, and the body whose specific gravity is to be determined is dropped into the liquid, the exact position at which it floats is noted, and by reference to the curve already made its density is at once determined. Microscopic specimens can thus be examined in small tubes. The specific gravity beads can be readily made from bits of capillary glass tubing sealed and one end enlarged by blowing ; the density of a number of these is determined beforehand by comparison with bodies of known density. The specific gravity of drops of aqueous liquids can also be found by this method, using methy- lene iodide diluted with benzol ; for oily Liquids a dilution of cadmium-boro-tungstate with water may be used. See Appendix, § 6 (3). SPECIFIC GRAVITY 73 Experiment 30. — Determination of the specific gravity of liquids by means of various hydro- meters : method v. Instruments required. — Various hydrometers. (i.) Tlie ordinary hydrometer (Fig. 28) is an instru- ment of variable immersion, and consists of a glass tube with a bulb at the lower end, containing mer- cury to make it in stable equilibrium when floating in the liquid under experiment. The tube or stem is graduated so that the instrument measures through a definite range of density. When immersed in the liquid under trial, the density of the liquid is shown by the depth to which the instrument sinks, the mark on the stem, level with the surface of the liquid, being read off (Appendix, § 6 (2)). (ii.) In the case of TwaddeU's hydrometer, instead of one instrument with a very long stem being employed to cover the range of the specific gravities of ordinary liquids, several short-stemmed instruments are made with mer- cury in each, so adjusted that the -range of one ends where the next one hegins. Fig. 28. A formula is given for use with this form of ordinary hydrometer. Thus for TwaddeU's hydrometer the rule is, " Multiply the reading on the instrument by 0-005 and add 1." (iii.) Sike's hydrometer (Fig. 29) is the form adopted by the Inland Eevenue Department for determining the percentage of alcohol in spirits. It is made of gilt 74 PEACTICAL PHYSICS brass and consists of a bulb with stem divided into ten parts or degrees; at the end of a lower submerged stem is a collar for tbe purpose of supporting a series of movable weights contained in the box. The instrument floats at on the stem without weights in strong spirits of density 0-825, in weaker — I Fig. 29. spirits weights are added to make it sink; and with the weight marked 90 (corresponding to 90 divisions of the scale) the instrument floats at 10 in distUled water; there is therefore a range of 100 degrees {i.e. 90 + 10 on the stem) between 0'825 and TO 00, each degree there- fore corresponds to 0-00175 variation of density. Example. — Find the specific gravity of a solution of copper sulphate by means of Twaddell's hydrometer. The reading on the instrument = 38, .-. density= 1 + 38x0-005 = 1-19. SPECIFIC GEAVITY 75 Exercises. 1. Pind by means of Twaddell's hydrometer the density of a mixture of sulphuric acid and water. 2. Find by Sike's hydrometer the percentage of alcohol in the sample given you. (iv.) Nicholson's hydrometer (Fig. 30) is a constant im- mersion hydrometer, and consists of a hollow metallic tube, having a scale pan at the upper end A, and at the lower end B a metallic basket with movable perforated lid, in which is put the solid body when being weighed in water. A bead C on the stem is the standard mark to which the instrument must be sunk at each observation.* (1) To find the specific gravity of an insoluble body heavier than water. First, when the hydrometer is immersed in water, find what weight (W) put on the upper pan will sink it to the standard mark ; this is the standard weight, then put the body on the upper pan together with weights so as to sink the instru- ment to the standard mark (the body under trial » "Pic ^fl must always weigh less than the standard weight of the instrument). Then put the body in the lower pan and weigh again. Then if W = standard weight, Wj = weight put on the upper pan along with the body to sink the instrument to the standard mark, * To prevent the instrument sinking too far through overloading, a collar of wood or stiff card, with a slot for the stem, should alviays be 76 PRACTICAL PHYSICS Wg = the weight in the upper pan, when the body is in the lower pan, required to sink the instrument again to the standard mark ; then "W — W^ = weight of the body in air, W2 — Wj = apparent loss of weight of the body in water, p = specific gravity of the body, _W -W, Por an insoluble body lighter than water the formula requires to be slightly changed (see Example 2 below). Example 1. — Find the specific gravity of a piece of bismuth. 8-9 -2-03 '^ = 2-73- 2-03 = ^"^^- Exaniph 2. — Find the specific gravity of a piece of mahogany. W-Wi P = Wj-Wj + W 8-9 -6-07 6-07 -10-65 + 8-9 = 0-655. The advantage of a Nicholson's hydrometer is that an ordinary balance is not required with it, unless it be employed to determine the specific gravity of a liquid, when the weight of the hydrometer must be found first. (2) To find the specific gravity of a liquid heavier than water by Nicholson's hydrometer. placed on the top of the hydrometer jar. The water used should hare been boiled, and air-bubbles adhering to the hydrometer must be removed. SPECIFIC GRAVITY 77 Let W = weight of the hydrometer, Wj = weight on the top pan when the hydrometer is in water, W2 = weight on the top pan when the hydrometer is in the liquid whose specific gravity is required, p — specific gravity of the liquid, _ w+w, ^ W + Wj" Example 3. Find the specific gravity of a solution of sulphate of copper. The hydrometer weighs 1 O^S 5 grammes. 10-35 + 9 '' 10-35 + 5-77 1. Find the specific gravity of a piece of brass. 2. Find the specific gravity of a piece of cork. Experiment: 31.— To determine the specific gravity of a liquid by the specific gravity bottle : method vi. Instruments required. — A balance, a box of gramme weights, together with a specific gravity bottle. •The specific gravity bottle is a bottle made of thin glass, with a ground stopper having a capillary bore ; or one with an ordinary stopper, having simply a mark on the narrow neck of the bottle. The bottle when quite full up to the top of the stopper, or to the level of the mark, is made to contain a certain even number of grammes of 78 PRACTICAL PHYSICS ■water, such as 50 or 100 grammes, at standard tem- perature. In place of a bottle a small beaker with ground lip and a ground glass lid may be used with advantage (especially for the determination of the specific gravity of fragments of mineral). The beaker is filled to over- flowing with the liquid to be tested, and the ground hd slipped carefully over the top, so as to exclude all air bubbles. The beaker is then wiped clean and dry, and weighed. Sprengel's sp. gr. tube for liquids is an instrument easily made by the student, and affords the simplest and most accurate method of determining the sp. gr. of liquids at any given tem- perature. It consists of a piece of glass tubing, say 6 mm. diameter (A A, Fig. 31), bent as shown, the ends drawn to a capillary bore, and it is convenient to have one arm a finer bore than the j^ 3j other. The liquid is drawn into the tube by suction, and its temperature can be rapidly brought to the required point by placing the tube in a beaker of water B, as shown ; a mark m is made on the wider capillary, and up to this mark the capacity of the tube is once for all determined by weigh- ing it empty and then full of water at the standard temperature.* It is necessary to let the liquid stand at the mark m, on the wider capillary bend, so that it can expand without loss on weighing. Then if * The tube is weighed by suspending it from the hook of the balance by a thread slipped over the bent arms of the tube. SPECIFIC GRAVITY 79 X = the number of grammes of water at standard temperature that the bottle, tube, or beaker contains, W = the weight of the empty bottle or beaker, W = the weight of the bottle or beaker when full of liquid, p = specific gravity of the liquid, W - W Example. — Find the specific gravity of petroleum, using a 50 gramme bottle. 81-9 -40-6 50 0-826. Exercise. Find the specific gravity of glycerine, also of copper sulphate solution. Experiment 32. — Determination of the specific gravity of a powder or small fragments of mineral by the sp. gv. bottle. Instruments required. — Same as in Experiment 31. To find the specific gravity of a powder by the bottle or beaker. Find the weight of water held by the bottle at the temperature of the air. Weigh the powder to be used and pour it into the clean empty bottle, fill up the bottle with distilled water at the temperature of the air. Carefully dislodge all air-bubdles held by the solid either by 80 PRACTICAL PHYSICS shaking or by the air-pump. Note the temperature. Let X = weight of water which the bottle or beaker contains at standard temperature, W = weight of the powder in air, Wj = weight of the poivder and water in the bottle or beaker when quite full of water, p = specific gravity of the powder. W ''"a; + W-Wi' Example. — Find the specific gravity of a sample of crushed glass, using a 50 gramme bottle. 7-64 P = 50 + 7-64 -54-61 = 2-52. Exercise. Find the specific gravity of a sample of sand. Mr. Moss has modified Sprengel's sp. gr. tube and made it suitable for minute specimens, such as precious stones, as c D A B bfr—- \i'y, 1^^ ^^^ Fig. 32. shown in Fig. 32. Two small pieces of glass tube, A B, are ground into one another, which can readily be done with SPECIFIC GEAVITY 81 fine emery powder and water. The other ends of the tube are then drawn off to a capillary bore (one wider than the other, for the reason already given) and bent twice at right angles, as at C and D. The weight and capacity of the empty tube being determined once for all, the tube is taken apart and the weighed specimen, whose specific gravity is to be determined, is placed within A B; the tube being closed, is again filled with water and care- fully weighed. The specific gravity is then calculated as in the previous experiment. To keep the temperature of the water in A B constant, or reduce it to 4° C, the tube can be placed in a beaker of water, the arms C D afford- ing a convenient support. In refitting the tube the same pressure must be used, or its volume may alter. Another method of determining the specific gravity of minute fragments is described in Appendix, § 6 (3). Corrections and Precautions. (i.) Since the specific gravity of a body is the weight of the body divided by the weight of an equal volume of water at 4° C, it is necessary to make a correction for the difference in temperature of the water above 4° C. This is easily done by reference to the tables of the density of water at different temperatures, and multiplying the specific gravity of the body as found by the specific gravity of the water at the temperature of the experiment, for the same reason as in weighing a body in a liquid other than water (see Table VIII.). (ii.) As the weighings are not made in vacuo the appar- ent weight of the body in air is less than the true weight PART I G 82 PEACTICAL PHYSICS by an amount equal to the weight of the air displaced (Experiment 36). Hence the apparent specific gravity, p, must be corrected for this and also for the temperature and different displacement at that temperature of the equi- valent bulk of water. Hie true specific, gravity, corrected for both temperature and for displaced air, is sufficiently given by the formula p {d — h) + h, where d is the density of water at the temperature of the room, and S the density of air compared with water, which may be taken as '00 12. It will be seen that the correction for buoy- ancy of the air vanishes when the density of the body is 1, and becomes larger the more the density differs from 1. (iii.) The foregoing corrections also apply to the results obtained by the specific gravity bottle, and in this case when greater accuracy is required a correction must be made for the expansion of the glass. This correction may become important in using a Nicholson's hydrometer, as the volume will alter with the temperature. (iv.) In weighing a body in water the submerged por- tion of the thread or wire which suspends the body loses weight, and in accurate work this must be allowed for by noting the length submerged, the weight and specific gravity of the wire being known. (v.) The removal of all adhering bubbles is obviously most important. If shaking and brushing the object fail, the water should be heated or an air-pump used ; especial care is needed in this direction when the specific gravity of a powder is determined by the specific gravity bottle. Dis- tilled water freed from air by boiUng should be used. The specific gravity bottle should be carefully cleaned and SPECIFIC GRAVITY 83 I k.WW^JW'MTTTl dried. The latter may be accomplished by warming it and blowing in air through a narrow tube from a foot- bellows. Einsing with alcohol hastens matters. Experiment 33. — To determine the volume and density of a solid by the stereometer. Instrmnent required. — A stereometer. A stereometer, or as it is sometimes called a volume- nometer, is an apparatus for determining the volume of solids which are affected by water, or the density of which cannot be obtained in the ordi- nary way : such, for example, as wool, cotton, gunpowder, etc. The instru- ment is an application of Boyle's law (see Experiment 43), the mass in grammes divided by the volume in cubic cms. giving the density of the body. There are several forms of stereo- meter, but the modification shown in rig. 33 is simple and effective. The substance whose volume is required is placed in a perfectly dry glass tube A, which is contracted in two or three places. On the ground top of A a ground glass cover S can be securely fastened by a screw; a flexible tube B connects A with a second glass tube C. Clean dry mercury is poured into C and allowed to rise in A by lifting the tube C until it reaches a fine mark m. 84 PRACTICAL PHYSICS The greased cover is then fastened down, and the tube C lowered until the mercury stands at a mark n in the lower neck of A. Fix the tube C and read the level of the mercury in A and C ; this can be done by the scale D, or by a cathetometer. The volume v between the marks m and n can now be calculated if we know the volume V of the receiver above the mark m. This can be found by puttiag a body of known volume, such as a piece of glass rod, into the receiver, closing the lid when the mer- cury is at m, and proceeding as before. The volumes of V and 17 are, however, best found directly by allowing mercury to run into A from a burette, the rubber tube B meanwhile being closed by a strong pinch -tap, nipped close to A to avoid extension. Greater accuracy can be obtained by running off the mercury in A and weighing it, but this necessitates a stop-cock or two. The solid whose volume is required is now weighed and introduced into the receiver A ; the quantity of the solid taken should be large enough nearly to fill the receiver. When the mercury stands at m, the greased hd is closed and the tube C lowered as before until the mercury stands at n ; the difference of level between the mercury in the tubes A and C is now accurately read. This enables us to find the volume of the receiver minus that of the sohd ; since the former is known the latter is at once obtained. Let V = volume of receiver, V = volume of tube between marks m and n, H = barometric height at the time of experiment, h = diEference of level of mercury in A and C when the body is Twt in the receiver, SPECIFIC GRAVITY 85 h' = difference of level of mercury in A and C when the body is in the receiver, X = volume of body to be found. Then, by Boyle's law, to find v, V:V + t) = H-A:H. And to find x, Y—x:{Y + v)-x=^'H.-h':TL. Hxample 1. — Find the volume v. V = 53-5 c.c, H = 76-4 cms., 'h= 14'0 cms., then 53-5 : 53-5 +■;;= 76-4 -14'0: 76-4; hence «= 12'0 c.c. By direct measurement v was found to be 12'1 c.c. Example 2. — To find x, the volume of a sample of gunpowder. v=53-5c.c., v = 12-1 C.C., H = 76-4 cms., K = 18'7 cms., then 53-5 -«: 65-6 -a;= 57-7 :76'4; hence a;= 16'2 c.c. The weight iv of the gunpowder was 13 '7 7 grammes, hence ^y 13-'77 „„, Exercise. Find the density of cotton-wool by the stereometer. 86 PRACTICAL PHYSICS Experiment 34. — To determine the weights of constituents in a mechanical compound, when the specific gravities of the constituents and compound are known. Instruments regiiired. — The same as in Experiment 23. If there are two constituents in the compound first determine the specific gravity of the compound body in the usual way, then if W, W^, Wj be the weights of the compound body and the two constituents respectively, and p, pi, p2 their respective specific gravities, then (^2 - Pl)P (Pl - P2)P Example. — A brass tube of specific gravity pi = 7'6 is filled with lead of specific gravity 11 '3. To find the weights of tlie brass and lead. The weight of the whole in air is 34'12 grammes, and in water 30'9 grammes. By the ordinary method 34-12 ^ = 34-12- 30-9 = ^^'^' (11-3 -10-6)7-6 ^^ = (11-3-7-6)10-6 ^^'^^ = ^"^^^ grammes, (7-6 -10-6)11-3 "^2 = (-17.0 _ ii-3-\i0-6 ^^'^^ ^ 29-49 grammes. Exercises. 1. Find the relative weights of gold of specific gravity 19-3 ; and quartz of specific gravity 2-65 in a nugget. 2. Find the weight of a bullet of specific gravity 11-3 buried in a mass of wax of specific gravity 0-87. SPECIFIC GEAVITY 87 Experiment 35. — To determine the sp. gr. of a mixture of two liquids, which contract on mixing, the volumes and sp. gr. of the liquids being known. Instruments required. — A burette and a graduated tube. Measure the volumes of the liquids taken by means of the burette, then run them into the graduated measuring glass and observe the contraction. Let V and uj = volume of the liquids, p and pi = sp. gr. of the liquids, P2 = sp. gr. of the mixture. Let f + ui, when mixed, be reduced Ifnth, as found by experiment. Then "■ ■ ■ "P + '^iPi "^^ n—1 v + Vj^ and if the specific gravity of the mixture be known, to find the contraction n {v+v^)p^ Example 1.— -Find the specific gravity of a mixture of equal parts of water and alcohol at temperature 15° C. Here 00^^ v = v^ = AA c.c, p= 0-9998, and /3^= 0-7995, and n = ^= 25-81, 50 21 23-81 25x-9998 + 25x-7995 P^ 22-81 50 23-81x44-98 22-81 X 50 -=0-938. 88 PEACTICAL PHYSICS Example 2. — If the specific gravity of the above mix- ture has been found to be 0'939, then 1 25 x-9998 + 25 x-7995 n^^ 50 X -939 44-95 2-14 = •0428 50 X -939 50 or the contraction is about ^ of the whole volume. Exercises. 1. Mix 30 c.c. of water with 45 c.c. of alcohol, and measure the contraction and specific gravity, correcting for temperature (see Table VIII.). 2. Mix 10 c.c. of sulphuric acid of specific gravity 1'84 with 50 c.c. of water. Note the temperature hefore and after mixture, and when the mixture has cooled to the temperature of the air, find its specific gravity, and estimate the contraction. Experiment 36. — To determine the true weight of a body— i.e. its weight in vacuo. Instruments required. — The same as in Experiment 23. Proceed as in that experiment. If Wi = weight of the body in air at t" C, W2 = weight of the body in water at If C, W = the required weight iu vacuo, p = the specific gravity of air compared with water at t° C, SPECIFIC GRAVITY 89 then (see p. 66) W-Wj .-. w = — V-^— -• i-p Hxample. — Find the true weight of an ivory ball. Wj = 73'447 grammes, ^2= 34-248 grammes, p = 0-0012 (from Table VI.), 73-447 -(0-0012x34-248) ■'■ ^" 1-0-0012 = 73'51 grammes. Exercises. 1. Find the true weight of a piece of wood (coat the wood with a thin layer of shellac varnish to prevent the absorption of water). 2. Find the true sp. gr. (p. 82) of a platinum crucible, correcting for the temperature of the air and water. Experiment 37. — To determine the density of a gas. Instruments required. — A balance and a light glass globe with stop-cock. The globe is first exhausted of all air and carefully weighed, it is then fiUed with dry air at the temperature and pressure of the surrounding atmosphere and weighed 90 PRACTICAL PHYSICS again. The globe is now carefully exhausted by means of the air-pump, and the gas to be tested allowed to flow in ; the globe is once more exhausted and again fiUed with the same gas, so as to insure that it is free from traces of air. Care should be taken that the gas in the globe is at the temperature and pressure of the atmosphere, which is noted ; the globe full of gaa is now carefully weighed. Then the ratio of the weight of the gas to the weight of the air gives the density of the gas compared with air. If the gas and the air are not at the same tempera- ture and pressure in the two experiments, a reduction must be made. The relative density of a gas may also be found by diffusion or effusion, as described in Chapter X. The experimental determination of the absolute density of a gas requires great precautions, the descrip- tion of which is beyond the scope of this book. ExartifU 1. — Find the weight of 1000 c.c. of dry air. By immersion in water the volume of the globe used was found to be approximately 3060 c.c, and by weigh- ing the globe empty and fuU of air, it was found to contain 3'95 grammes of air. .•. 1000 c.c. = 1-29 grammes. Exam-pU 2. — Find the relative density of oxygen gas, taking air as unity. By experiment, using a different globe, it was found that — SPECIFIC GRAVITY 91 127'290 grams. = weight of globe full of air, 125"505 „ = „ „ empty, 1'785 „ = weight of the air m globe. 12 7-4 "7 6 grams. = weight of globe full of oxygen, 125'505 „ = „ „ empty. 1-971 „ = weight of the oxygen in globe. 1-971 P = TT -=1-104. 1-785 The temperature and pressure remained constant throughout the experiments. Hxercise. Find the weight of 1000 c.c. of coal gas and its relative density compared with air. Note. — In testing coal gas a globe with a stop -cock above and below may be used, one cock fixed to a gas bracket, the upper having a gas burner screwed to it. Both cocks being open the gas is turned on and allowed to stream through for a minute or two when the jet is lighted ; after having burnt for, say, half an hour, the two cocks are turned off and the globe weighed. A similar proceeding may be adopted to fiR it with dry air, which can be driven from a gas holder through drying tubes; temperature and pressure being the same in both cases, or corrected if different. CHAPTER V MEASUREMENT OF FLUID PKESS0EE The Barometer The characteristic property of gases is their power of indefinite expansion under diminished pressure. Boyle's law expresses the relationship between the pressure and the volume of a gas as follows : the volume (V) of any portion of a gas varies inversely as the pressure (P) to which it is subject, the temperature being constant, or VP = VP'. The law of Charles expresses the relation- ship between the volume and the temperature of a gas, the pressure being constant, as follows : the volume of any given portion of gas is directly proportional to its absolute temperature (T) where T = <° C. + 273, or YT' = V'T. Combining the two laws, VPT' = V'P'T. The atmospheric pressure (11) is that pressure in dynes per square centimetre which is equivalent to the pressure of the mercury in a barometer tube. If H be the standard barometric height in cms. at 0° C, and " g" the force of gravity, which in Dublin = 981'3, and p the density of mercury = 13'6, then Tl^'S.gp dynes per square centimetre,orn= 76 x 981-3 x 13-6 = 1,014,000 MEASUREMENT OF FLUID PRESSURE 93 dynes, that is a little over a megadyne (a million dynes) per square cm. It would be convenient, and it has been proposed to adopt exactly a megadyne per square cm. as the standard pressure instead of 7 6 cms. For Dublin this would mean a barometric height of 74'94 cms., or 2 9 '5 in. Experiment 38. — To prove that the pressure at any point in water is proportional to the depth of the point below the surface. Instruments required. — A bent glass tube with one long and one short limb, and a millimetre scale. Take a glass tube about 1 metre long and O'o cm. in diameter, make a bend at one end, the short limb being about 10 cms. long. Pour some mercury into the bend and it wUl stand at the same level, since the tube is open at both ends. Now take a tall jar full of water and place the tube at various depths in the water, the long limb of the tube passing up through the surface of the water, and being open to the air. Measure the difference of the level of the mercury in the two limbs at the various depths. This difference will be a measure of the pressure due to the depth of the end of the short limb below the surface of the water, the pressm-e of the air is the same on both Umbs, and can therefore be neglected. Then if d = depth of the end of the short limb below the surface, h = difference in the levels of the mercury in the limbs. ,". ■r- = & constant. h 94 PRACTICAL PHYSICS Example. — Proof of the above law. Enter results thus : — d h d in Cms. in Cms. h 4 0-17 24-1 14 0-57 24-5 24 0-95 24-3 30 1-22 24-5 35 1-42 24-7 40 1-60 25-0 45 1-80 25-0 50 2-00 25-0 55 2-20 25-0 Uxercises. 1. Make a U-tube and repeat the above experiment, plot your results in a curve on millimetre paper. 2. Make several tubes with the ends of the short limbs pointing in various directions, and show that the pressure in a fiuid is the same in all directions. The following experiment affords another proof of the foregoing law, but is one more suitable for lecture illus- tration than accurate measurement. Lower a glass cylinder open at both ends, such as a tall Argand lamp chimney, into a vessel of water ; the lower end of the cylinder is ground and covered with a ground disc of metal or glass, to the centre of which is fixed a thread or MEASUREMENT OF FLUID PRESSURE 95 wire. Holding the disc by the thread or wire against the bottom of the cylinder, lower it to the bottom of a tall jar of water. Now release the thread and slowly raise the cylinder, at a certain depth below the surface of the water the disc falls off, measure this depth, weigh the disc and find the sectional area of the cylinder. The weight of the disc should be equal to the upward pres- sure of the water at the depth where the disc fell. Example. — The disc fell at a depth of 10 '5 centi- metres from the surface of the water. Diameter of metal disc 7 '4 cm., area = 43 square cms., weight of the disc 445 grammes. Hence as 1 c.c. of water weighs 1 gramme, the depth at which the disc ought to have 445 fallen is --rir- = 10-3 centimetres below the surface. 43 Experiment gave 10 '5. Hxercise. Eepeat the foregoing experiment with the disc loaded ; or in a liquid of greater density than water, and estimate the pressure corresponding to the depth at which the disc falls. Experiment 39. — To determine the pressure of the atmosphere by the barometer. Instruments required. — Barometer tube and mercury, also a standard barometer. The construction of a barometer is shown in Fig. 34. A tube of glass about 1 metre long, closed at one end, is 96 PRACTICAL PHYSICS nearly filled with pure warm mercury, the tube itself being dry and warm. By closing the open end with the finger the air bubbles attached to the sides of the glass can be removed by allowing a large bubble of air to travel up and down the tube, and afterwards tapping the A Fig. 34. B closed end of the tube on a soft surface. The tube is then completely filled with mercury, and inverted as shown in A, Fig. 34. Eemoving the finger the mercury falls until it stands at a height corresponding to the atmospheric pressure, as shown in B. Fortin's standard barometer (Fig. 35) consists of a barometer tube dipping into a vessel of mercury A, the bottom of this vessel is made of wash-leather, which can be raised or lowered by means of the screw B, and so bring the surface of mercury to coincide with an ivory MEASUREMENT OF FLUID PRESSURE point, which is the zero of the barometer scale. An enlarged view of the cistern is shown in Fig. 36. Instead of the ivory point a platinum pin may be fixed over the mercury in the cistern, with one end joined to a galvanometer and single cell, and a wire from the other pole of the cell to the mercury in the cistern. By pressing a key in the circuit the precise moment of contact between the mercury and platinum point is indicated by the galvanometer. When there is any difficulty in illuminating the ivory point, this method is convenient. A brass tube, containing a scale near the top, encloses the glass tube of the barometer, and has at its upper end two vertical slots through which the top of the mercury column can be seen. In these slots a slider with vernier moves by means of a rack and pinion. In reading the height of the barometer the eye of the observer is placed on a level with the top of the mercury column, and the vernier moved until the light which passes be- tween the top of the mercury and the vernier is just excluded. If the eye be not level with the top of the mercury the reading wiU be too high, owing to the front and back edge if m PAET I Fig. 35. 98 PEACTICAL PHYSIOS of the slider and the top of the mercury not being in one straight line. The height of the mercury is now read off by means of the scale and vernier. There are usually two scales on the standard barometer, a scale of inches, with "least count" on the vernier = '002 inch, and a scale of cms. with " least count " = -05 mm. Correction for capillarity, which causes a depression of the mercury column, must be allowed for if the diameter of the tube be less than -| of an inch (see Table X.). The correction is, however, a little uncer- tain, and it is better to avoid it by using wide tubes. A correction must also be made for the expansion by heat of the brass tube and of the mercury, when great accuracy is required. Then if H = barometric height in inches or cms. at 0° C, h( = barometric height in inches or cms. at f 0., t = temperature of mercury and brass, H = AX1- 0-0001620. Pig. 86. (See Appendix, § 7). In making an observation (i.) read the temperature of the attached thermometer before the observer's body has altered it ; (ii.) adjust the level of the mercury in the cistern by the screw B until the ivory point coincides MEASUREMENT OF FLUID PRESSURE 99 with its image reflected from the mercury ; (iii.) gently tap the barometer tube to overcome the " stiction " of the mercury ; (iv.) set the vernier, avoiding parallax, and illu- minating the background by a sheet of white paper; (v.) carefully read both verniers, using a lens if necessary. Example. — Eead the height of the barometer at 15° C, and reduce to 0° C. Here ^(=29'48 inches, .-. H = 29'48(l- -000162x10) = 29-41 inches. Exercise. Kead the barometric height in inches and millimetres, and reduce to 0° C. Note. — To reduce the barometric reading to sea-level at latitude 45°. Let Hi = height in metres of the place of observation above sea-level, 6 = latitude of the place, h = observed barometric height, Ao = height corrected to sea-level ; then Ao = A(l-'0026 cos 2^- ■0000002Hi). Experiment 40. — To determine the pressure of the atmosphere by simple means. Instruments required. — A narrow glass tube, pure mercury, and a millimetre scale. Take a clean, dry, glass tube, say 40 cms. long and 3 mm. in diameter, closed at one end, heat the tube 100 PRACTICAL PHYSICS gently over a Bunsen flame to expand the enclosed air, then plunge the open end into a beaker of pure mercury. When it has cooled to the temperature of the air lift the tube up vertically and measure the length of the air column and the length of the mercury thread sustained in the tube by the atmospheric pressure, and which is prevented from falUng out by capillarity. Now turn the tube upside down, and again measure the length of the air column. In the first instance the enclosed air is dilated by the mercury column, and in the second instance compressed. Hence, by Boyle's law (see Experiments 42 and 43), if H = the barometric height, h = length of mercury in the tube, I = length of air column in first case, V = length of air column in second case, then lQ3.-K) = l'{B. + h); Example. — With a tube 35 cms. long and 3 mm. in diameter, closed at one end, we obtained A = 58-2mm., /=265mm., Z' = 227mm. 58-2x492 .". H = T^ = 753'6 mm. The height of the standard barometen in the laboratory at the time of the experiment was 753'2 mm. MEASUKEMENT OF FLUID PRESSURE 101 Exercise, Eepeat the above experiment, and compare the result with the standard barometer.* Experiment 41. — Determination of heights by the barometer. Instruments required. — A portable mercurial, or a delicate aneroid barometer. To measure a small vertical height between two stations, read the barometer at the two places, and the difference in the readings gives the pressure of the column of air between the stations in mUlimetres or inches of mercury. If the mercurial barometer is employed, care must be taken that the tube is vertical, which can be ensured by a plumb-line or by taking the lowest reading of the mercury column. If the aneroid be used the instrument must be held in the same position at the two stations, that is to say either vertically at both places or horizontally.t Let H = height required between the two places, h = difference of the barometer or aneroid readings expressed in centimetres of mercury, p = density of mercury at t° C, S = density of the air at f C, * Mr. Joly, of Trinity College, Dublin, by means of a plug of ivory or boxwood that nearly fills the tube, enables a glycerine or long-range baro- meter to be made of moderate height (see Appendix, § 8). t A new and portable form of aneroid, devised by Captain Watkin and made by Messrs. Hicks, of Hatton Garden, London, enables a difference of level of 6 feet to be easily and accurately read. 102 PEACTICAL PHYSICS then hp = difference in height between the stations ex- pressed in centimetres of a water barometer, H = -^ = height of the air column between the o two stations which are not far apart. Example. — Find the height from the bottom to the top step of a staircase. Temperature of air 1 2° C. By using a Watkin's aneroid the readings at the two places were At bottom . . 29-938 inches At top . . 29-905 inches .-. the difference = -033 inches = "084 cms. and since /3 = 13-56, and S = -001239 (see Tables V. and IX.), „ -084X 13-56 ,,,. . oc^^a f <■ .-. H = = 919-4 cms. = 30-16 feet. •001239 The height as measured by a tape was 30 feet 2 inches. Exercise. Find by the barometer the height from the basement to the top storey of a house or tower, and check your result if possible by direct measurement. Note. — For an accurate determination, corrections must be made for the different density of the air at the observed height, and the hygrometric state of the air (see Maxwell's Theory of Heat, chap. xiv.). For the measure- ment of greater heights by the barometer, Laplace's formula, as modified by recent determinations of the density of mercury and of air, can be used (see Jamin and Bouty, Cours de Physique, vol. i. chap. iv.). MEASTJEEMENT OP FLUID PEESSUEE 103 f?" Experiment 42. — To prove Boyle's law for pres- sures above that of the atmosphere. Instruments required. — A U-tube, a centimetre scale, and pure dry mercury. In Pig. 37 ABC is the tube about 100 cms. long, open at C and closed at A. S and S' are two centimetre scales for measuring the height of the mercury in the two Umbs of the tube, which must be clean and dry. In making the experiment a small quantity of mercury is poured into the bend of the tube, and adjusted till the levels in the two limbs stand at the zeros of the scales S and S'. Then if the section of the short limb of the tube be uniform, the volumes are pro- portional to the lengths ; if the tube be not uniform, then it must be calibrated (see Experiment 17, p. 45). The length of the tube occupied by the gas is read off on the scale S', and the atmo- spheric pressure noted by the barometer; now poured in at the end C, and the gas thereby compressed : when the mercury is being poured into the tube it should be tapped with the fingers so as to cause any air bubbles taken down by the mer- cury to rise to the top. The new length is now read off on the scale S', and if the mercury stand at the level E in the short limb and at F in the long limb, the new pressure will be the barometric height H Fig. 37. mercury is 104 PRACTICAL PHYSICS plus the height Er = A, and so on for other points. Then if V = volume proportional to length of gas columns, P = H + A the total pressure, . •. VP = a constant. Example. — Prove Boyle's law for atmospheric air. Enter results thus : — Volume in c.c. or Length in Cms. V. Pressure in Cms. of Mercury. P. VP. 14-3 77-4 1106-8 12-3 90-1 1108-2 9-3 119-0 1106-7 7-3 151-2 1103-7 Exercise. Eepeat the above experiment and plot your results in a curve, see Fig. 19, p. 42. Experiment 43. — To prove Boyle's law for pres- sures below that of the atmosphere. Instruments required. — A strong iron tube, a glass tube, a centimetre scale, and clean dry mercury. In rig. 38, BC is an iron tube or gun barrel, closed at the lower end, having a wide glass funnel or cistern fixed firmly to the top, and supported on a tripod stand, not shown in the figure. The tube is filled with mercury up to near the middle of the glass cistern. MEASUREMENT OF FLUID PRESSUKE 105 A is a uniform glass tube 100 cms. or so long, closed at A and open at the lower end, which dips in the mer- cury. S is a scale for measuring the length of the gas column, and also the height of the mercury in the glass tube. In making the experiment the glass tube is entirely filled with mercury, then a suffi- cient quantity of the air or gas to be tested is introduced ; the open end is now closed with the finger, and the tube plunged into the mercury in the iron tube EC. The glass tube is now pressed down till the level of the mercury in it is the same as that in the cistern, and the length of the gas column in the glass tube measured, which wiU. be proportional to the volume of the enclosed gas at the pressure of the atmosphere then indicated by the barometer. The glass tube is next raised a little, the enclosed gas dilates, and the mercury in the tube stands at a certain level above that in the cistern. The volume or length of the column of the expanded air and the length of the mercury in the glass tube are again measured. Air, tube, and mercury must all be dry. The new pressure will be the atmospheric pressure H minus the height EF = h, and so on for other points. Then if V = volume proportional to length of gas column, P = H — A = total pressure, .'. 'VP = a constant. 106 PRACTICAL PHYSICS Example. — Prove Boyle's law for atmospheric air. Enter results thus : — Volume in c. o. or Length in Cms. V. Pressure in Cms. of Mercury. P. VP. 10-0 69-2 692-0 12-4 560 694-4 18-2 38-1 693-4 23-3 29-3 692-7 Exercise. Eepeat the above experiment -with hydrogen gas. Experivient 44. — Determination of very low pres- sures of gas by means of the M'Leod gauge. Instrument required. — The M'Leod gauge attached to the Sprengel pump. The M'Leod gauge (Fig. 39) is an arrangement con- nected with the Sprengel pump, whereby pressures below the indications given by a barometer can be determined. It consists of a globe A, at the upper end of which is a graduated tube B, called the volume tube. A T-piece connects the globe with another graduated glass tube C, called the pressure tube ; this tube is in direct communi- cation with the pump and the tube under exhaustion. Attached to the globe A is a vertical glass tube D, about 80 cms. long, which is connected with the mer- cury reservoir E by means of a flexible tube. The capacity of the globe A, from a platinum wire o to MEASUREMENT OF FLUID PRESSURE 107 the lowest division of the volume tube B, having been determined, and the volume of B being known, the ratio of the capacity of B to A is found. In making the experiment the reservoir E is lowered, so that C is in communication with A and B; after ex- haustion, when a measure- ment has to be made of the pressure of the residual gas, E is raised, the mercury rises in C and A until the whole of the gas in the globe is com- pressed into B ; its pressure is then found by measuring the differences of level of the columns of mercury in the volume and pressure tubes. Dividing this difference by the ratio of the capacities of the globe and volume tubes we obtain approximately the pressure of the gas. If the residual gas is compressed up the volume tube we must of course find the new ratio from the known capacity of the volume tube per division. On adding the pressure so found to the mercury head (or difference of level in and B), and once more Pig. 39. 108 PRACTICAL PHYSICS dividing this number by the ratio of B to A, the exact pressure of the residual gas is determined. Example. — After exhaustion the mercury head was found = 5 cms. The ratio of the capacities of B to A was 1 : 42'5, therefore, since VT' VP = V'F, orP = -— -. i.e. P= X 5 = 0-117 cms., 42-5 5*117 then the corrected pressure = =0"12 cms. ^ 42-5 Note. — The M'Leod gauge is subject to error arising from the condensation of the residual gas on the surface of the gauge tube and bulb (see Appendix, § 9). Exercise. Examine the construction and working of the Sprengel pump, and make a vacuum tube for experiments with the induction coil or spectroscope as follows : Draw off a piece of glass tube at one end, and then close the other end, fuse a short length of platiuum wire into and near each end of the tube, exhaust by the Sprengel pump, and when the mercury falls with a metallic click find the pressure of the residual gas by the M'Leod gauge. Fill a second similar tube with pure hydrogen, exhaust, and determine the pressure in the same way. CHAPTEE VI MEASUREMENT OF FORCE Force is the mutual action of two bodies upon one another, and its definition is implicitly contained in Newton's First Law of Motion — viz., force is any cause which alters, or tends to alter, a body's state of rest or of uniform motion in a straight line. The sense of muscular exertion gives us our primitive idea of force, and whatever else is capable of producing a similar effect is said to exert force.* The Second Law of Motion expresses the action of force on matter, and teaches us how to measure force by observation of its effects. The Third Law of Motion shows that wherever force exists in nature it is invariably accompanied by an equal and opposite force, the two forces constituting a stress. A body in motion encountering no resistance is not exert- ing force. Force is in every case a transference of, or a tendency to transfer, energy from one body to another. The effects of force on matter are twofold, either (a) producing change of motion, that is, acceleration, * ' ' The least ambiguous meaning of force is simple pressure or tension." 110 PRACTICAL PHYSICS or (6) producing change of size or shape, that is strain. Either the amount of acceleration or of strain produced affords a measure of force. In the following experiments forces will first be measured by the strain they are able to produce, as in a spring balance, and then the force of gravity wOl be measured by the acceleration " g " it can produce * (see Table IV.). Experiment 45.— Measurement of force by means of a dynamometer. Instruments required. — A spring balance and other forms of dynamometer. Force can be measured indirectly by means of a spring balance, if the value of the acceleration due to gravity at the place be known. Thus if ^ = 980 cms. per second per second, then 40 grammes weight = 40x980 = 39200 dynes (force); or if "^" = 32 feet per second per second, then a weight of 3 pounds = 3x32 = 96 poundals (force) (see pp. 9 and 10). Uxercises. 1. Take a spring balance or other form of dynamometer graduated in grammes or kilogrammes, and make a series of observations by putting on pound weights, tabulate the resTilts, and reduce them to dynes. 2. Take a piece of thin steel wire and make it into a spiral spring by winding it on a small mandril ; fix one * For a fuller discussion of the subject of force, see Professor 0. Lodge's Elementary Mechanics, or Principal Garuett's admirable text-book on Elementary Dynamics. MEASUEEMENT OF FOECE HI end of this spring to a firm support, and hang a scale pan with pointer on the other end. Now add gramme weights to the scale pan, noting the extension of the spring for every additional gramme, and plot the results on millimetre paper. Experiment 46. — Measurement of centrifug'al force, etc., with a whirling' table. Instruments required. — A whirling table with accessories. Owing to inertia, as defined in the First Law of Motion, a moving body continues in a state of uniform motion in a straight line; hence to keep a body moving in a curve a force must act upon the body, which can be shown to be directed towards the centre of curvature, if the speed be uniform ; this is the so-called centripetal force, as seen in the tension of a string that holds a whirl- ing body. But by the Third Law of Motion an equal force tending outwards must be exerted on the body to which the other end of the string is attached; this is the so-called centrifugal force, the two forces being oppo- site aspects of the stress on the string. If F = the centrifugal or centripetal stress, V = the velocity of rotation of the body, m = the mass of the body, r = its distance from the axis of rotation, then F = r 112 PRACTICAL PHYSICS Exercises. 1. "With the whirling table revolving uniformly, find by means of a registering dynamometer the centrifugal force exerted by a body of known weight placed at different distances from the centre of the table. 2. In the same way determine the centrifugal force exerted by bodies of different weights when placed at the same distance from the centre of the table. 3. Fix a strong wide glass tube half full of liquid on the centre of the table, and note the form of the surface of revolution of the liquid when the table is revolving uniformly. Eepeat with water, oil, and mercury. 4. Fix a glass globe full of water on the centre of the table. Float a wax ball on the water ; when the table is revolving uniformly, note that the ball can be sunk to any depth and will remain there in equilibrium. Experiment 47. — To determine the brake horse- power by a friction dynamometer. Pig. 40. Instruments required. — A Prony brake, a stop-watch, and a speed-counter. The Prony brake is a form of absorption dynamometer MEASUREMENT OF FOECE 113 and consists (Fig. 40) of two pieces of wood A B hollowed out to fit a pulley or wheel on the engine or machine to be tested. These can be clamped to any desired extent by means of the two screw nuts m n, a stout rod CD is attaclied to the upper piece of wood having a scale pan at one end D. Before making the experiment the scale pan must be counterpoised by a weight iv ; a known weight W is now put on the scale pan, the engine started and brought to the desired speed, the screw nuts m n being at the same time tightened or the weight W altered until it is just kept balanced, that is when the rod CD is between the stops a h. The number of revolutions per minute is at the same time taken with the speed- counter and stop- watch. If n = revolutions per minute, I = distance in feet from the centre of the pulley to where the weight W acts on the lever, W = weight in pounds in the scale pan, then 27rnrW TT p _ . 33000 Example. — Find the horse -power of the laboratory water motor when driving a lathe. TO = 162, Z = 2ft., W = 4 1bs. „^ 27rxl62x2x4 , , .• H-P. = 33000 ^ * horse-power nearly. A simpler mode of determining the horse-power is to take a stout cord or rope and lap it one or two times round the fly-wheel or pulley of the engine to be tested, PART I I 114 PRACTICAL PHYSICS as in Fig 41. A Salter's balance S is fixed to the floor and to one end of the rope ; a small weight W being Pig. 41. hung at the other end of the rope. The wheel revolves as shown by the arrow, so as to stretch S, and the speed taken as before. If n = revolutions per minute, E = radius of the fly-wheel, W = reading on the Salter's balance, W' = weight on the other end of rope, then 27mE(W - W) ■^'■^" ~ 33000 Note. — If N = revolutions per second, E = radius of fly-wheel in cms., W and W = weights in grammes, then 27rEN(W - W) HP. = 7-6 X 10« MEASTJEEMENT OF FOECE 115 Exercises. Find the horse-power under varying loads of — 1. An electric motor. 2. The Laboratory gas-engine. Q Experiment 48. — To determine the value of 'g' by Atwood's machine. Instruments required. — Atwood's machine, a stop- watch, and a centimetre scale. Atwood's machine (Fig. 42) con- sists of a frictionless pulley A, with a fine string passing over it, and having two equal weights M attached to it. Another small weight m, called the rider, is put on one of the large weights, and thus sets the system in motion. When the weight M -f m has fallen a certain known distance, a ring E takes the rider m off and allows the large weight M to travel onwards through another known distance to the platform P ; the time of travelling this latter dis- tance being carefully noted. In making the experiment first find by trial the smallest weight, w, that will set the wheel- work in motion, this gives the correction for the friction of the wheel- work. Pig. 42. 116 PRACTICAL PHYSICS Now add the rider m, and note the time taken for it to fall from R to P ; repeat with a heavier rider m', noting the time again. This double experiment enables the momentum of the wheel-work to be taken into account. A seconds pendulum is usually attached to the machine ; the time may be taken by its means, or by an ordinary watch — Experiment 21 (2) ; or more accu- rately by a stop-watch reading to fifths of a second, or by a chronograph. ISTow measure BE and RP. In order to start the motion, one weight should rest on the platform C, and, when aU is steady, the hand removed from the string AC. If g = acceleration due to gravity, s = distance fallen before m is removed, Sj = distance fallen after m is removed, ti and t2 = times taken to fall from E to P, after m and m' are removed respectively, w = weight required to overcome the friction, {m — m')si^ then g ■■ (See Appendix, § 10.) Example. — Find the value of " g " by Atwood's machine. s = 50 cms. m = 31'2 grammes. Si=110 cms. m'=15'6 grammes. ti = 1'7 sees. w = 0'57 gramme. ^2 = 2'4 sees. MEASUREMENT OF FORCE 117 Putting these values in the above equation and reducing we get 15-6x110' „,, ^^ 100x1-956 ^ '^^^' P®^ ^^°' P®^ ^®°' As "g" in Dublin is 981'3, this is at least 1'6 per cent too low. It will therefore be seen that this method of determining g is far less accui-ate than the pendulum experiments which follow ; this is mainly on account of the experimental difficulty in obtaining the exact value of w, the friction of the wheel-work. Thus, making w 0'58 gramme instead of 0'57 brings the result as much as 4"2 per cent too low. Hxercise, Determine the value of " g." Experiment 49. — Determination of the laws of falling bodies. Instruments required. — The same as in the last experi- ment, or Morin's apparatus. The preceding experiment shows that as an instru- ment of precision the ordinary form of Atwood's machine is of small value. For the purpose of investigating the laws of falling bodies it is, however, of great service to the student, as by its means the force of gravity is diluted, so that its effects can be observed within a limited fall. The relationship between the space traversed by a falling body and the time taken to traverse it can also be verified by means of Morin's apparatus. This consists of 118 PRACTICAL PHYSICS a vertical drum or cylinder covered with paper, and made to rotate on its vertical axis with a uniform velocity. The falling weight, kept parallel to the cylinder by giddes, has a pencil attached to it which marks the drum as it falls. A curve is thus drawn, due to the composition of the uniform horizontal motion of the cylinder with the uniformly accelerated vertical motion due to gravity. On removing the paper from the cylinder this curve is found to be a parabola OP (Fig. 43). With one or other of these instruments the following relations may be proved : — Space (s) = ^t^ = ^vt, Velocity {v) =gt== ^2gs = —> TimeW=71f = r = ^. ^ g 9 V Uxample 1. — Show by Atwood's machine that s=--^t^, taking g= 32'2 in British units. The mass on one side of the pulley was M and on the other side M + m = M', hence the total mass to be moved was M' + M,and the moving force M' — M. The acceleration force M'-M 23-22 , „ * ,^, ,. a = = = = ■r't- 01 Q. The tune " mass M'+M 23 + 22 ^ ^ taken to fall from rest through the space, s, was found to be 3 seconds. As a = ^^^g^^', therefore s = ^ x-^x 322 X 3^ = 3'22 feet. By direct measurement the space was 38 inches. * To the denominator of the fraction must be added the small weight necessary to overcome the friction of the wheel-work, and the value of which must be found by trial before making the experiment. MEASUREMENT OF FORCE 119 Example 2. — Prove by Morin's machine that the space described by a falling body is proportional to the square of the time taken to fall. The figure obtained on the paper was the curve OP, shown in Fig. 43. Draw the vertical OY and the horizontal line OX. Divide OX into equal parts, as 1, 2, 3, etc. ; from these points draw vertical lines cutting the curve at the points a, h, c, etc. ; from OY draw per- Pig. 43. pendiculars meeting the curve at these points. As OX represents the time taken for the body to fall through the space OY, the successive spaces from along OY will be found proportional to the squares of the distances 120 PRACTICAL PHYSICS from along OX. Hence, as 5* is constant, s is pro- portional to t^. Exercises. 1. By Atwood's machine prove that v = 2s jt. 2. Show that the velocity is being accelerated before and remains uniform after the rider is removed from the falling weight. 3. Show by Atwood's machine that the velocity generated by a constant force is proportional to the time during which the force has acted. 4. Show also that, the mass remaining the same, the velocity generated ia unit time varies directly as the force, and with the force constant the velocity varies inversely as the mass. 5. With Morin's apparatus prove that the distance fallen from rest, divided by the square of the time taken to fall, is a constant. Note. — With a freely falling lody, and a chronograph, the value of g may also be determined, and the laws of falling bodies deduced (see Experiment 21). For this purpose simple electro-magnetic appliances attached to Willis's apparatus * give excellent results. Experiment 50. — To prove the isochronism of the vibrations of a torsional pendulum. Instruments required. — A wire clamped above, and a weight with pointer moving on a graduated circle below. By Hooke's law, within the limits of elasticity the " torque " of restitution is proportional to the amount of * See Sir Eobert Ball's Experimental Mechanics. MEASUREMENT OF FORCE , 121 distortion. This causes the oscillations which are set up on the removal of constraint to be performed in equal times, that is to say they are isochronous. Exercises. 1. Prove the isochronous character of the oscillations of a torsional pendulum. Twist the wire through an angle of 45°, and, by the method of passages (Experiment 19), find the time of a single oscillation, noting the angle at the close of the experiment; now twist the wire through 90°, and again determine the time of a single oscillation, noting the angle at the close. 2. Prove by a similar experiment the isochronism of an ordinary simple pendulum vibrating through a small arc. Experiment 51. — To pFove experimentally the laws of the simple pendulum ; determination by this means of the value of ' g.' Iiistruments required. — A simple pendulum, a centi- metre rule, and a stop-watch. A simple pendulum is theoretically a massive point suspended from a rigid support by a massless thread, and swinging through a small arc. This condition is approached in practical work by suspending a lead bullet, say 1 cm. in diameter, by a fine silk thread, the centre of gravity of the bullet being taken as the centre of oscillation. 122 PRACTICAL PHYSICS In making the experiment, the length of the pen- dulum (i.e., the distance from the point of support to the centre of the bullet) is varied, and the corresponding period of vibration observed. The pendulum is made to vibrate through an arc not exceeding 3° on each side of the vertical — i.e., a hori- zontal length of about 5 cm. on each side in a pendulum 1 metre long. A great number of observations, or sets of observations, are taken, and the mean time of vibration determined. Then if T = period of vibration in seconds,* I = length of the pendulum in cms., g = acceleration due to gravity, 7r= 3-1416, T = 2.^ ^ 9 .: — = — (a constant). (See Appendix, § 11.) Example 1. — To prove experimentally the pendulum laws. * We use the term vibration to indicate a double oscillation, that is a complete movement to and fro, a " swing-swang," as it has been called. MEASUREMENT OF FORCE Enter results thus : — 123 Number of Experiment. T Sees. T= I Cms. I 1 1-84 3-38 83-4 24-67 2 1-76 3-11 76-7 24-66 3 1-65 2-73 67-6 24-76 4 1-59 2-54 62-7 24-68 5 113 1-27 31-5 24-69 Example 2. — To find the value of " g" by the simple pendulum, -we have -^ The time and length being observed very accurately, we obtained Z=230 cms., T= 3'04 sees., which was the mean of 10 sets of 10 vibrations each, 4x9-87x230 ■'■9 = 3-04' 982-5. The value of "g" in Dublin is 981-32 cms., hence the result in the foregoing experiment is 0-12 per cent too large (p. 38); thus 982-5 - 981-32 = 1-18, and 981-32: 100 = 1-18: 0-12. For an accurate determination the result thus obtained needs to be corrected — 124 PEACTICAL PHYSICS (i.) Should the mean amplitude of the swing 6 exceeds 3°. (ii.) In order that the length may be reduced to that of the equivalent simple pendulum ; thus if the bob be a sphere the corrected length is ^ + -gy (see Appendix, §12). Hence the formula for " g " becomes Other corrections are — (iii.) Buoyancy of the air on the bob. (iv.) Eesistance of the air. (v.) Dragging of the air by the bob. (vi.) Want of rigidity of the support, JSxercises. 1. Prove by experiment the laws of the simple pen- dulum, taking I as 20, 40, 80, and 100 cms. respectively. 2. rind the value of " g " by the pendulum, applying corrections (i.) and (ii.) Experiment 52. — Determination of the value of « g ' by Rater's pendulum. Instrwnunts required. — A Kater pendulum, a milli- metre scale, a clock or stop-watch. A simple form of Kater's compound pendulum (Fig. 44) consists of a bar of seasoned hardwood AB, with 9 c MEASUEEMENT OF FORCE 125 two weights attached to it, M and m. The larger weight M is fixed perraanently to the bar, the other weight m can be moved up or down the bar and clamped at any desired position. C and D are two steel knife-edges fixed in the bar, and the pen- dulum can be made to vibrate about either one of them, the knife-edges resting on agate plates. In making the experiment the pendulum is caused to vibrate about the knife-edge C, and the vibrations accurately observed. It is then made to vibrate about the knife-edge D, and the vibrations again carefully observed. If the number of vibrations in equal times are not the same about the two knife-edges, the small weight in is moved slightly and the observations again made ; the weight m being always moved towards that knife-edge about which the time of vibjation is the greater. Having by adjustment got the times of vibration about the two axes equal, measure very accurately the distance between the knife- edges, which will be the length of the equivalent simple pendulum. (A simple form of Kater's pendulum, which with care gives fairly good results, can be made with a thick, flat, piece of heavy wood, about 1 metre long and 5 cms. broad. A hole is made near each end and uniform pieces of glass rod inserted into the holes, which serve as knife-edges, and in place of the agate plates, two pieces of the same glass rod are fixed firmly on a suitable support. 6 D 126 PEACTICAL PHYSICS The adjustment to equal times of vibration can be done by slipping small strips of lead up or down the rod.) If I = distance between the knife-edges, T = period of vibration, g = acceleration due to gravity, then g = -^ • Example. — To find the value of " g " by Kater's pen- dulum. By trial and error the times of vibration about the two axes were found to be 1'74 sec, and the correspond- ing distance between the knife-edges 75"3 cms. 47r^x75-3 „„, „ ■■■9= 1.742 =981-9. If the times of vibration about the two centres are not perfectly the same, it is convenient in an experiment to use the following formula — Let h and Aj be the distances of the centre of gravity of the pendulum from the two knife-edges, t and t^ the corresponding times of vibration, then 9 = (hf - li^t^^) Exercise. Pind the value of " g" by means of a Kater's pendu- lum (1) when the periods are in perfect agreement; (2) when the periods are slightly different. * For the proof of this equation, see Routh's Rigid Dynamics, Part I, p. 80. MEASUREMENT OF FOECE 127 Experiment 53. — To illustrate the composition of two simple haFmonic vibrations in rectangular directions by means of Blackburn's compound pendulimi. Instruments required. — A Blackburn pendulum and a centimetre scale. Blackburn's pendulum (Fig. 45) consists of a com- bination of two pendulums on the same string suspended from the one support M. The pendulum of length AC vibrates in the plane perpen- dicular to the plane of the paper, and the other of length BC vibrates in the plane of the paper. If BC is vibrated alone, it does so from B as a fixed point. Now if the bob C be drawn aside in the direction of the bisector of these two planes, it will take the path due to the compounding of the two separate motions. In making the experiment the bob C is usually a disc of lead with a fine nosed funnel inserted into it, through which a stream of sand flows and thus traces out the path of the bob.* Fig. 45. * These curves, togetlier with the mode of drawing them geometri- cally, will be given in the part on Sound, and for the theory of the Blackburn pendulum see Everett's Vibrating Motion and Sound, chap. vi. 128 PKACTICAL PHYSICS From the simple pendulum law we have T=27ry-, /T ^ Example. — Find the lengths of a compound pendulum when the periods of the component pendulums are in the ratio 5:6. The length / = AC = 100 cms., 100 x52 .-. /i = BC = ^2 — = 69-44 cms. Uxercise. Find the lengths and obtain the tracings of the paths of the bob when the periods are in the ratio of (i.) 2 : 3, (ii.) 3 : 4, (iii.) 4:5. Experiment 54. — To find the centre of percussion of a body. Instruments required. — A rough plank of wood, thread, and a bullet. The centres of oscillation and suspension in a com- pound pendulum are reciprocal. The centre of oscillation is identical with another remarkable point called the centre of percussion, which may be defined as follows : If a body be suspended from any point in it, the centre of percussion is such a point in the body that when an MEASUREMENT OF FORCE 129 impulse is received whose line of action passes through this point in a direction perpendicular to the straight line joining this point to the centre of suspension, no reaction or impulse is caused at the centre of sus- pension. This point is of much practical importance, for a blow deHvered through the centre of percussion produces simply rotation about the centre of suspension of the body without instantaneous strain on the support. Hence, for example, a door -stop should be placed at the centre of percussion of the door, otherwise the hinges will be strained. In making the experiment, suspend the body by a pin so as to oscillate about the axis of suspension, and from the same support hang a simple pendulum, and adjust its length until its period of vibration synchro- nises with that of the body. Then the length of this equivalent simple pendulum is the same as the distance between the centres of suspension and oscillation of the body, and the latter point gives us the centre of percussion. Example. — Find the centre of percussion of a piece of wood 120 cms. long, 3 cms. broad, and 2 cms. thick, when suspended by a pin at a distance of 2 cms. from one end. By trial and error the length of the simple pen- dulum which synchronised with the rod was found to be 78*5 cms., therefore the centre of percussion is 78'5 cms. from the point of suspension of the rod, that is about two-thirds its length from the support. PART I K 130 PRACTICAL PHYSICS Hxercises. 1. Find tlie centre of percussion in the Ballistic pen- dulum used in the next experiment. 2. Find the centre of percussion of an irregular shaped plank. Ex'periment 55. — To determine the velocity of a bullet by means of the Ballistic pendulum. Instruments required. — Ballistic pendulum, a gim, stop-watch, and centimetre rule. The Ballistic pendulum (Fig. 46) consists of a heavy bob of wood C, supported by a light frame -work from a horizontal knife-edge A. D is a point on the bob in the line of the axis to which a tape is attached for measuring the chord of the arc of oscillation of the pendulum when it has been disturbed by the impact of the bullet. A clip N is fixed on the table or stand, quite close to D, when the pendulum is at rest, and serves to put a small amount of friction on the tape when it is being drawn out by the swing of the pendulum ; on the return of the pendulum the length of tape drawn out remains unaltered. Fig. 46. MEASUREMENT OF FOEOE 131 In making the experiment the bullet is fired in the direction B, perpendicular to the line through the knife- edge and centre of percussion of the pendulum. If M = mass of the pendulum bob with the bullet in it, m = mass of the bullet, h = distance from the centre of percussion of the bob to the knife-edge, hi = distance of the line of fire from the knife-edge, L = distance of attachment of tape from the knife-edge, S = length of tape drawn out, T = half period of equivalent simple pendulum, 5^ = acceleration due to gravity, 7r= 3-1416, V = velocity of the bullet, then V = /^ (Appendix, § 13). mhiLv Example. — Find by the Ballistic pendulum the velocity of a bullet shot from an air-gim. By experiment we have M = 6000 grams., m = 6 grams. A = Ai = 59'5 cms., L = 7lcms. S = 5-2 cms., T = 0'75secs. 6000X 59-0x980 X 0-75 x5'2 ■'■ 6x59-5x71x3-1416 = 17140 cms. per second. 132 PRACTICAL PHYSICS Hxercises. 1. Vary the air charge in the gun and repeat the above experiment. 2. Hang a heavy ball up by a thread and let it fall against the pendulum, then calculate the velocity with which it strikes the bob. Experiment 56. — To determine the moment of in- ertia of a body about a given axis. Instruments required. — A body of known moment of inertia, a stop-watch, and a centimetre scale. Hang up the body whose moment of inertia is required by a wire iirmly fixed to it and passing through the axis of rotation and centre of gravity of the body, and take the period of vibration. Then add the body of known moment of inertia, so that this compound body will vibrate round the same axis, and again note the period of vibration. Now if K = moment of inertia required, k = moment of inertia of known body, ti = period of unknown body, <2 = period of compound body, m = couple exerted by the wire when twisted one radian (57-3°) from rest, then Jh, or -r- is a constant. Example. — Make the above experiment with a vessel containing water, the orifice being of the same area. Enter results thus : — h Cms. W Grams. h 84 25 708 385 5967 5930 180 PRACTICAL PHYSICS Exercise. Repeat the above experiment with various sizes of orifice. Note. — It will be observed that the deyisity of a liquid does not affect its rate of flow (see Experiment 79). The law of Torricelli does not strictly hold if the orifice is at the end of a pipe or tap instead of being in the thin wall of the vessel, owing to resistance of the pipe, as the following results of an experiment show. There were three taps in the side of a tall vessel at different heights, one nozzle fitted all the taps, so that the area of the orifice was the same in all cases. h Cms. W Grams. W2 h' 86 50-5 15 103 76 38 123-3 114-4 96-3 Experiment 75. — To determine the height of a head of water by efflux. Instruvunts required. — The same as in Experiment 73. A pipe leading from the bottom of a tall reservoir of water, or from a cistern of known height, has an orifice of known area : receive in a beaker the water flowing from the orifice in a given time, and from the weight of water calculate the velocity, thence deduce the height of the MECHANICAL PROPERTIES OF LIQUIDS 181 head. Experiment gave the value of K as 0-73, owing to the nature of the orifice. If W = weight of water in grams, which flows out in t sees., t = time of flow in sees., A = area of orifice in sq. cms., h = height in cms., W then, from p. 179, \'^^ = ^JJ(2g)^' Example. — Find the height of the water surface in a cistern in the laboratory. A = 7r7-2 = 7rx-153^ W = 1 2 6 grams., < = 3 sees., 1260 ' ■ '^^ 0-73 XTTX -1532x30(2x981)^' .-. A =312-2 cms. The height of the surface of the water in the cistern was by direct measurement found to be 310-8 cms. Exercises. Eepeat the above experiment with different sizes of orifice. Care must be taken that the orifice is small relatively to the pipe. Experiment 76. — Comparison of two methods for determining' the parabola formed by a spouting- jet. Instruments required. — -A vessel of water with a small orifice at the side, a beaker, and a millimetre scale. (1) Take a tall vessel of water (Fig. 52), with an orifice in the side near the bottom, which may be closed temporarily by a plug, and as before keep the head of 182 PEACTICAL PHYSICS ,...p.-. water constant. Place the vessel so that the distance AB = OA, and set the experiment going by re- moving the plug from 0. Now measure from the level 00 the distance mn to the jet of water, say for every 1 cms. of 00, and plot a curve with 00 as axis of abscissae, and OA as axis of ordinates. Eoaample 1. — Find the curve for a water jet issu- ing from a lateral orifice. In the experiment the distances from the orifice are given in column x below, and the fall of the curve from the level of the orifice in column y. The distance OA = AB = 70 cms. Fig. 52. X y Cms. Cms. 10 1-2 20 4'9 30 10-0 40 18-0 50 32-5 60 44-0 70 68'0 MECHANICAL PROPERTIES OF LIQUIDS 183 By plotting these values we get the curve A in Fig. 53. (2) Find the volume of water flowing from the orifice in unit of time, by receiving it in a beaker and weighing and calculating the volume. Let W = weight of water run out in t seconds, A = area of orifice, V = velocity of efflux, W .•. v = . pA.t Also, from dynamical considerations,* x = vt, y = Ifjf ; .'. y = -^x^, which is the equation to the parabola. Then, knowing v, plot the curve whose equation is Example 2. It was found that W = 9 9 grams, in 6 sees., A = TT X -0538^ sq. cms., p = 1, 99 7rX-0538^x 60 981 = 181'5 cms. per sec, and y= — ""-^^ — .a;' = •0149a;l ^ 2x181-5' Then by giving various values to x in the equation y — •0149a;^, we get the following table ; — * Garnett's Dynamics, chap. iii. , or Worraell's Dynamics, p. 40. 184 PRACTICAL PHYSICS :r.. y- 10 1-49 20 5-96 30 13-41 40 23-84 50 37-25 60 53-64 70 73-01 By plotting these values we get the curve B in Fig. o3.» m \i a; I'i^s^ ^s^ i5.__. X3 --»--r ■'■-->■ -Y- ■ '--X nr^ njj d±r^i;. eh; ^ "_;e ixtr: -'- - ^ j ■ x i y ^ t: tczs Trri 4im Ji^ 3t£ir ^T^ ^^ffi^ t±±it ±a ffWT r-M-. liT Fig 68. * A mistake has been made in engraving the above figurf; ; the crosses after 1 in the cnrves reqnire shifting one diviirion to the right. MECHANICAL PROPERTIES OF LIQUIDS 185 £xercise. Eepeat the above experiment with different heads ; see also Experiment 78 (ii.). ExpcriTnent 77. — Detepmination of the resistance to the flow of water throug-h pipes. I-nstruments required. — A cistern of water with a long outlet pipe having lateral orifices in which are fixed glass tubes. In Fig. 54 BC is a cistern in which the water can be kept at a constant head by suitable arrangements, such as a ball-cock. AB is the outlet pipe having orifices terminated by glass tubes a, h, c, etc. As the water flows from the cistern it wUl meet with resistance in the pipe AB, and rise in the glass tubes to heights proportional to the resistance in AB. The height of the water in the glass tubes will gradu- ally decrease from the inlet to the outlet A, where the resistance is a minimiun when the tube AB is horizontal. In making the experiment, the height of the water in 186 PRACTICAL PHYSICS the tubes a, b, c is measured when the water is flowing uniformly through AB, and at the same time the water flowing from A in a given time is received in a beaker and weighed, and from this weight the velocity of efflux is calculated. The results are now plotted on millimetre paper, with the distances of the glass tubes from the inlet as abscissae, and the heights of the water in the tubes as ordinates. Again, by varying the height of head of the water in the cistern, or the velocity of efflux by means of a tap, another curve is obtained, having the velocities of efflux as \' _ _ __ _ ■^ - ~_ ■ - __ . ^ - -^ - - -- - J^--. _ _ x — : Fig. 65. abscissa and the heights of the water in the tube farthest from the cistern as ordinates. Example. — The head of water being kept constant. MECHANICAL PEOPEETIES OF LIQUIDS 187 the following results were obtained, the distance between the tubes being 13 cms. Height of the water in the tubes in cms. a b c d e / ? 41-5 39-5 37-2 34-2 31-0 29-0 26-4 The results are plotted in Fig. 55, Exercises. 1. Kepeat the above experiment with AB horizontal, using different heads, and plot the result in curves. (a) With distances of the tubes from the cistern as abscissae and heights of the water in the tubes as ordi- nates. (6) With the velocities of efflux as abscissae and the heights of the water in the tube farthest from the cistern as ordinates. 2. With the head of water constant lower the end A of the outlet pipe, and find the angle which AB makes with the horizon, when the resistance throughout the tube is the same ; that is when the heights of the water in the glass tubes are all equal to one another. Experiment 78. — To determine the relation between the angle of elevation, velocity, and range of a spouting jet. Instruments required. — A vessel of water with tap 188 PRACTICAL PHYSICS and nipple which can be set at any angle, and a centi- metre scale. (i.) With the same arrangements for keeping the head of water constant as in Experiment 73, set the tap at various angles, and note in each case the horizontal range of the water jet. If V = velocity of efflux, 6 = angle of elevation, H = horizontal range, 2;2siu2^ then* H^ H v^ ■'■ ~ — K7i = — = a constant, sm 20 (J Example. — Prove the above law. Enter results thus : — e. Sin 18. H Cms. H sill 2d' 25 45 65 0-7660 1-0000 0-7660 26-2 34-4 27-6 34-3 34-4 36-0 Exercise. Eepeat the above experiment for every 10° from 10° to 70° * Worniell's Dynamics, p. 42. MECHANICAL PEOPERTIES OF LIQUIDS 189 (ii.) When the jet issues horizontally a vessel similar to that in Experiment 76 may be used, having several apertures vertically above each other. Then the head being kept constant as before the ho7'izontal ranges of the various jets are determined. Now, by putting in the value of v'^ = 2gh in the equation y = gx^/2v^ (see p. 183), we get x^ = 4h7/, where X is, in this case, the whole horizontal range of the water jet, y the height of the orifice above the horizontal plane, and h the height of the water surface above the orifice. It is evident from the above equation that the value of X will be unaltered by interchanging the values of h and y ; this simply means that the highest aperture will give the same range as the lowest, and the highest but one the same range as the lowest but one, and so on. Also, since a; is a maximum when h = y, the greatest range is given by the central orifice, that is when the water surface is as miich above the orifice as the hori- zontal plane is below it. Again, it follows from the properties of the para- bola that the focus of the parabolic jet will be as far below the orifice as the surface of the water is above it. Example. — In an experiment with the jets issuing horizontally. (1) A =57 cms.; 2/=7'5 cms. gave the range a; =41 cms. (2) h=1'b cms.; 2/= 57 cms. gave a; = 40 '5 cms. Prom the theory x=2 s/Tiy= 2 Jhl X 7-5 = 41-4 cms. 190 PRACTICAL PHYSICS (Experiment gives a value slightly less than this theory, as air friction affects the former.) Exercise Eepeat the above experiment with different distances for h and y. Experiment 79. — To prove that, under a given head, the velocity of efflux of a liquid from an orifice is independent of its density. Instruments reqidred. — A burette tube, liquids of dif- ferent density, beaker, and a stop-watch. It will have been seen (p. 178) that the expression for the velocity of efflux of a liquid is the same as that which would be acquired by a body falling freely from the upper surface of the liquid to the centre of the orifice whence the liquid escapes, the atmospheric pressure being regarded as equal at the surface and at the orifice, or V = y/2gh. Inasmuch as the loss of potential energy mgh, by a certain mass m of the issuing liquid, is equal to the gain of kinetic energy ^mv^ by that mass, .•. mgh = -J-mw^ ; hence v'^ = 2gh. The velocity, therefore, as in falling bodies, is inde- pendent of the mass, and accordingly of the density of the liquid. A simple experiment is sufficient to prove this. Take a piece of glass tubing about 60 cms. long and 1 or 2 cms. internal diameter, and by means of the blow- pipe contract one end of the tube to an orifice about MECHANICAL PROPERTIES OF LIQUIDS 191 2 mm. in diameter, not drawn off, to a long narrow tube, as the viscosity, etc. of the liquid then affects the flow. Make two marks on the tube some 50 cms. apart, and fix the tube vertically in a cUp, with the small orifice below. Close the orifice with the finger, and fill the tube with the liquid under experiment. The finger is now removed, and whilst the liquid runs into a beaker the time when the surface of the liquid passes the marks is taken by means of a stop-watch, or by listening to the beats of an ordinary watch. The tube is cleaned and the same operation gone through with another liquid, when it will be foimd that (the liquid head in each case corresponding) the time taken for equal volumes of different liquids to flow out is the same, however great their difference in density. Eocanvple. — To show that the rate of flow from an orifice of water, mercury (/? = 13'59), and sulphate of zinc {p = 1"36) is the same. The stop-watch was started when the liquid surface passed the upper mark, and stopped when it passed the lower mark ; this gave the time of efflux of equal volumes of each liquid from this particular tube to be 21 seconds, with a variation of less than half a second in several dif- ferent experiments with each liquid. Uxercise. Repeat the above experiment, taking the mean of three observations with each hquid. CHAPTER X MOLECULAR PEOPEIITIES OF FLUIDS: VISCOSITY DIFFUSION SUEFACE TENSION In the last chapter and iu Chapter V. masses of fluids have been considered, chiefly as acted upon by the force of gravity, without reference to any motion or mutual action of their molecules. In the present chapter, instead of forces acting through sensible distances on large masses of matter, we must consider the action of those forces which only become sensible at insetmblc distances. To those molecular motions and forces are due the phenomena of the capillarity of liquids and the viscosity and diffusion of fluids in general, as well as aU chemical action, solution, etc. The distance at which these forces become sensible is too small for microscopic observation or for any method of direct measurement. From observations on capillarity Quincke has deduced this distance — which may be called the radius of molecular action — to be about 50 /a/t (micromillimetres) ; a number which lies between the upper and lower limits of molecular action determined MOLECULAR PROPERTIES OF FLUIDS 193 by Professors Eeinold and Eiicker by a different method of experiment.* Within the radius of molecular action a great devia- tion occurs from the ordinary law of force as expressed by gravitational attraction ; though the law of variation of molecular attraction is at present unknown, the magnitude of the force which comes into play is, ceteris paribus, vastly greater than the known forces which act upon ordinary or molar masses of matter, increasing from a small amount at the outer limit of molecular attraction to some unknown enormous amount, " which may be hundreds or thousands of kilogrammes weight before the molecules come into absolute contact." t The cohesion of bodies is due to the action of these molecular forces, and hence also elasticity, though it has been more convenient to deal with this subject in the mechanics of solid and liquid masses. In like manner the viscosity of bodies as stated on p. I7l is due to internal or molecular friction, and we shall therefore begin by a few measurements of this property. Experiment 80. — To determine the internal friction or viscosity of a liquid. Instruments required. — As described below. Even the most limpid liquids exert a certain resistance to change of shape, depending on the rate of change ; this * See on this subject Professor Riioker's important memoir "On the Range of Molecular Forces," Journal of the Chemical Society, March 1888. Young in 1816 estimated this distance to be ^f^ millionth of an inch. — Young's Works, vol. i. p. 461. t See Lord Kelvin's Lectures and Addresses, vol. i. p. 11. PART I 194 PEACTICAL PHYSICS is due to the viscosity or internal friction of the liquid. The higher the internal friction the more nearly the liquid approaches the solid state, such as, for example, treacle, pitch, and cobbler's wax. The coefficient of internal friction, rj, is the force required to slide two parallel surfaces of unit area, within the liquid, unit distance apart, through one cm., in one second of time.* The dunensions of viscosity differ from rigidity by the time unit only, the same difference that occurs between acceleration and velocity. For many liquids the coefficient of viscosity is very small, eg. water being 0'013 C.G.S. units; a rise of temperature rapidly diminishes this coefficient. (i.) The most convenient way of determining the coefficient of viscosity is to allow a known volume of the liquid to run through a capUlary tube, and take the time required. If rj = the coefficient of viscosity, h = height of the column of liquid in cms., r = radius of the capillary tube in cms., I = length of the tube in cms., V = volume of liquid that flows out in unit time, p = density, ^ = 980, thent , = ^. This formula only holds true if the liquid wets the tube so that the liquid layer ia contact with the tube is * That is to say, it is the tangential force per sc^uare em. required to maintain a constant difference of velocity of one cm. per sec. between two parallel layers of liquid moving in parallel directions and one cm. asunder. + Jamin et Bouty, Cours de Physique, vol. i. part ii. p. 132. MOLECULAR PEOPEETIES OF FLUIDS 195 at rest, the moving liquid therefore rubbing against itself. The whole cross -section of the liquid does not move with the same velocity, the axial portion flowing faster than the other parts. A small correction requires to be applied from the fact that the liquid leaves the tube with a certain velocity, all the work done not being spent in overcoming internal friction. As the corrections involve the square of the velocity it is best to diminish the speed by using a long fine capUlary tube, which ought to be kept horizontal, the pressure height of the liquid in no case including the capillary. The student can easily make a simple and efficient apparatus for the verification of this formula by drawing off the tube of, say, a 20 c.c. pipette to as uniform a capillary bore as possible, and then bending the capillary tube at right angles to the bulb. The capillary can afterwards be broken off and its mean diameter found by measuring the bore at its two ends, either by the Eeading microscope or by means of a mercury thread (p. 46). To obtain the necessary constant hydrostatic pressure the upper part of the pipette may be connected with a cistern, when water is used, or to a reservoir of com- pressed air, of known pressure, for other liquids. Example. — Find the coefficient of viscosity of water at temperature 15° C. 7i = 300cms., r = '03ocrns., /3=1, ^ = 49 cms., v = Q-2 c.c, 300 XttX 0-035* X 980 ^^ . •■• '' = 8 X 49x0-2 = *^'^^^- 196 PRACTICAL PHYSICS As might be expected, the number obtained is some- what larger than that given in Table XIX. The diameter of the tube used should have been less in the case of water, and the corrections referred to above would also . reduce the value.* Exercis&s. (1) Find the coefficient of viscosity of alcohol, and (2) of glycerine with 1 per cent of water. Note. — The mean velocity of a liquid in a capillary tube is proportional to the pressure (not to the square root of the pressure as in wider tubes, p. 178) and to the square of the radius. Hence as the volume dis- charged in a given time is jointly proportional to the velocity and to the area of the tube, it is proportional to the fourth power of the radius, not to the square as in wider tubes. This law was found by Poiseuille, and is called by his name.t By using capillary tubes of different diameter the student can verify this law, but he wiU find it necessary to use considerable pressure with fine tubes. This pressure may be obtained as described above, or by a column of mercury acting upon the liquid under trial through a communicating U tube, so that the mer- cury does not flow into the capillary. For water, a tube about half a millimetre, or under, in diameter may be considered a capillary tube, but for * For these corrections, see Ostwald's Chemistry, English ed., p. 113 j see also a paper by Mr. L. R. Wilberforce, Phil. Mag., May 1891. t Poiseuille's law ceases to be obeyed when the velocity x section of the stream -r the viscosity of the liquid reaches a certain critical value. MOLECULAR PROPEETIBS OF FLUIDS 197 liquids more viscous than water the diameter of the tube may be proportionally larger. (ii.) Owing to experimental difficulties, such as obtain- ing a capillary tube of perfectly uniform bore and circular cross-section, the accurate determination of the absolute value of viscosity is very troublesome ; but by taking the internal friction of water at 0° C. as unity, the relative value of other liquids can readily be found by the apparatus shown in Fig. 56, p. 198. Cisa capillary tube with a bulb B, and above is an india-rubber tube with a pinch tap D, which enables the liquid to be drawn up and held in the bulb. The tube and bulb are enclosed in a vessel of water the temperature of which is obtained by the thermometer T. The pinch tap is opened and the liquid drawn up to a mark o above the bulb ; the time is taken that the liquid falls from this mark to another mark o below, the pinch tap being closed till the experiment begins. If t = time of flow of the given volume of liquid, f = time of flow of a corresponding volume of water, density = 1, p = density of the liquid, K = the relative coefficient of viscosity, t then ■^""?^' Uxample. — Find the relative viscosity of ether of den- sity 0-7. <=106secs., !!' = 276 sees., p = "7. 198 PRACTICAL PHYSICS 7x106 K 276 0'27. Exercise. Determine the relative viscosity of petroleum, olive oil, and turpentine at 15° C. and 50° C. Note. — The effect of temperature on the viscosity of water is shown in the following series of experiments with a wide capillary tube. With a fine capillary the viscosity at 70° was found by Graham T ^^^x— '^^ to be only one-fourth that at 0° C. Temperature. Time of Efflux 1-5° C. . 95 seconds. 5°C. . . 89 „ 17° C. . . 74 „ 28° C. . . 67 „ 44° C. . • 59 „ 72° C. . . 50 „ g^^b'fe=:3s^ Plot the above numbers in a curve. The following measurements of relative viscosity also show the effect of temperature. Substance. Alcohol. J) Turpentine. Temperature. 0° 60° 19° 53° Time of flow. 275 sees. 76 „ 1331 „ 83 „ MOLECULAR PROPERTIES OF FLUIDS 199 Experiment 81. — Determination of the relative viscosity of liquids by a vibrating disc or vessel. Instruments required. — A metal disc suspended horizon- tally by a fine wire, and a vessel containing the liquid; also a hollow cylinder with wire suspension and long glass tubes. (i.) If a metal disc, wetted (but not acted on chemi- cally) by the liquids under experiment, be suspended from a firm support by a wire, and the disc made to vibrate torsionally in the liquids successively, it will be found that the periodic time of vibration will be sensibly the same in each liquid ; but the amplitude will diminish like a vibrating tuning fork or pendulum, and the ratio of the amplitudes of any two successive swings in the one direction, whilst constant for the same liquid, will be different for each liquid. Thus if in any one liquid this ratio be 0'8, this means that the second swing is only ^ of the first, and the third only ^ of the second, and so on. In general, the constant relation of an arc of oscillation to that next fol- lowing is called the ratio of damping, and the logarithm of this ratio is called the logarithmic decrement of the vibrating body. In making the experiment fix a pointer at the lower end of the wire, and allow it to move over a scale. Then observe the period of vibration of the disc in the respec- tive liquids, and from the ratio of the observed amplitudes of the successive swings the relative viscosities of the liquids may be deduced.* * The relative viscosity of wires (Experiment 67) may be found in a similar way by noting the log. dec, wires of the same length and cross- 200 PEACTICAL PHYSICS If t = periodic time of vibration of disc, k = a, constant proportional to the viscosity, d — ratio of the amplitudes, e = base of the ISTeperian logarithms, then d = e-'^* i.e. log d= —tk log e, t ' Example. — Find the relative viscosities of water at 15° C. and at 80° C. Since t =1'2 sees, for both temperatures, and d=0'75 at 15° and = 0'9 at 80° C, -logO-75 0-2877 hence k = -^^r = — — - = 0-24 at 15° C, -logO-9 0-1054 and k'= — -^-— = = 0-088 at 80° C, k' 0-088 1 •■•I = -0^ = 27 (- Table XIX.). section being successively attached to the same vibrator ; or the time taken to reduce an oscillation of given magnitude by a deiinite amount may be noted. Thus it was found that whilst it required 8 minutes to reduce the arc of oscillation of a steel wire from 20° to 1 5°, it required only 6 minutes with a copper wire of the same length and thickness to reduce the arc the same amount. By this means the striking effect produced by temperature on the viscosity of iron wire (p. 157) may be verified by the student. * For a discussion of this formula see Perry's admirable manual on Practical Mechanics, chap, xviii. The student should, on millimetre paper, make a series of curves showing the results of his experiments on free and damped vibrations. The manner of drawing these curves with figures is fully explained by Professor Perry, and will be referred to in the next volume, in the part on Sound. MOLECULAR PEOPEETIES OF FLUIDS 201 Exercises. 1. Find the relative viscosities of water at 10° C. and 40° C. by the foregoing method. 2. Find the relative viscosities of water and a solu- tion of zinc sulphate, density 1"2 at 15° C. (ii.) The viscosity of liquids may also be compared by suspending a vessel containing the liquid by a fine wire, with attached pointer moving over a graduated circle. The wire is turned through a given angle and released, so that a torsional vibration is set up, and the logarithmic decrement found as above, or the time taken to reduce the vibration through any definite angle is accurately determined, the same vessel being employed for the liquids compared. An unboUed and a hard-boiled egg thus compared form an instructive and striking experiment, the former coming to rest immediately. The reason for this is obvious. A boiled egg acts as a solid body, but an unboiled egg behaves like a vessel enclosing a very viscous liquid. The inertia of the liquid opposes the changing motion of the oscillating vessel, and the greater the viscosity the greater the frictional resistance. The amplitude of successive oscillations being reduced in a geometric ratio, the logarithmic decrement measures the relative viscosity. Employing a hollow cylinder with plane ends suspended axially by a fine wire, or by a bifilar 202 PRACTICAL PHYSICS suspension, accurate absolute measurements of viscosity have lately been made by 0. E. Meyer and others * and some interesting relations between the coefficient of viscosity of solutions and their chemical properties have been found. But as yet no general relation between the internal friction of liquids and their chemical constitution has been arrived at, though much labour has been spent on this subject. Rvercise. Compare in this way the viscosity of water and engine oil, also boiled and unboiled eggs as above. (iii.) Another method of exhibiting the difference in the viscosity of liquids is by filling a long glass tube, closed at one end, with the liquid under experiment, and note the time taken for a bubble of air, just large enough to touch the sides of the tube, to rise to the top of the tube. In this way various liquids may be compared with water. It is best to cork the tube, leaving a small air space over the liquid, then, stop-watch in hand, invert the tube. The ascent of bubbles in a tube of liquid presents many interesting points, such as a remarkable periodic fluctuation in the speed of ascent as the length of the bubble of air is" increased; these phenomena have been investigated and ingeniously explained by Dr. Trouton of Trinity College, Dublin. * See 0. ,E. Meyer and Miitzel ia Wiedemann's Annalen, vol. xliii., where the student will find a complete expression for the investigation of this subject. MOLECULAR PROPERTIES OF FLUIDS 203 Uxercises. 1. Compare, by the rate of ascent of a bubble of air in a tube of the liquid, the viscosity of water containing various proportions of glycerine, and plot your results on millimetre paper. 2. Keeping the solution constant, vary the size of the bubble and note the speed of ascent : plot your results in a curve. EyTperiment 82. — Determination of the relative viscosity of gases by their rate of transpiration. Instruments reqidrecl. — Similar to that in the succeed- ing experiment, the pinhole being replaced by a long capillary tube, and the pressure and temperature of the gas alike in each experiment. When a gas flows through a capillary tube the process is called transpiration, and, like the corresponding pheno- menon in liquids, the rate of flow is independent of the nature of the tube ; a film of gas adheres to the sides of the tube, and the gas flows in a stream along the axis, being impeded by its own internal friction or viscosity. The rate of flow is inversely proportional to' the length of the tube, and to a constant 77, called the coefficient of viscosity, which is peculiar to each gas. Graham has found the singular result that " any cause which promotes the density of a gas increases its velocity of transpiration, i.e. decreases its viscosity, whether the increased density be due to a lower temperature, to compression, or to the addition of an element in combination, — e.g. the viscosity of CO2 is less than O2." Hence whilst rise of temperature 204 PRACTICAL PHYSICS decreases the viscosity of liquids it increases that of gases. The rate of transpiration is very different from that of effusion (Appendix, § 16). As in the case of liquids the determination of the absolute coefficient of viscosity involves difficulties, which do not exist in a relative estimate ; this latter may easUy be made by an arrangement such as that described in the next experiment, a length of fine tlifirmometer tube furnishing the transpiration tube to be used in each experiment. When the same volume of different gases are transpired through the same tube under the same pressure, we have rj/rj' = t/t', where t and t' are the tran- spiration times. Example. — Find the relative viscosity from the tran- spiration times of hydrogen and oxygen. For H2 the time was found to be 51 sees., ;, '-'2 >' " ■■• -L O )) or calling O = 1, as 1 : 0-44 (see Table XX.). Uxercise. Find the relative viscosity of air, coal gas, and hydrogen. JVote. — The viscosity of gases may also be determined by the vibration of discs in the gas ; this method was employed by Maxwell.* The viscosity of gases compared with liquids is much greater than their relative densities would suggest. Thus whilst water is 770 times denser than air, its viscosity is * The student should read Maxwell's Heat, new ed., chap. xxi. MOLECULAR PROPERTIES OF FLUIDS 205 only 100 times greater. Barus* has measured the change of viscosity which occurs in passing from the gaseous to the liquid state, thus he finds r] for ether vapour at 0° C. = 6-8 X 10-^ for liquid ether at 30° C. = 9 x,10-*. He also finds tj for marine glue = 2 x 1 0^ for steel =6 x 10"; note the enormous range from hydrogen to steel. Experiment 83 — To determine the velocity of effu- sion of a gas. Instruments required. — As described below. When a gas flows through a small aperture (not more than 0'013 mm. diameter) in a thin plate, say of metal or glass, the process is called effusion, and the velocity of efflux obeys the same law as liquids, viz., V = Jigli. Air at 0° and 76 cm. pressure, rushing through such an aperture into a vacuum, will therefore move at the rate of ^2 x 981 x 790000, =39380 cms. per second, the value of h in this case being the height of a homo- geneous atmosphere, or 790,000 cms., nearly 5 miles. With different gases the velocities will be inversely pro- portional to the square root of their densities, e.g. at the same temperature and pressure the rate of effusion of oxygen to hydrogen will therefore be as 1:4; accord- ingly the densities of two gases are inversely as the squares of their velocities of effusion. By this means the relative densities of gases and vapours may be determined as follows : Into a tall vessel (A) containing mercury (Fig. 57) a glass tube (B) is plunged.t To the upper end of * Phil. Mag., April 1890. t A lamp chimney closed at the upper end with a piece of tinfoil, in which a fine hole has been pricked, and the whole immersed in a vessel of 206 PEACTICAL PHYSICS C-n- the tube is cemented a piece of platinum foil C, in which a fine hole has been pricked by a needle point. Closing this with the finger, the tube is filled with mercury, and the gas whose effusion rate is to be measured is allowed to displace the mercury. A bit of glass rod (F), drawn out with a long stem, is introduced, and floats on the mercury in the tube B. Ou the stem are two marks ; the upper mark is brought level with the mercury in the outer vessel by allowing a little of the enclosed gas to escape from the pin- hole ; now removing the finger from C the time is noted which elapses until the lower mark is level with the mercury. The tube B must be rigidly held in a clip during the experiment. This method can be used for testing the density of different specimens of coal gas, and may be varied in this case by allowing the gas to stream from a constant pres- sure gasholder through a small gas-meter connected with the tube B by flexible tubing. The quantity of gas that escapes in a given time being thus measured. Exam-ple. Find the relative effusion rate of hydrogen and oxygen (see Table XXI. and Appendix, § 16). Time for hydrogen =20 sees., „ „ oxygen = 77 „ water or brine, serves very well. The finest needle must be used to prick the hole, which may be made smaller by a gentle tap with a hammer, the foil lying on a smooth surface. If the hole is too large the motion of the gas is tumultuous, and the law of effusion is not obeyed. Fig. 67. MOLECULAR PROPERTIES OF FLUIDS 207 The ratio of the squares of these numbers should express . 20^ 1 the relative densities of the two eases, i.e. — - = —-—■ ^ 77^ 14-8 would be, in a more accurate experiment, 1/16. Exercise,. Determine the relative densities, from the rates of effusion, of hydrogen, coal gas, and CO2. Experiment 84. — To determine the law of the diffusion of g-ases. Instruments required. — A diffusion tube or bulb, closed at one end by a porous septum as described below, a balance, gramme weights, beaker of water and bell jars or small holders containing a few different gases. At temperatures above the absolute zero, the molecules of all bodies are in rapid motion; whilst in solids this motion is restrained within very narrow limits, in liquids and gases it is not so. Hence any two or more gases, or any miscible liquids put in contact rapidly intermingle. This molecular intermingling is due to the transference of individual particles from place to place throughout the mass, and is called diffusion. This process is far more rapid in gases than in liquids, owing to the greater molecular freedom in the former, but it is a slow process compared with the intermingling of masses of liquid or gas by convection, which is a molar and not a molecular motion. The rate of diffusion, as of effusion, of any gas is inversely proportional to the square root of its density.* * See Daniell's Physics, p. 233, and Maxwell's Seat, chap. xxii. 208 PEACTICAL PHYSICS To illustrate diffusion the following experiments may be made. (1) Connect two soda-water bottles with a long narrow glass tube and well-fitting corks, fill one of the bottles with hydrogen and the other with oxygen gas, and place the bottles (vertically connected together) aside, the Hj being uppermost. The next day test the gas in each bottle by a lighted taper ; an explosive mix- ture will be found in each, showing that the hydrogen has intermingled with the lower heavier gas. (2) Take a small cylindrical porous jar, such as is used for Daniell's batteries, and a wide-mouthed bottle half full of water ; close both by a cork, through which pass a glass tube, so that the porous pot is supported vertically over the bottle by the glass tubing, which may be a foot or so long. Fix a second piece of glass tubing in the cork of the bottle, so that one end dips below the surface of the water, the upper end being drawn off to a jet. Fill a bell jar by displacement with H^ or coal gas and hold it over the porous pot ; immediately diffusion begins, the H^ passing more rapidly in than the air out, hence a pressme is produced inside the porous pot in excess of the atmosphere; this causes the water in the bottle to be driven with considerable force up the second tube, so that a fountain plays from the jet. To verify the law of diffusion it is best to allow the gas to diffuse through a porous septum, such as plaster of Paris, compressed graphite, or better still, " biscuit ware," i.e. any fine kind of unglazed earthenware, like the porous pot in the last experiment. For this purpose a diffusion tube is made as follows. A glass tube, some 20 or 30 MOLECULAR PROPEKTIES OF FLUIDS 209 cms. long and some 2 cms. diameter, graduated like a burette (p. 47), is closed at one end by a disc cut from a broken porous pot, the disc being cemented in with sealing-wax a little below the end of the tube and a cork tightly fitted above it. The tube is now filled with the gas under trial as follows : pass one limb of a bent glass tube (like an inverted siphon) up the diffusion tube till it touches the disc, the air within can thus escape when the tube is lowered in a vessel of water, and in the same way the gas under trial can be made to enter and displace the water. Care must be taken not to wet the disc. When the tube is full of gas remove the cork and allow diffusion to take place ; with gases lighter than air water will rapidly enter the tube, and after half an hour the volume of the remaining gas is read off, the tube being depressed until the level of the water within and without is the same.* With gases heavier than air the tube should only be half filled, and the increased volume of gas read off in the same way. The temperature should remain constant, and a loose cone of damp paper should be placed over the disc, so that the humidity of the entering gas may be the same as that of the escaping. A more accurate experiment may be made with a diffusion lull (Fig. 58). The bulb B may be some 5 or 6 cms. diameter, having a neck above, say 3 cms. diameter, and for convenience the neck below may be narrower. As before, close the upper neck or opening by a disc of biscuit-ware A, cemented a little below the top, so that when filling the bulb it can be stopped by a rubber cork ; * This should be the case throughout the experiment, see next page. PART I P 210 PRACTICAL PHYSICS fiUthe whole with gas as before by means of a bent glass tube, and aUow diffusion to occur under constant pressure and temperature. The weight and capacity of the bulb being known by weighing it empty and full of water, the weight of water which enters or leaves the bulb, and hence the " diffusion volume," is found by closing the lower end with a cork and weighing the bulb and water in it. To preserve the hydrostatic pres- sure constant during dif- fusion suspend the bulb by a thread t from one arm of a balance, and having counter- poised it allow the lower end to dip in a beaker of water, as shown in Fig. 58. As water enters or leaves the bulb the movement of the balance preserves a constant water-level within and without. If Vg be the original volume of the gas, and Vj the volume remaining after diffusion {i.e. the volume of the return air), then the diffusion volume of the gas is Vj/Vj, and the density of the gas compared with air is o = (V /V y. Example. — To find the diffusion volume and density of hydrogen compared with air. Fig. 58. MOLECULAR PEOPEETIES OF FLUIDS 211 A bulb diffusion tube was filled with hydrogen in the manner described above. The weight of the bulb con- taining air =41-4 grammes; when full of water = 147'9 grammes. The capacity of the bulb was therefore 147'9 — 41-4 = 106-5 c.c. approximately. After one hour diffusion ceased, and the bulb was weighed again, when 7 8 '7 c.c. of water were found to have entered the bulb. Hence 106-5 - 78-7 = 27-8 c.c. of air had replaced 106-5 c.c. of hydrogen. . - . 106-5/27-8 = 3-83 = comparative rate of diffusion, or diffusion volume of Hg, and (27-8/106-5)^ = -076 = density of hydrogen. From Table XXI. the density of Hg compared with air is seen to be -0693 = , . , „ : the calculated diffusion 14-43 volume would therefore be 3-795, the square of which is 14-4.* (See Appendix, § 16.) Exercise. Find the rate of diffusion, and the density compared with air, of Hj, COj, and coal gas. Experiment 85. — On the diffusion of liquids. Instruments required. — Solutions, glass jars, and sp. gr. beads. Diffusion takes place between all gases and between those liquids which mix with each other. When, for * See Graham's Researches, collected and edited by Dr. Angus Smith, p. 64. To these famous investigations our present knowledge of the diffusion of gases and liquids is chiefly due. The number found (3-83) happens to be the same as the mean of Graham's observations. 212 PRACTICAL PHYSICS example, a layer of pure water rests over brine, the dense salt solution immediately begins to rise into the water and continues to diffuse until the salt is uniformly distributed throughout the whole mass of water. Differ- ent soluble bodies have different rates of diffusion. Their diffusibility may be compared, and the laws of liquid diffusion studied as follows : take a small wide-mouthed bottle or beaker with a ground top, fill this " diffusion jar " with the solution to be tested, cover it with a ground glass lid, and lower it to the bottom of a large beaker of distilled water. When the water has come to rest gently slide off the lid by means of a wire, note the time, and leave the arrangement undisturbed and in a uniform temperature, which note. After an hour carefully draw off by a pipette a measured amount, say 20 c.c. of the water from two different and measured distances above the top of the diffusion jar. Evaporate each sample to dryness in an evaporating basin and weigh the residue ; this corresponds to the amount of the substance diffused in an hour. Take other samples at other longer intervals of time from corresponding layers above the diffusion jar and test these ; and repeat, if possible, when the tempera- ture of the air differs considerably. In this way the quantity of the substance diffused wiU be found to depend on (1) the length of time ; (2) the strength of the solution in the diffusion jar; (3) the temperature — being greater with a high temperature— and (4) the coefficient of diffusibility peculiar to each substance. The diffusivity of one substance in another may be defined as " the number of units of the substance MOLECULAR PEOPERTIBS OF FLUIDS 213 which pass in unit of time through unit of surface, across which the gradient of concentration is unit of substance per unit of volume per unit of length." * Making the unit of time a day, instead of a second, to avoid incon- veniently small numbers, the following numbers have been deduced from Graham's experiments on diffusion. Temp. C. Gramme per day. Katio of Times. Hydrochloric acid 5° 1-74 1 Common salt . 5° 0-76 2'33 )) 10° 0-91 Sugar 9° 0-31 7 Albumen 13° 0-06 49 Caramel 10° 0-05 98 The last column gives the ratio of the times required for diffusion into water of equal amounts of the sub- stances named. The third column means that if the vessel of water were divided into horizontal layers 1 centim. apart, and each layer had 1 gramme per c.c. more of the substance named than the stratum above, the upward diffusion of the substance would be the number of grammes given through each sq. cm. pe7' day. The process of liquid diffusion may also be investi- gated by using small sp. gr. beads of differing densities and observing from day to day the position where each floats. The beads can be made with paraffin wax having a little tail of copper wire stuck in ; they require careful adjustment — first by nipping off bits of the wire, * Tait's Properties of Matter, p. 257. 214 PRACTICAL PHYSICS and finally by removing shreds of paraffin. As air- bubbles are apt to collect on them it is desirable that the liquids under diffusion should first be boiled^ and the whole covered with a layer of oil to exclude air ; a smart tap will, however, generally remove the air -bubbles without disturbing the liquids. Hydrometer jars or glass tubes closed at the lower end may in this case be used for diffusion. The jar or tube is first partly filled with water, and then the solution, if denser than water, is slowly poured down a funnel -tube reaching to the bottom of the jar ; the funnel- tube need not be removed, but if it is, care must be taken to close the upper end before withdrawing it. When highly coloured salts, such as bichromate of potash or sulphate of copper, are employed, the process of diffusion can be watched day by day, the upward progress of the dense layer forming a continuous gradation of tint, analogous to the fall of temperature in a metal bar heated at one end — the law of diffusion of heat being the same as that of the difi'usion of matter. It will be found that the process of diffusion is very slow, going on for weeks and even years in a long glass tube. The rapid intermixture of liquids produced by stirring arises from the enormously increased area of diffusion and the greatly thinner layers across which diffusion has to take place. Various optical methods of estimating the rate of diffusion have been tried, such as enclosing the liquid in a hollow prism and taking the refractive index at the vertically placed refracting angle. In coloured solutions the absorption of light by the liquid, placed in a MOLECULAR PROPERTIES OF FLUIDS 216 vessel with parallel glass sides, might be tried by the student. Exercise. ♦ Pind the rate of diffusion into water of saturated solutions of (1) common salt, (2) sulphate of zinc, (3) sulphate of copper, and also (4) of alcohol. Experiment 86. — On the osmose of liquids. Instruments required. — As described below. When a semi-permeable diaphragm, such as parch- ment paper or bladder or a porous earthenware partition, separates two miscible liquids, the process of free diffusion is modified. Thus if a bladder be partly filled with alcohol, or a piece of bladder tied over the end of a glass tube or lamp chimney and alcohol poured in, and the bladder then immersed in a vessel of water, the water enters and distends the bladder or raises the level of the liquid in the tube ; the reverse effect occurring if the liquids be interchanged. If, instead of bladder, a piece of thin caoutchouc or a toy rubber balloon be used, the alcohol and not the water passes through the rubber. The reason is that whilst water moistens or is soluble in bladder, alcohol hardens, and is not soluble in, bladder ; whilst the reverse is true of caoutchouc. The solution of the liquid in the septum creates a pressure — osmotic pressure — which drives the liquid into the solvent. The following experiment further illustrates this action : — Tie a piece of bladder over the mouth of a glass 216 PEACTICAL PHYSICS funnel ; to the shank attach say two feet of glass tubing by means of a rubber connection. Pour some thin treacle, or syrup, or a saturated solution of sulphate of copper into the funnel before attaching the tube; by means of a clip support the funnel within a beaker ; poitr water in the beaker till it rises above the mouth of the funnel, and mark the level of the treacle which should stand in the stem of the funnel. In the course of a few hours the liquid wiU be found to have risen in the tube attached to the funnel, and after a day or two may even reach the top of the tube an^ overflow. Simultaneou-sly some of the solution has diffused out into the beaker of water, though at a slower rate. The difference in the rate of diffusion between crystalloids and colloids * enables a separation to be made between a mixed solution of these bodies. This process is termed dialysis. A dialyser can be formed by floating a smaU toy tambourine, or suspending an inverted funnel with a piece of bladder or parchment paper tied over its mouth, in a beaker of pure water. Into the hoop or funnel pour a solution of gum arable mixed with bichromate of potash or common salt ; after a few days the salt, together with saline impurities in the gum, will have entirely diffused through leaving a pure solution of gum on the dialyser. The following results were obtained by Graham with 100 c.c. of different solutions, each containing 10 grammes of the substances named. * Gmhsim's Researches, p. 552; ox Phil. Trans., 1861. DifFusate in Grammes. Relative Diffusate. 7-50 100 3-30 44 2-10 27 1-61 21 1-39 18 0-03 0-4 MOLECULA.K PROPERTIES OF FLUIDS 217 Dialysis through parchment paper in 24: hours at 10° to 15°0. Ten per cent Solution. Chloride of sodium Glycerine Staroli sugar Cane sugar Milk sugar Gum arabic Exercise. Make two solutions, one containing sugar and the other an equal quantity of comnaon salt; after 24 hours compare their rates of diffusion through two dialysers, by evaporating in a water bath the liquid on each dialyser and weighing the residues. Note. — A difference of electromotive force on the two sides of the porous septum also causes osmose, the liquid passing in the direction of the electric current ; in this way with liquids of high electrical resistance, like water and alcohol, a rapid molecular transport is pro- duced. Surface Tension In the interior of a mass of liquid the molecular forces acting on any particle of the liquid balance one another, but the particles forming the external surface of the liquid will not remain in the same kind of equili- brium. These particles may be considered as forming 218 PRACTICAL PHYSICS " a thin coating of the liquid surrounding a substance which resists only in a direction perpendicular to its surface; . . . this coating must exert a force on the points in contact with it precisely similar to that of a flexible surface, which is everywhere stretched by an equal force; and from this simple principle we may derive all the effects which have been denominated capillary attraction."* Hence the surface of a liquid behaves as if under the influence of a contractile force, which tends to reduce it to the smallest possible area, and exerts a pressure on the interior which is greater when the surface is convex and less when it is concave than when it is plane. The surface of every detached portion of a liquid must, in fact, everywhere have such a curvature as to be able to withstand the hydrostatic pressure which acts against it. If the surface of water be supposed to be divided into two parts by an ideal boundary, each of the two surfaces tends to enlarge the other and to contract itself. The force exerted by each is at right angles to the boundary and equal to nearly 0'08 grammes per lineal centimetre. Over the free surface of the liquid this force is uniform, and its amount depends on the nature and temperature of the liquid. The magnitude of this force is however modified by the contact of the liquid surface with the surface of another liquid or with the surface of a solid. All capillary phenomena can be explained by assuming * Young's Lectures (delivered in 1807), p. 475. To the genius of the learned and famous Dr. Thomas Young we largely owe the development and application of the surface tension of liquids. For the modem treat- ment of the subject of capillarity the ai-ticle in the Ericy. Brit, should be read, or chap. xii. in Tait's Properties of Matter. MOLECULAR PEOPEETIES OF FLUIDS 219 the existence of this force and the ordinary hydrostatic laws. As capillarity occurs in vacuo a definite molecular pressure appears therefore to be exerted by the body of a liquid. This internal pressure, E, may be regarded as a measure of, and due to, the cohesive forces within a liquid, the range of which is limited to the " radius of molecular action." The value of E is probably enormous, Young estimated it at 20,000 atmospheres. In addition to the elastic skin which surface tension appears to confer upon the free surface of a liquid, the superficial film may possess a certain degree of toughness which renders it hard to displace or break ; this was supposed to arise from the fact that the surface of most liquids had a greater viscosity than the interior mass. The frothing of liquids and the formation of a soap bubble were attributed to this surface viscosity. Certain substances, such as albumen and saponine (infusion of horse-chestnuts) do appear to confer on water a kind of surface rigidity; the surface film having more the properties of a solid than a liquid. A bubble blown with saponine wrinkles up when the air within is with- drawn, behaving therefore quite differently from a soap bubble, the tension of which is the same in all directions of its surface. A sewing needle can easUy be floated on a solution of saponine, and if magnetised the superficial rigidity of the solution will prevent it obeying the directive force of the earth. The apparent " superficial viscosity " of water has recently been shown by Lord Eayleigh * to be due to the almost invariable contamination * Ptoc. Royal Society, March and June 1890 ; see Appendix, § 18. 220 PEACTICAL PHYSICS of its surface by a kind of greasy film, which can only be removed by special precautions (see p. 236, and Ap- pendix, § 17); when these are taken the water is then found to be devoid of " superficial viscosity." The addi- tion of soap to water greatly diminishes its surface tension and increases its " surface viscosity." Both these effects are doubtless due to the formation of a film or pellicle on the surface, which cannot be skimmed off, for it is renewed from the interior as fast as it is removed. But as the formation of this pellicle takes a certain, though very small interval of time, when soapy water is tested before its surface is -j-J^ of a second old, it will be found to have the same tension as pure water. In the feebler tension of this outer renewable pellicle is to be found the probable explanation of why soap and some other sub- stances confer on water the property of enabling a fairly durable bubble or stable extended film to be formed.* Liquid veins also afford beautiful illustrations of superficial tension ; their constitution and sensitiveness to sound and electricity will be studied experimentally in the next volume. Experiment 87. — On the direct measurement of surface tension. Instruments required. — A balance and weights, wire bent as described, and a beaker of water and soap solution. (i.) The following experiment directly illustrates the existence of surface tension and affords a measurement of * For a further explanation see Maxwell's Heat, new ed., p. 298, and Appendix, § 18. MOLECULAR PROPERTIES OF FLUIDS 221 its amount, well adapted for soap films ; other methods of determination will be subsequently described under the capillary constants of liquids. Bend a piece of wire into a rectangular fork, thus, the width being from 2 to 3 cms. and the length, say, 5 or 6 cms. Suspend it by a thread (attached to the cross piece) from one arm of a balance, allowing the legs to dip into a beaker of pure clean water. Counterpoise the wire so that the cross piece is about 2 cms. above the surface of the water when the balance is in equilibrium. Next, depress the balance beam so that the wire dips below the water surface — the counterpoise will now be found unable to restore the equilibrium ; add weights gently. A film of water will be drawn up in the rectangle of the fork until equilibrium is restored. Find the greatest stretching weight the film will bear without breaking ; this can be done, after one or two trials, by altering the level of the water in the beaker and readjusting the counterpoise. Note that as the film thins the stretching force required is just the same (allowing for the very slight loss of weight it sustains by evaporation, etc.) till the breaking point is reached. A thin sheet of india-rubber would, of course, behave differently, requiring less force to stretch it than a thicker sheet ; and the tension of such a sheet would grow less as it shrinks, whereas the tension of the liquid' film remains the same and has therefore a certain definite value. Next, try in the same way a soap solution (Appendix, § 20). A much larger and tougher film can be obtained ; 222 PEACTICAL PHYSICS but though the film is bigger, a less stretching force is required, that is, the contractile force of the film pulling the fork back into the solution is less. As the film has two surfaces the tension per cm. is double that of the free surface of the liquid in the beaker. The stretching force F is distributed over the breadth AB of the film ; hence if T be the surface tension fer unit of length F = 2T X AB, .". 2T = F/AB, where F is the stretching force in dynes. Example. — Find the surface tension of pure water and of soapy water. A fork 3 cms. broad was suspended in a beaker of pure water and counterpoised. At this temperature, 19° C, the maximum pull of the water film was 0'37 gramme. The same fork being used in the glycerine- soap solution the maximum pull of the film was 0'18 gramme. (The weight of the film itself, which was 5 times the area of the water film, should be deducted.) Hence for water at 19° C. ^ 0-37x981 ^„ ^ , T = ;- — = 60'4 dynes per lineal cm., ^ X o and for soap solution, neglecting the weight of the film, 018x981 „„ , ^ T' = — - — ^ — = 29-4 dynes per lineal cm. ^ X o The value of T for water is here rather low, doubtless owing to some impurity on the surface ; but it will never- theless be seen that the soap has reduced the surface tension of the liquid one-half. MOLECULAE PEOPEETIES OF FLUIDS 223 Exercise. Eepeat the above experiment. (ii.) Another method of measuring the surface tension of a soap film is as follows : * Make two wire rings, one about 5 cms. diameter, the other a little larger ; support both by three wire legs, the legs of the larger ring about 5 cms. long and the smaller about 2 cms. long ; fix to the legs of the smaller tripod, and parallel to the ring, a paper disc or tray. Now wet the rings with the soap solution and form a film on the larger ring ; raise the lower ring till it is in contact with the film and concentric with the larger ring ; let it go, and break the film within the smaller ring ; absorb by filter paper any drops of liquid, and gently load the paper tray with sand. The annular film is drawn down into the shape of a catenoid. Continue loading with sand, preserving the lower ring horizontal, till the tangent to the base of the curved film is little removed from the vertical. When it is practically vertical, the surface tension pulling the tray up acts wholly vertically and is equal to the weight w (which is then at its maximum value), pulling the film vertically down ; hence Airr T being in dynes per lineal cm. and acting along the circumference of the ring whose mean radius is r in cms. {i.e. one-half the sum of the radii of the external and * Van der Mensbrugghe in Phil. Mag., April 1867, p. 280. 224 PRACTICAL PHYSICS internal rings). When the legs of the lower ring rest on the table, break the film and weigh the whole load. Example. — Find the surface tension of a soap film by Van der Mensbrugghe's method. The mean of five experiments gave a load of I'Ol gramme, 27rr=16'75. ■ ■ -^- 16-75 ~^^^' and T= 29'6 for a glycerine-soap solution. Experiment 88. — Various illustrations of surface tension. Instruments required. — As described below. The following additional experiments, which should be repeated by the student, afi'ord instructive illustrations of the action of surface tension. (1) Make a wire ring some 5 cms. in diameter, leaving a straight piece of wire for a handle ; dip the ring in a soap solution, and by removing it obtain a film, covering the ring. Throw a thread wetted in the soap solution across the film ; it will move about freely, being drawn equally on all sides ; now break the film on one side of the thread, — instantly the thread is pulled across the ring by the contractile force of the remaining portion of the film (to make the soap solution, see p. 261). (2) Tie the ends of a short length of thread, place the loop thus formed, wetted, on the film, and break the film within the loop; the uniform tension of the film outside instantly pulls the loop into a perfect circle. MOLECULAE PROPERTIES OF FLUIDS 225 (3) Blow a soap bubble from the end of a wide glass tube, close the free end of the tube with the finger ; on removing the finger the bubble contracts and can be made to blow out a candle. (4) Blow a bubble at the end of a glass T-piece, to the other horizontal limb of which is attached a small U-tube containing water, to indicate the pressure ; note that the greatest pressure (about J inch) is when the bubble is small — owing to the greater curvature of a small bubble. From this pressure and the size of the bubble the surface tension of the soap film can be deduced. (5) Make a solution of zinc sulphate of the same density as bisulphide of carbon, and with a pipette allow a drop of bisulphide of carbon, coloured by iodine, to enter the solution; note the spherical form of the drop. A liquid sphere of olive oil of considerable size may be similarly obtained in a mixture of alcohol and water. The drop being freed from the influence of gravitation the spherical shape is the result of the action of the molecular forces alone. By increasing the density of the solution, a large drop may be floated on the surface ; note its flattened form, as gravitation now plays a part. (6) Spread some coloured water on a plate, drop into it a little alcohol, or hold a red-hot rod, or a rod dipped in ether, over the water ; immediately a bare spot is left beneath the rod. The stronger surface tension of the pure water draws the liquid away on all sides from that portion which has had the contractile force of the film weakened by the solution of alcohol or ether, or by a rise of tem- perature. PART I Q 226 PRACTICAL PHYSICS (7) On the clean surface of pure water contained in a wide beaker, place a drop of creosote ; the globule floats, and a partial solution in the water begins. Instantly the surface tension is lowered and the globule is pulled to pieces and drawn rapidly about by the higher tension of the water around ; hence also ensues a rapid vibratory and singularly life-like motion of the drop, particles being shot out in radial lines like the scattering of spores. Touch the water with a rod dipped in turpentine ; immediately the whole is struck lifeless, as a turpentine film of much lower surface tension than water flashes over the surface. Each essential oil has its own " cohesion figure." These have been investigated and drawn by Mr. Charles Tomlinson,* and afford a method of discriminating and testing the purity of essential oils. (8) Fragments of camphor floating on clean water exhibit similar lively motions, owing to the lowering of the tension of the water surface at those points where -the camphor dissolves most freely. A trace of oil or turpentine on the water stops the movement. The im- portance and difficulty of obtaining a water surface free from contamination is referred to in the Note on p. 236.t Before making more exact measurements of surface tension it is necessary to investigate the law of the ascent or depression of liquids in capillary tubes. * Phil. Mag., June 1867. t Other striking and beautiful illustrations of surface tension are con- tained in Professor Boys' charming book on Soap Bubbles, and in a paper by Lord Kayleigh in the Phil. Mag., April 1892. MOLECULAR PROPERTIES OF FLUIDS 227 Experiment 89. — To prove that the height of ascent of water in capillary tubes is inversely as the radius. Instruments required. — A vessel of water, a millimetre scale or cathetometer, and several capillary tubes. (i.) Take three or four capillary tubes of different sizes and clean them internally by dipping them into strong sulphuric acid several times, wash with distilled water, then alcohol, and finally blow air through them, heating them at the same time in a Bunsen flame. It is necessary that the tubes should be of uniform bore and circular cross-section. Ordinary thermometer tubes answer very well, but the student had better pre- pare his own tubes by softening a piece of quill- glass tubing in a blow-pipe flame ; then remove the tubing from the flame and draw it out quickly and steadily to a long length. Selecting about six inches of the central part of the capillary, test it for uniformity of bore, and measure its diameter as described in Experiment 17. Place the tubes vertically in the vessel of water (Fig. 59); raise the liquid in the tubes to a height above that which it will ultimately take (by first lowering the tubes and then raising them) ; this ensures that the inside of the tubes are thoroughly wet. Now measure by means of the millimetre scale or the cathetometer the heights of the liquid in the tubes above Fig. 60. 228 PRACTICAL PHYSICS that in the vessel, the latter can be given hy a fine point P, screwed down to the level of the liquid L : it wUl be found that the heights are inversely as the radii of the tubes. Then the height h, multiplied by the radius r, should be a constant* (see Appendix, § 19). Example. — Prove the law of ascent, using three capillary tubes. Enter results thus : — Number of Tube. h Cms. r Cms. hr. 1 2 3 4-2 3-1 2-7 •0317 •0435 •0489 •1332 •1349 •1320 The water used in the foregoing experiment was slightly coloured. When very clean water is used, and a perfectly clean vessel and tube, the value of hr is higher and will be found to be nearly 0*1 5 — a constant higher than that given by any other liquid. The ex- periment should be repeated in water at different temperatures; the capillary elevation will be found to diminish as the temperature rises ; the surface tension of water, on which capillarity depends, diminishing rather more than -g^ of its value for each degree centigrade above 0°. Without special precautions it * For exact measurement it is necessary to take into account the height of the liquid meniscus in the tube. By taking the reading a little above the-bottom of the meniscus (say one-third of the radius of the tube), this error vanishes compared with the greater errors due to want of perfect cleanliness of tube and liquid surface. MOLECULAR PEOPEKTIES OF FLUIDS 229 to keep the liquid in the tubes at above that of the air ; but by using tubes and allowing them to attain a somewhat higher than the water used, experiments may be made if quickly is difficult temperatures thick glass temperature comparative performed. (ii.) Instead of using tubes, the law of elevation (some- times called Jurin's law) may be determined by using glass plates. When two parallel glass plates, a small distance apart, are immersed in water, the liquid rises to a height which is inversely proportional to their distance asunder, hence the height of ascent is one half what it would be for a tube whose diameter corresponds to the distance between the plates.* The best way of keeping the plates a measured distance apart is to cut two short lengths of wire, say 1 mm. diameter, place the wire be- tween the plates near the edges which are vertical when in the liquid, and bind the whole together by a couple of elastic bands. Wires of other thick- nesses enable the distance to be varied. Glass plates inclined at a small angle may also be used as follows : — Take a shallow vessel of coloured water and place two plates of glass vertically in the vessel, as in Fig. 60. The two plates touch along one edge as at A, and are a little apart at the opposite edges B. When they are placed in position the water will rise * Deschanel's Physics, edited by Everett, p. 128. Fig. 60. 230 PRACTICAL PHYSICS between them in the form of an equilateral hyperbola. Now measure the ordinates of a few points of the curve, and show that their lengths are inversely proportional to the distances between the plates at these points, that is the same law as in parallel plates. In liquids which do not wet glass, like mercury, a corresponding depression of the liquid is produced by capillary tubes and plates of glass. Jurin's law will here be found to be only approximately true owing to the disturbing influence of the unwetted walls of the tube. The capillary depression of mercury in glass tubes is given in Table X. Uxercise. Eepeat the above experiments with capillary tubes and parallel and inclined plates of glass. Experiment 90. — Determination of the capillary constants of liquids. Instruments required. — As in the last experiment, with specimens of different liquids. The most convenient and accurate way of determining the surface tension of a liquid |which wets glass, is by measuring the height to which the liquid rises in a perfectly clean and vertical capillary tube, the temperature being noted. Quill tubing should be cleaned as described in the last experiment (p. 227), and the cleaned tube used by the student to make his own capillary tubes — a number of which may be prepared and the ends sealed tiU they are required. After the rise of the liquid in the tube has been determined, — and for this purpose the reading MOLECULAR PEOPEKTIES OF FLUIDS 231 microscope is most suitable for the wider tubes, though its range is not long enough for the finer tubes, when a cathetometer or millimetre scale to which the tubes may be attached by an elastic band must be used, — the portion of the tube up which the liquid has risen should be broken off and its exact diameter found. Measuring the bore at the upper end by the reading microscope is the quickest way, but the student should satisfy himself that this result is reliable by also measuring one tube (which must be dry and clean) with a thread of mercury (p. 45). The force which holds up the liquid is the vertical component of the surface tension T, of the liquid, acting along the upper rim of the liquid in the tube ; if a be the angle of contact (see next Experiment), this force is therefore 27rrT cos a. The weight of the cylindrical thread of liquid sustained in the tube, equivalent to this force, is irrVipg, where h is the mean height of the liquid in the tube in cms. and p its density ; hence , 2Tcosa rpg In those liquids which wet the tube, cos a=l, we have simply 2T = rhpg ; T being in dynes per lineal cm. The value rhp, at any given temperature, is thus a constant for each liquid and is termed its capillary constant. Example. — Find the capillary constants at 20° C. of water, soap solution, amylic alcohol (fusel oil), and ether. 232 PEACTICAL PHYSICS Enter results thus : — r and h in cms. and T in dynes per cm., see Tables XXII. and XXIII. Name of Substance- P- r. h. rhf>. T. Water . 1-00 •033 4-55 •149 731 Soap solution 1-01 •038 r32 •051 25 •O Amylic alooliol . 0-82 ■035 1-60 •046 22 •& Ether . 0-73 •033 1-5 ■036 17^6 Exercise. rind the capillary constants and surface tension of water, soap solution, turpentine, and alcohol. The size of drops and bubbles also affords another method of measuring the capillary constant of liquids. A convenient method * is to measure the radius of curvature r of a drop hanging in stable equilibrium from a tube, say a millimetre in diameter and under a small definite hydrostatic pressure h, then 2T = rh. The relative surface tension of liquids may be readily found by comparing the weight of the drops of different liquids issuing under similar conditions from the same orifice, or by counting the number of drops delivered from a pipette, as in the following example. Take a 5 c.c. pipette and bend the delivery tube at right angles. Filled with water at 15° C. it delivered 100 drops; with 1 per cent alcohol mixed with the water it delivered 107 * See Lord Kelvin's Lectures on the Constihdion of Matter, vol. i. p. 45. MOLECULAR PROPERTIES OF FLUIDS 233 drops; with 2 per cent alcohol 113 drops; with 5 per cent 127 ; with 10 per cent 145 drops ; with 50 per cent 242 drops ; the temperature being the same throughout.* Experiment 91. — To measure the angle of con- tact of mercury and glass. Instniments required. — Clean mercury, level glass plate, spherometer, and reading microscope. The immediate cause of the elevation or depression of a liquid in capillary tubes arises from the curvature of the surface due to the surface tension of the liquid in contact with the solid. With water and liquids which wet the tube, the liquid surface inside the tube is always concave outwards ; with mercury which does not wet the tube, it is convex, the curvature and therefore the eleva- tion or depression being greater the finer the bore. Owing to surface tension the film tends to reduce the surface to the smallest area within the boundary, that is to a level surface ; and so may be considered as pulling the liquid outwards when the surface is concave, and pressing it inwards when the surface is convex, with a force per unit area which is proportional to the tension and the curvature (or sum of the principal curvatures) of the film. Hence immediately under the concave surface the pressure is less, and under the con- vex surface greater than the atmosphere by the amount of this force per unit area.t * See Jaminet Bouty, CoKre rf«PAi/si2't<«, vol. i. part ii. p. 65. Obviously this method might be used for commercially testing the alcoholic strength of spirits. + For a fuller knowledge of this subject the student should especially 234 PRACTICAL PHYSICS If a tangent be drawn to the curved liquid surface where it meets this side of the tube, the angle enclosed between this tangent and the side of the tube is termed the angle of contact. This angle has a definite value for each liquid in contact with a particular solid : for pure mercury and clean glass the angle of contact a is about 130°,* if we reckon a as the angle formed by the wedge of liquid in the tube ; or the supplement of this, say 40° to 50° (according to different observers), if we measure the external angle, i.e. from the tangent to the side of the tube above the liquid. With pure water and clean glass the angle vanishes or becomes equal to 180°. The value of this angle in the case of mercury may be derived from two measurements — first of the depth of a large drop of mercury on a clean glass plate; then of the capillary depression produced by a clean strip of glass in mercury. Proceed as follows : support a clean plate of glass, say 10 cms. diameter, on a small table provided with levelling screws ; after levelling carefully pour on the glass plate some pure clean mercury till a disc of at least 5 or 6 cms. diameter is obtained (the glass plate should rest within a small dish to catch any mercury). Next, by means of the spherometer, find the depth of the disc of mercury ; by noting the reflection of the point, or the dimple formed in the liquid surface on screwing read Professor Tait's lucid exposition in Properties of Matter, p. 234, et seq. ; and Maxwell's Heat, chap. xx. * According to Young the angle of contact of mercury in a barometer tube is 140°. Young determined the angle by the reflection of light from the convex surface of the mercury inside the tube (see Young's Works, vol. 1., Essays 19 and 20). MOLECULAR PROPERTIES OF FLUIDS 235 down the central point, a very accurate determination can be made. Next pour clean mercury into a small clean glass trough with parallel sides, and hold a narrow strip of clean glass vertically in the trough by means of a clip. With the reading microscope measure the depth of the depression ; a careful adjustment of light will be necessary to avoid confusing reflections from the mercury surface. The reading microscope may also be used, instead of the spherometer, to measure the depth of the drop. The angle of contact can then be calculated from the following equations,* where T = surface tension of mercury in dynes per lineal cm., a = external angle of contact, = capillary depression in cms., e = thickness of a large drop of mercury in cms., „ 2T(l-sina) 0^ = , P9 2_2T(l + cosa) Pff whence, tan -^ = 1 - V2-, and T = o^pg 2(1 -sin a) Examjple. — Find the surface tension and angle of contact of mercury with glass. "* Jamin et Bouty, Cows de Physique, vol. i. part ii. p. 40. 236 PEACTICAL PHYSICS By measurement of a very large flattened drop of mercury, 0=015 cm., and c=0-34 cm., whence as above a=41°15' and T = 439-6 dynes per lineal cm. Uxercise. Eepeat the above exp^iment and calculation. Note. — The high surface tension of mercury and water renders it extremely difficult to obtain, or to keep, a clean surface to these liquids ; in the last experiment the mercury was not quite clean. Dipping the finger into water instantly contaminates the surface with a greasy film and lowers the surface tension. Lord Eayleigh has shown that distilled water poured from a stock bottle gathers impurities from the side of the bottle, and has a dirtier surface than water drawn from a tap. The easiest way to obtain a fairly clean surface of water or mercury is to draw off the liquids through a tube that dips below their surface, rejecting the first portions ; a wash bottle well cleansed answers the purpose. More perfect methods of obtaining a clean water surface are described in Appendix, § 17. The activity of fragments of camphor affords a test of a moderately clean water surface, and the absence of " surface viscosity " in water is a test of perfect cleanliness. APPENDIX § 1. Theory of the vernier. If n divisions on the vernier be equal to n + 1 or ?i - 1 divisions on the scale ; and if S be the length of a scale division and V the length of a vernier division, then (7i + l)S = nV, (a). -jj. 71 + 1 „ n n n , S = «(V-S) Also («-l)S = mV, n n ' 7i(S - V) = S - S is the least count of the vernier, n m- § 2. Theory of the spheFometep. In Fig. 61 A, B, C are the positions of the three fixed feet of the spherometer, and that of the movable foot, when they are all in the same plane. A3 = l, and angle ABC = 60°, . •. AD = Z sin 60° = I—, 2 ' 2 2 *^3 Z and A0 = ^AD = ^xl~2' - ^5' 238 PRACTICAL PHYSICS In Fig. 62 FK X KE = KH', but FK = FE - EK = 2R - a, and AO in Fig. 61 is the same as KH in Fig. 62 ; p .-. (2R-a)a = AO'=3' and Ga 2 Fig. 62. APPENDIX 239 § 3. Reading mieroseope and telescope. The cathetometer is a costly and often defective piece of physical apparatus ; unless well made it is seldom reliable to the "least count" of its readings, which may therefore give a misleading appearance of accuracy. A simpler and more trustworthy method of measuring small differences of length is to employ a microscope, in the eye-piece of which is a trans- parent micro-photograph of a well-divided scale. The value of each division can be ascertained by focusing on a milli- metre rule ; 10 or 20 divisions of the micro-photograph to a millimetre can easily be seen, and reliance can therefore be placed on readings to the -^V of a mm. This method has the advantage of enabling readings to be taken more quickly than with the cathetometer, is a much less costly arrangement, and more reliable. Moreover, by merely turning the eye-piece, horizontal as well as vertical displacements may be read. Obviously, however, its use is limited to observations where the total space to be measured is small. Quincke's catlie- tonuter microscope is an instrument of this kind ; it is mounted on a small glass table with levelling screws. To enable longer vertical spaces to be measured a reading telescope may be employed. This is simply an ordinary tele- scope capable of being focused upon objects within 10 or 15 feet distant, and having a cross wire in the focus of the eye-piece. The telescope slides on a vertical rod, to which it can be clamped at any height. A millimetre scale is placed beside the object to be measured, the cross wire brought to coincide with the upper mark, the corresponding scale reading taken, and then the telescope is slid down to the lower mark and the scale reading again taken ; the difference of the two readings gives the space between the marks. Better results will often be given with this method than by a cathetometer, as small differences in the level of the telescope do not destroy "the accuracy of the result, as with the cathetometer. 240 PRACTICAL PHYSICS Theyeading microscope is one of the most accurate and con- venient instruments for measuring small quantities, such as the linear coefficient of expansion of a rod by heat, the Ttnijap^ Fig. 63. diameter of a wire or the bore of a capillary tube, if of circular section. Two reading microscopes sliding on a firm bed, to which they can be clamped (Fig. 63), form an accurate optical beam compass, and are thus used for deter- mining the linear coefficient of expansion, etc. This arrange- ment will be more fully described in the next volume. In the eye-piece of the microscope is a single fine hair or wire, stretched on a frame which can be moved across the field of view by a fine screw attached to a divided circle A. Fixed in the eye-piece is a scale (shown at C in Fig. 64), the distance between each of its divisions being equal to the pitch of the screw, hence a complete rotation of the circle A (usually divided into 100 parts) moves the hair line H (Fig. 64) through one division of the fixed scale. To find the value of one division the instrument is focused on a millimetre scale, which is Pig. 64. APPENDIX 241 shown magnified in the field of view, at S (Fig. 63), where five divisions of the fixed scale C are seen to coincide with 1 milli- metre, hence each division = 0'2 mm., and as the divided circle enables -j^ of this space to be read, the instrument can measure to 0'002 of a millimetre. § 4. The so-called hopizontal or micrometer-pendulum. This instrument was first devised, in 1832, by Hengeller {Phil. Mag. vol. xlvi. p. 416), then independently by the Rev. M. H. Close, of Dublin, and a little later by ZoUner. The instrument here de- scribed is the form devised by Mr. Close, the mode of sus- pension being different from and preferable to Hengeller's or Zollner's. It consists essentially of a rod AB sup- ported horizontally for con- venience by two silk threads AC and BD, as shown in Fig. 65, so that the rod oscillates about the axis CD. The threads should be as fine as possible, so that their stiffness should interfere as little as may be with the behaviour of the pendulum. In order that the rod AB may be in stable equilibrium the axis CD is inclined slightly towards A. If its inclination 6 to the vertical be very small, and if we could neglect the stiffness of the fibres at C and D, then an exceedingly small tilt fi of the axis CD at right angles to the plane of rest of AB would produce an angular movement of the pendulum equal to ;8/sin d. Thus if 6 were 1', the magnifying power of the pendulum would be 1/sin 1' or 3438. The stiffness of the PAKT I R Fig. 65. 242 PRACTICAL PHYSICS threads makes this less — how much can only be determined by direct experiment. Obviously the best way of using the pendulum is to attach a mirror to the rod, and view either a reflected scale or a reflected spot of light. In this way the pendulum rod may be kept as short as desirable, and the sensitiveness enormously increased. The instrument is mounted on a firm base with two fine graduated levelling screws : one to regulate the inclination of CD towards A, and thus to modify the sensitiveness of the instrument as desired ; the other to tilt the axis in a direction at right angles to the vertical plane of rest of the pendulum, in order to bring the pendulum, when necessary, to its zero point, and also to measure its delicacy by observing what angular movement would be produced in it by a given trans- verse tilt. The micrometer -pendulum can be easily made by the student, and so delicate is the instrument that it will be found necessary to place it on a support free from vibration, and to enclose the whole in a box, with glass front, lined with tinfoil joined to earth, to prevent any electrical or air disturbance. The pendulum rod also should not be made of any magnetic metal ; a stout brass or platinum wire answers very well. Mr. Close has found that in a moderate gale, the micrometer- pendulum, placed on a rigid support in the basement of a house, shows that at every gust of wind the whole ground floor of the house tilts over to leeward through an angle con- siderably greater than can be measured by the micrometer- pendulum when it is finely adjusted. APPENDIX 243 § 5. Theory of the balance in the ease when the points of suspension of the beam and the points of support for the scale pans are all in the one straight line. In Fig. 66 let W = weight of the balance beam, I = CB the half length of the beam, I' = CGr the distance from the point of suspension to the centre of gravity of the beam, P = weight in one pan, T +p = weight in other pan, 6 = the angle which the beam is turned through by the weight p. Fig. 66. Now taking moments round C we get (P + p)B'M. = W X G'R + P X A'N, (P +p)l cos e = Wr sin 9+Fl cos 8, pi cos 6 = wr sin 0, tane = =^„ which expresses the sensibility of the balance. 244 PRACTICAL PHYSICS § 6. Speeifle gravity — Hydrometers. (1) For the density of an insoluble body lighter than water let W = weight of the solid in air, W = „ sinker „ W, = „ „ in water, Ws = „ solid and sinker together in water, then W - W, = weight of water displaced by the sinker alone, and (W + W) - W^ = weight of water displaced by both sinker and body ; hence (W + W-W,) - (W'-W,) = W + W,-W, is the weight of water displaced by the body alone ; W ■ ■ '' ~ w + w, - w; (2) All hydrometers are subject to a slight error, arising from the surface tension of the liquid in which they float ; this causes a film of the liquid to be drawn up the stem, and the hydrometer floats at a higher position than it otherwise would do ; this would not matter if all liquids had the same surface tension, but this is not the case. The error due to this cause may be reduced by making the stem of the hydro- meter very thin, as in Nicholson's hydrometer, and if, before use, the stem be wiped with a clean dry cloth or with a cloth damped with alcohol, the error is still further reduced. Mr, Joly, F.RS. (Proc. Royal Dublin Soc, 1886) has devised a form of Nicholson's hydrometer which is practically free from this error, and at the same time forms a delicate balance for ordinary purposes. Joly's hydrometer is a suspended metal globe, made in two pieces, having a narrow tubulure below ; the globe is fiUed with water, and within it floats a smaller empty metal or glass globe from which depends a fine wire passing through the tubulure and carrying a pan below. The instru- ment is therefore an inverted Nicholson, with the weights APPENDIX 245 acting in tension and not in compression on the fine wire stem. Owing to the narrowness of the tubulure the water is unable to escape from the globe. The outer globe if of metal may be made in two hemispheres, fastened together after the inner globe has been inserted ; or if of glass the inner globe can be blown inside it and a brass cap and narrow tube for the wire cemented on the neck. (3) Mr. Joly has also devised a method for determining the speciiic gravity of minute specimens,* which has many advan- tages : it is simple, accurate, expeditious, and can be applied to porous bodies. The method consists in embedding the specimen in a little disc of paraffin wax, cut from a paraffin candle, about 1'5 mm. thick and 3 or 4 mms. diameter, the edges smoothed and the size of the disc adapted to the quantity of the solid to be tested. The disc is carefully weighed, and the fragment or fragments to be tested placed on the paraffin and bedded in it by holding a hot copper wire over the specimen ; the compound body or pellet is again weighed, the increase in weight gives the weight of the specimen. A saturated solution of common salt and water having been prepared, the pellet is dropped in and water added till the pellet is just balanced in the liquid ; the final adjustment and stirring of the liquid is best accomplished by a camel's-hair brush. By a Sprengel's tube or hydrometer, the specific gravity of the solution is then found which gives the specific gravity of the pellet of wax enclosing the specimen ; from this the specific gravity of the specimen can be found, the specific gravity of paraSin wax having been previously determined. If W = weight of the specimen in air, w= „ ,, paraffin used, cr = specific gravity of the paraffin used, s= ,, „ mixed substance, * Proa. Royal Dublin Soc, January 1886. 246 PRACTICAL PHYSICS , , . w then the specific gravity of the specimen p = ^ — - s cr Was or p = pf^ 1 • '^ ( W + W)cr - WS Example. — A specimen of magnetite from the Krakatoa ash bedded in paraflSn. W = 0-0202 grm., w = 0-0654 grm., 0- = 0-9206 „ s= 1-1361 „ .-. p = 4-71. One great advantage of this method is the complete extrusion of the entangled air by the melted paraffin soaking into the specimen, and, moreover, a specimen of any density above unity can be determined, the method described on p. 72 being limited to bodies below 3-4. § 7. Corpeetion of the height of the barometric column for temperature. Let Vt and Vo = volume of the mercury at t° C. and 0° C. respectively, ht and hg = height of mercury column at t° C. and 0° C. respectively, S = coefficient of cubical expansion of mercury, a = coefficient of linear expansion of brass, H = true corrected height of mercury column, then K = hJ^, and V^ = V„(l + Si!). ■ ^" ^ ,= \-U + ^e, etc., Yt 1 + Si ■ and by neglecting S^f and following factors, we have li„ = ht{\-?,t). APPENDIX 247 In the same way the length of the column will be apparently increased by the expansion of the scale in the ratio oi\ + at to 1, . •. H = KO- + oi) = K'^ + <«!)(1 - ^ •=>,{! -(S-a)^}, and since 8 = 0-000182, a = 0-000020, .-. H = Ae(l-0"0001620. § 8. Joly's mereury-glyeerine barometer. Various devices have been made to increase the sensitive- ness of a mercurial barometer, but what is gained in an open scale or long range is usually lost by increased friction and irregularity of action. The common weather glass or wheel barometer is an illustration of this. A simpler plan is to bend the upper part of a long barometer tube to an obtuse angle ; a vertical rise or fall of an inch will thus cause the mercury to move over a foot or more of the inclined tube, but the same objection applies. Using a liquid of less density than mercury enables a proportionally longer range to be obtained ; thus in a water barometer the range is obviously 13| times that of a mercurial barometer, but the tube has to be some 35 feet long, and the considerable vapour pressure of water, variable with temperature, destroys its usefulness. Glycerine is a far better liquid to use, owing to its non- volatility, but the difficulty of construction and inconvenient length of the tube is a drawback. Mr. Joly's device enables a glycerine barometer to be made of moderate length.* As the arrangement may be useful for laboratory purposes the following details are given. A plug or disc of ebonite or ivory, having a diameter a little less than the barometer tube, is made with a steel pin projecting from it, a small sphere or cylinder of wood or ivory called the float • Proc. Royal Dublin Soc, 1892. 248 PRACTICAL PHYSICS being fastened to the end of the pin, as shown at B, Fig. 67. Placed at the bottom of a column of mercury, the float tends to rise with a force equal to the difference in weight of the float and the mercury it displaces. If, however, the float be made of the right bulk, it will be unable to pull the plug after it through the mercury. The lower surface of the mercury being covered by the disc or plug remains un- broken by the weight of the mercury column above, and through the very narrow annular space around the plug the mercury does not pass. A column of mercury may thus be sus- tained in a wide tube closed at the upper end only ; moreover, the elasticity of the air permits pj ^j the column to be oscillated with considerable violence without any mercury escaping or air creeping up through it. The space below the mercury may be filled with glycerine, and the tube may dip into an open vessel of glycerine ; the mercury will thus appear to float on the glycerine, and the latter will enter or leave the tube as the mercury rises or falls. If the space above the mercury be a vacuum and the mercury column say 27 inches long, the glycerine column below the tube must be of such a length that the joint pressure of the mercury and glycerine shall be equal to the atmospheric pressure. If now the atmospheric pressure rise or fall, as much glycerine must enter or leave the tube as if it were a glycerine barometer of corresponding bore. Hence the range of the instrument is the same as that of a glycerine barometer, though its length need not be more than a few feet, whilst its sensitiveness and promptness is probably greater, as there is a much shorter length of the viscous glycerine to move. It is important that the bore of the tube be uniform, or the scale will be incorrect ; if not quite uniform the tube should be calibrated and a corrected scale applied. APPENDIX 249 The tube is filled as follows : A glass tube about ^ or | of an inch in diameter, and say 8 feet long, has one end closed and filled to a height of some 27 inches with clean, dry, warm mercury ; adhering bubbles of air may be removed by tapping (see p. 96) or by means of an air-pump. An ebonite plug or cylinder (A), with its length equal to its diameter, which latter should be nearly equal to that of the tube, is now dropped in. The float (B) attached to the plug may be a small sphere of ebonite, as in the figure, which is drawn to scale; or for a barometer tube | of an inch diameter an ebonite cylinder ^ an inch long and ^ of an inch diameter will be found suitable.* Mercury is then poured on the plug till the float is completely submerged. Glycerine that has been previously warmed and its dissolved air extracted by an air-pump is now poured in. When the tube is full it is care- fully corked, so that no air bubble is entrapped, and then the tube is inverted in a bath of glycerine and the cork cautiously withdrawn. The column sinks slowly until the atmospheric pressure is balanced, when the operation is complete. The bath may be a glass dish about 4 inches deep and 8 inches diameter, though a larger rectangular trough would be better. A wooden cover excludes dust, and a little sperm oil floating on the surface of the glycerine in the bath prevents absorption of moisture from the atmosphere. If the bath be placed on the floor, the top of the column is about level with the eye and thus convenient to read. The scale must be set by an ordinary barometer, and the graduation made from the density, say 1'26, of the glycerine employed; about Tl inch would therefore be equivalent to one-tenth of an inch in a mercurial barometer. Creosote, with some advantage, can be used in place of glycerine. * The proportion of the float to the plug is important ; if too large it will drag _up the plug. The plug A must be of sufScient depth ; if too shallow it will be unstable and apt to jam in the tube. 250 PRACTICAL PHYSICS § 9. The measurement of small fluid pressures. The M'Leod gauge is subject to an error arising from tlio condensation of the residual gas on the surface of the glass gauge tube and bulb ; with gases like COj this condensation is considerable, and vitiates the indications given by the instrument. By cautiously heating the glass the condensed gases may be expelled, but, unless removed, the residiial gas will be re-absorbed on the glass cooling. The highest vacuum may, however, be obtained by making the residual gas CO,, heating the vessel under exhaustion whilst the pump is at work, and absorbing the last traces of gas by fused caustic potash. When the vacuum is very good the spark from an induction coil ceases to pass between two adjacent platinum wires fused into the vessel under exhaustion. The difficulty of measuring small pressures arises loss from the smallness of the force (as a millionth of an atmosphere is equal to the weight of a milligramme per sq. cm.) than from the difficulty of making the surface pressed on, such as a liquid in a U-tubo, freely movable. The best method appears to be to observe through a microscope the motion of a thin flat film or membrane, such as glass, mica, or india-rubber, stretched across a tube and subject to a small difference of pressure on its opposite faces. In this way, as Fitzgerald, to whom this method is due, suggests, the relative densities of gases may be measured by balancing one column of gas against another ; the density thus found would bo free from the error arising from the condensation of the gas on the sur- face of the vessel in which it is weighed. § 10. Proof of the formula for the Atwood's machine. Let g = acceleration due to gravity, a = acceleration produced by tho mass m, S = distance fallen before m is removed, APPENDIX 251 S, = distance fallen after m is removed, t = time taken to fall through S, h = J) ;) ^5lj K = a constant, equivalent to the moment of inertia of the wheel-work, w = a mass whose weight is equivalent to friction, then 2M + m + K may be called the mass moved, . •. (2M + m + K)a = {m- w)g, ■ ' " ~ 2M + m + K' also S = \af. And if V be the velocity at the instant that m is removed, v" = 2aS, S, = rf„ _ S,^ _ {m-w)g 2SC " 2JVI + m + K' 2^t,Hm - w)g , ,. — !-^gi — ^ = 2M + TO + K . . (1). Now by using a heavier rider m', and keeping the distances the same, we also get ?M^^=2M + m' + K . . (2), and by subtracting the former of these two equations from the latter to eliminate K, we arrive at the equation in the text. § 11. Proof of the simple pendulum formula to the first approximation. (1) To find the acceleration of a body towards the centre of a circle. In Fig. 68 if a body be moving in a circle with uniform 252 PRACTICAL PHYSICS velocity V, and P be the position of the body at any instant, if no force act upon it, in time t it would be at M, when PM = Yt. Therefore MN represents the distance through which the body has been moved by the central force ; and if a be the acceleration towards the centre, MN = ^af. But PM'' = 20P X MN in the limit {Euclid III. 37), .-. Vy = 2rx|af, (2) To find the relation between acceleration and displace- ment in simple harmonic motion. In Fig. 69 a point P moving uniformly in a circle with velocity V, the foot M of the perpendicular from P on the diameter AA' wUl move to and fro along AA', and this motion of M is called simple harmonic motion. If OP = r, OM = d, angle POM = 9, then from (1) the acceleration of P towards the centre is APPENDIX 253 r a = — , and a the component of this along AA' or the acceler- 6 = (-^)— cos 6,ii T be the periodic ation of M is — cos r time of P. And the displacement of M from is (i = r cos 6, ■ a iirV I ^ i-/ ■ '■ ~j= "?p5;7 cos 6 1 r cos = -?p5- (For harmonic motion, see Daniell's Physics, chap, v., or Everett's Vibratory Motion and Sound.) (3) To find the relation between the acceleration and dis- placement in a simple pendulum. Let I = length of the pendulum, T = the periodic time of a vibration, g = acceleration due to gravity. In the simple pendulum (Fig. 70) OP = I and the angle POM = 6 the angle of displacement, and PM = lO = d the linear displacement, the effective acceleration a is in the direction PN, and equal to ^ sin 6 ; a g sin 6. g ,^ , . ,,v . •. - = ^— J- — = f (p bemg very small). a Id 'I 254 PRACTICAL PHYSICS Therefore, for a small displacement, the motion of the boh of a simple pendulum is very approximately a simple har- monic motion, and for a point performing such a motion we have seen in (2) that — = -— j-. Therefore from (3) we get T being the periodic time of a vibration, an oscillation or single swing is t = Tr ^J L. 9 For a nearer approximation T = 27r/v/in+l sin^ -)• g\ 2/ § 12. To find the length of a simple pendulum whose time of oscillation is equal to that of a given compound pendulum. This can be done by experiment, as described in the middle paragraph of p. 129; or if the bob of the pendulum be a sphere of uniform density and radius r, the length I of the pendulum from the centre of the sphere to the point of sus- 2 r^ pension must be increased by -.-, so that the corrected D t 2/ length becomes I + —. (See moment of inertia of a sphere, Table XIV., and Hicks' Dynamics, p. 347.) § 13. Proof of the formula for the Ballistic pendulum. Let M = mass of the bob of the pendulum with the bullet in it, m = mass of the bullet. APPENDIX 255 h = distance from the centre of inertia of the bob to the knife-edge, \ = distance of the line of fire from the knife-edge, ^2 = distance which the centre of inertia of the bob rises by impact, L = distance of attachment of tape from the knife-edge, 5 = length of tape drawn out, T = half period of equivalent simple pendulum, V = velocity of bullet just before impact, (0 = angular velocity of pendulum when impact is complete, 6 = angle of displacement, K = radius of gyration of pendulum, g = acceleration due to gravity. Now, when a body is struck the change of its moment of momentum about any axis is equal to the moment of the impact about the same axis ; or the whole moment of momentum about the axis of the pendulum is the same before impact and after it. mYhj mV/i, = MKw, .-. (o = MK' And when the deflection of the bob is greatest its kinetic energy is changed into its equivalent potential energy, and since h, = h (1 - cos 6) we have M^;»(l - cos ^) = JMKV = 1'^'. By substituting the value of w above, MK' ^ 2 ' MK X 2 sin - X ^/^ 256 PKACTICAL PHYSICS But 2sin- = ^, and T = tt / — , V ah 2"L' ~" " V ^h' .-. K= — "—^ ■ Substituting we get TT y _ M%TS 77lA,Lir (See Thomson and Tait's Natural Philosophy, small edition, p. 91.) § 1 4. To find the moment of inertia of a vibrator ; for example a right cylinder with the axis of figure the axis of rotation. Let I = moment of inertia of cylinder, M = mass, I = length, E = radius, p = density. Suppose an infinitely thin cylinder of radius r and mass dm, its moment of inertia with respect to the axis is t'dm ; and dm = ^-n-rlpdr. And considering each thin cylinder as an element of the whole cylinder, we have E I = 27rlp r^dr = i^-n-lpR'; J and since M = -n-Vvlp, I = m'K'. § 15. Theory of simple rigidity. A wire is fixed firmly at one end to a support and hung vertically downwards with a vibrator at the other end. APPENDIX 257 Let n = simple rigidity, T = moment of torsional couple due to unit twist on the wire, I = moment of inertia of vibrator, t = half period of vibrator, I = length of wire, r = radius of wire. Now if X be the distance from the centre of the wire to an elementary ring of breadth dx, the area of this ring will be ttx^ - 7r(x + dx)" = 2Trxdx ; by neglecting ird^, and when the vibrator is twisted through unit angle or one radian, we have a; , . . .... -T = shearmg stram at a point in the ring, and since n = stress/strain, nx . • ,. -y = stress per unit area at distance x from the centre ; i-n-nMx ■ ■■ -J — = stress for the whole elementary area, 2 irflxdx and — -J — = moment of this stress ; •• ^--TJo """If' which is the moment of the stress for the whole section. But <=-y^; •■•^='?' 4 2T and by substitution we get —j- = -^ ; 2wll PART I « = -:3;5- 258 PRACTICAL PHYSICS § 16. Effusion, transpiration, and diffusion of gases. The passage of gases througli (i) a fine hole in a thin plate, effusion, or (ii) through capillary tubes, transpkatim, or (ill) through a porous solid, diffusion, are distinct processes, as first pointed out by Graham. Though the rate of effusion and diffusion both vary inversely as the square root of the densities of the gases, yet they are processes essentially different in their nature. When the aperture of efBux becomes a tube the efiusion rate is distiu-bed, and becomes the tran- spiration rate when the length of the capillary tube exceeds its diameter by at least 4000 times. The rates of transpiration are singularly unlike the rates of effusion, as will be seen by comparing Tables XX. and XXL ; note, however, that in Table XX. tiTnes of transpiration are given which are of course inversely as the rates. A plate of compressed pliunbago or of " biscuitware," though impermeable to gas by effiision or transpiration, is readily penetrated by the molecular or dilfusive movement of gases. Through such a plate a separa- tion of mixed gases follows as a consequence of the movement being molecular, thus a mixture of 67 of Hj and 33 of 0^ before diffusion became 9 of H2 and 91 of Og after diflftisioa This gaseous analysis by diffusion has been termed akmlym. A very different behaviour occurs if different gases are separated by a membrane, such as caoutchouc, in which the gas may be more or less soluble. The kinetic theory of gases gives, as Clerk Maxwell points out {Heat, new ed., p. 331), a simple relation between the dififttsion, viscosity, and conductivity of heat in gases ; these phenomena express " the rate of equalisation of three properties of the medium, — the proportion of it? ingredients, its velocity, and its temperature. The equalisation is effected by the same agency in each case, namely, the agitation of the molecules. In each case, if the density remains the same, the rate of equalisation is proportional to the absolute temperature." APPENDIX 259 Eecent experiments have shown that the exact law of the variation of the viscosity of gases with temperature is still uncertain. § 1 7. Removal of surface eontamination from water. In order to obtain a perfectly clean surface on water the following procedure, due to Lord Eayleigh, is necessary. Draw off water from a tap, or distOled water from a tube dipping below the surface, reject the first portions, receive the water in a well cleansed tin tray. Now bend a long strip of thin clean sheet-brass, about two inches wide, into as small a hoop as possible ; dip its edge just below the surface of the water and open out the hoop. By this means the surface contamina- tion, will be swept aside, and wUl take some time to return ; repeat this process two or three times, each time the con- tamination increases on the outside and diminishes within the hoop. Finally, by means of a Bunsen flame held beneath the tin tray, slightly warm the cleansed portion of the water ; by this means the colder water surrounding wiU have a relatively higher surface tension, and draw off the remaining trace of any greasy film. A gentle current of air along the surface may be used to cleanse the surface where the vessel contain- ing the water cannot be warmed. The student should try the capUlary constant of water in such a cleansed surface (see Table XXIII.). The activity of the motion of fragments of camphor dropped on the surface of water is a convenient test for a moderately clean surface (p. 236). Lord Eayleigh has shown that a greasy film two miUionths of a millimetre (2 fj.fi.) in thickness will arrest these movements ; the black and thinnest parts of a soap bubble are about 12 /i/x. Still thinner probably, about ^ fifi, is the film which gives rise to the effects of "surface viscosity." 260 PRACTICAL PHYSICS § 18. Superficial viscosity, formation of soap films. On the surface of ordinary water a vibrating compass needle comes to rest in half the time it requires when vibrat- ing wholly within the water. This apparent surface viscosity is caused by the oscillating needle sweeping the contamination on the surface before its advancing edge, and therefore leaving less behind it. The tension is consequently higher behind than in front of the moving needle, hence a force comes into play which damps the vibration. This quasi-surface viscosity disappears when the water surface is cleansed, as described in § 1 7. Motes or sulphur dusted on the surface of water reveal the apparent viscosity of the surface ; on an unprepared surface of water a vibrating compass needle will be seen to set the whole in motion, but with a surface freed from impurity the motes do not move until the vibrating needle almost strikes them. Lord Rayleigh's investigations render it highly probable that the power, which soap confers upon water, of forming a fairly durable bubble and extending into a stable film, is really due to the creation of an outer coating or pellicle, the surface tension of which is less than that of the purer water withia " The stability of the film requires that the tension be not absolutely constant, but liable to augment under extension. If the central parts of a vertical film were suddenly displaced downwards, an increase of tension above and a decrease below would be called into play, and the original condition would be restored " (Eayleigh). The stronger tension of the purer water in the innermost part of a soap film not only tends to resist the rupture of the outer pellicle, but helps to repair it when broken and to pull it up against the force of gravity, which is thinning a vertical film. A self-acting adjustment therefore comes into play preserving the equilibrium and creating the comparative stability of a soap film. The behaviour of oil or turpentine on water (see Experiment 88, p. 226) illustrates what takes place. APPENDIX 261 § 1 9. Theory of the rise of liquids in capillary tubes. The capillary tube is placed vertically in a vessel contain- ing liquid. Let h = height from the lower part of the meniscus to the level of the liquid in the vessel, r = radius of the tube, T = surface tension of the liquid acting at angle 9 to the vertical, = capillary angle, p = density of the liquid, then T cos ^ = vertical component of T, 27rrT cos 6 = total force tending to raise the liquid in the tube, . and, neglecting the liquid forming the meniscus, the force in absolute units required to support the liquid in the tube is hirr'pg. Therefore for equilibrium we have 2irrT cos 6 = hin^pg, hrpg ■. T = 2 cos 6 - , 2T cos e and hr = = a constant. P9 If the liquid wets the tube, we have approximately ^ = and cos 9=1, , 2T .'. hr = — = a constant. P9 § 20. Soap solution for soap films. The following method of preparing a good solution for soap films is described and recommended by Mr. Vernon Boys : — " Fill a clean stoppered-bottle three-quarters full of water. Add one-fortieth part of its weight of oleate of soda, which 262 PRACTICAL PHYSICS will probably float on the water. Leave it for a day, when the oleate of soda will be dissolved. Nearly fill up the bottle with Price's glycerine and shake well, or pour it into another clean bottle and back again several times. Leave the bottle, stoppered of course, for about a week in a dark place. Then with a syphon draw off the clear liquid from the scum which will have collected at the top. Add one or two drops of strong liquid ammonia to every pint of the liquid. Then carefully keep it in a stoppered bottle in a dark place. Do not get out this stock bottle every time a bubble is to be blown, but have a small working bottle ; never put any back into the stock. In making the liquid do not warm or filter it ; either will spoil it. Never leave the stopper out of the bottle, nor allow the liquid to be exposed to the air more than is necessary. This liquid is still perfectly good after two years' keeping." The oleate of soda should be fresh, if dry it is not so good ; when it cannot be obtained, Castille or Marseilles soap, obtainable at any chemists, answers almost as well. In the Philosophical Magazine, January 1867, p. 40 et seq., Plateau describes the preparation of his soap - glycerine solution, bubbles blown with which have a great perman- ence ; a bubble 1 decimetre in diameter, supported on a wire ring, lasted 24 hours when freely exposed to the air, and upwards of 54 hours when covered by a vessel contain- ing dry air ; a soap film, 7 cms. diameter, formed in a bottle of the solution, lasted 18 days. TABLES Table I. Units arul Dimensions. (The numbers in the table are the indices of [L] [M] [T], thus the dimensions offeree are [L] [M] [T""]). Derived Units. Fundamental Units. Length [L.l Mass [M.] Time [T.] Surface .... 2 Volume .... 3 Angle .... Velocity .... Angular velocity Acceleration 1 1 -1 -1 -2 Force .... 1 _ 2 Pressure . -2 -2 Density .... Momentum -3 1 -1 Work 2 -2 Moment of inertia 2 Moment of momentum 2 -1 264 PRACTICAL PHYSICS Table II. Metrical and British Measurements. (Chiefly from Everett's Units and Physical Constants.) 1 Metre 1 Metre 1 Metre 1 Micron (/i) 1 Micro-millimetre (/x/x) 1 Inch 1 Foot 1 Mile 1 Square centimetre 1 Square inch 1 Square foot 1 Cubic centimetre 1 Cubic inch 1 Cubic foot 1 Pint 1 Gallon 1 Gramme 1 Kilogramme 1 Grain 1 Ounce (avoir.) 1 Pound 1 Litre 1 Poundal 1 Foot-pound 1 Foot-poundal 1 Gramme-centimetre 1 Joule 1 Horse-power = 1-094 yards. = 3-281 feet. = 39-3704 inches. = one thousandth of a mm. = 10"' m. = one-millionth of a mm. = 10"* m. = 2-54 centimetres. = 30-4797 centimetres. = 1-609 kilometre. = 0-155 square inch. = 6-4515 square centimetres. = 929-01 square centimetres. = 0-061 cubic inch. = 16-3866 cubic centimetres. = 28316 cubic centimetres. = 567-63 cubic centimetres. = 4-5435 litres. = 15-432 grains. = 2-205 pounds. = 0-0648 gramme. = 28-3495 grammes. = 453-59 grammes. = 0-22 gallon. = 13825 dynes. = 1-356 X 10' ergs. = 4-2139 X 10' ergs. = 981 ergs. = 10' ergs. = 746 X 10' ergs per second. TABLES 265 1 Force-de-cheval = 736 x 10' ergs per second. 1 Watt =10' ergs per second. 1 Poundal = 1 pound -e-'g'.' 1 Dyne = 1 gramme -=- 'g-.' 1 Megadyne = 10° dynes = 2-247 lbs. (force). 1 Pound (force) = 32-2 poundals = 4'45 x 10' dynes. 1 Gramme „ =981 dynes. 1 Gramme per square cm. = 0'01422 lb. per sq. inch. 1 Pound per square foot = 0'488 grammes per square cm. 1 Pound per square inch = 70'31 „ ,, The length of the metre given in the text = 39-37043 inches (p. 7) is according to the recent determination by Colonel Clarke. As the metric unit is taken at 0° C. and the British unit at 62° F., a metre used at the ordinary tempera- ture of 62° F. is equivalent to 39-382 inches. A convenient approximation is 1 metre = 39f inches, or 3 feet 3 inches and three-eighths of an inch ; a rough approximation is 1 inch = 25 millimetres, or I decimetre = 4 inches. Dr. Johnstone Stoney has pointed out {Proc. Roy. Dublin Soc, 1889) that, without sensible error (less than -001 to '002 per cent), we may take the folio-wing numbers for the conver- sion of British and metrical measures into one another : — Length. Weight. yard = 9144 mm. pound =453-6 grammes, foot =304-8 „ ounce = 28-35 „ inch = 25-4 „ grain = -0648 „ metre = 39-37 inches. gramme = 15-432 grains. And with an error of only -01 per cent we may take a gallon = 4544 c.c, pint= 568 c.c, fluid ounce = 28-4 c.c. For the description of a gauge for the appreciation of ultra-visible quantities, see a suggestive paper by Dr. Stoney, in Proc. Roy. Dublin Soc, 1892. 266 PRACTICAL PHYSICS Table III. Mensuration. A = area of curve, S = surface of body, V = volume of body. (1) Eectangle, sides a and J, A = ab. (2) Square, side «, A = a'. (3) Parallelogram, sides a and b, height h, 6 = angle between a and b, A = bh = ab sin 6. (4) Triangle, sides a, b, c, height h, s = |(a + b + c), A = ^bh = ^bc sin 6 = {s(s — a){s — b){s — c)}*. (5) Circle of radius r, circumference = 2irr, A = ttt^. e r' Segment of circle, A = ttt' t-q?. - "o sin 6. (6) Circular ring, r and r^ the external and internal radius, A = 7r{r'-r,'). 2 (7) Parabola, a = abscissa, 6 = double ordinate, A = -ab, o (8) Ellipse, a and b semi-axes, A = irab. (9) Eectangular parallelepiped of sides a, b, c, Y = ahc, S = 2(a5 + ac + be). (10) Cube, side a, V = a^ S = 6a'. (11) Prism and cylinder, radius r, height h, Y - ir/h, S = 2Trr{h + r). (12) Pyramid and cone, radius r, height h, slant height I, V = ^r%, S = 7rr(l + r). (13) Frustum of cone or pyramid, r and r, the radii of the ends, V = ^(r^ + r/ + ri\), S = 7r{/ + r,' + l{r + r,)}. (1 i) Sphere of radius r, V = |7r/, S = iwf. (15) Segment of sphere, r, = radius of base, ^ = height of segment, V = ^(A' + 3r/) = '!j(3r - /i), S = 2^rh + ttt/. D «5 TABLES 267 Table IV. Acceleration due to Gi'avity, (Everett.) Value of g Length of Place. Latitude. in cms. per sec. Seconds Pendu- per sec. lum in cms. Equator 0° 0' 978-10 99-103 Latitude 45° 45° 0' 980-61 99-356 Paris . 48° 50' 980-94 99-390 Greenwich . 51° 29' 981-17 99-413 Berlin . 52° 30' 981-25 99-422 Dublin 53° 21' 981-32 99-429 Belfast 54° 36' 981-43 99-440 Aberdeen 57° 9' 981-64 99-461 Pole . 90° 0' 983-11 99-610 Table V. Densities. Liquids AT 15° C. Substance. p- Substance. 9- Sea water . 1-026 Bisulphide of carbon 1-27 Alcohol 0-806 Ether . 0-72 Glycerine 1-26 Chloroform . 1-49 Mercury 13-56 Methylated spirit 0-83 Milk . 1-03 Sulphuric acid 1-85 Turpentine 0-87 Nitric acid . 1-56 Petroleum 0-82 Hydrochloric acid 1-17 Linseed oil 0-94 Sol. zinc sulphate " satd. 1-38 Olive oil 0-92 Fusel oil 0-81 Sol. copper sul- ' phate satd. 1-22 Benzol 0-883 268 PRACTICAL PHYSICS Table V. — Continued. Densities. Solids. Substance. P- Substance. P- Aluminium . 2-6 Tin . 7-3 Bismuth 9-8 Zinc . 7-1 Brass (wire) . 8-5 Gold . 19-3 Copper ,, . 8-8 Silver . 10-5 Iron „ . 7-7 Platinum 21-5 Steel „ . 7-72 Oak . 0-86 German silver 8-5 Beech . 0-75 Lead . H-3 Elm . 0-54 Nickel 8-6 Cork . 0-24 Diamond 3-5 Paraffin wax 0-89 Graphite 2-3 Bees' wax . 0-96 Gas carbon . 1-9 Phosphorus . 1-84 Wood charcoal 1-6 Sulphur 1-97 Granite 2-7 Sugar (cane) 1-6 Quartz 2-65 Rock salt . 2-24 Basalt . 2-86 Gunpowder (loose) 8 -4-9 -4 Slate . 2-5 Gutta-percha 0-97 Porcelain 2-4 India-rubber 0-99 Brick . 2-1 Ivory- 1-92 Sand . 1-5 Marble 2-8 Crown glass 2-6 Fluorspar . 3-2 Flint glass . 2-93 to 3-5 Calcspar Ice 2-7 0-92 Note. — The student must bear in mind that, independently of their temperature, the densities of many substances vary within a considerable range, owing to. Jtheir degree of purity and (in the case of solids) to their physical state.' A useful table of densities is given in the £ncy. Brit., Art. "Hydrometer." TABLES 269 Table VI. Density of Dry Air compared with Water at 4° C. (Kohlrausoh.) Temperature. Pressure. Centigrade. 750 mms. 760 mms. 770 mms. 0-001276 0-001293 0-001310 1 001271 001288 0-001305 2 001267 001283 0-001300 3 00126? 001279 0-001296 4 001257 001274 0-001291 5 001253 001270 0-001286 6 001248 001265 0-001282 7 001244 001260 0-001277 8 001239 001256 0-001272 9 001235 001251 0-001268 10 001231 001247 0-001263 11 001226 001243 0-001259 12 001222 001238 0-001255 13 001218 001234 0-001250 14 001214 001230 0-001246 15 001209 001225 0-001242 16 001205 001221 0-001237 17 001201 001217 0-001233 18 001197 001213 0-001229 19 001193 001209 0-001224 20 0-001189 0-001204 0-001220 270 PRACTICAL PHYSICS Table VII. Densihj of Gases at Temperature 0° C. and 760 mm. Pressure. Name. Weight of 1000 CO. Density Comparod in Grammea, with Air. Air . 1 '2932 1-0000 Oxygen 1-4298 1-1056 Hydrogen . 0-0896 0-0693 Nitrogen . 1-2562 0-9714 Coal gas (variable) 0-6666 0-5156 Marsh gas . 0-7270 0-5590 Carbon monoxide 1-2344 0-9569 Carbon dioxide . 1-9774 1-5290 Sulphur dioxide . 2-7289 2-1930 Chlorine . 3-1328 2-4216 Aqueous vapour . 0-6230 The reciprocals of the weight in grammes of one c.c. of the different gases give the volume in c.c. of one gramme at the standard temperature and pressure. At this pressure and at a temperature of 21°-2 C. the weight of a litre (1000 c.c) of air is exactly 1-2 gramme, or 1 gramme of hydrogen, 16 grms. of oxygen, and 14 grms. of nitrogen, at this tempera- ture and pressure, occupy 1 2 litres. From this Dr. Johnstone Stoney {Proc. Roy. Dublin Sac, 1889) has deduced the follow- ing convenient formula for laboratory use : — At 760 mm. pressure and say 21° C. temperature, The weight of a litre of any gas = D/1 2 grammes, The volume of a gramme of the gas = 1 2/D litres, where D is the density of the gas compared with hydrogen as unity, thus for air D = 1 4-4. At any other temperature and pressure, not far removed from the above, add or subtract 1 per cent for every 3° C. above or below 21° C, and 1 per cent for every 7 J mm. (-j^ of an inch) above or below the standard pressure. TABLES 271 Table vm Volume and Density of Water. (Everett's Physical Consianls.) Temperatoie Volume at 4==!. 1 Ateolate Density. Oentigiade. Giammes per cc. 1 000129 •999884 1 1-0000 72 •999941 2 1-000031 ■9999S2 3 1-000009 1-000004 4 1-000000 1-000013 5 1-000010 1-000003 6 1-000030 •9999S3 7 1-000067 ■999946 8 1-000114 ■999899 9 1-000176 •999837 10 1-000253 •999760 11 1-000345 -999668 12 1000451 -999562 13 1-000570 •999443 14 1-000701 -999312 15 1-000841 -999173 20 1-001744 -99S272 25 1 -002; SS -997133 30 1-004-2.-3 -995778 40 loorro 1 -99236 50 1-01195 ■9SS21 60 1-01691 •95339 ro 1 -02256 •97795 so 1-02SS7 -97195 90 1 -03567 96557 100 1-04312 ■95S66 A collation of the best determinations of the variation of density of water with temperature will be found in a paper by Mendel6e£F in the PM. Mag., Jan. 1892. 272 PRACTICAL PHYSICS Table IX. Volume and Density of Mercury. (Lupton's Tables.) Temporaturo Contigrade. Volume at 0° = 1. Density. Grammes per c.c. 1-000000 13-596 4 1-000716 13-586 5 1-000896 13 -.584 10 1-001792 13-572 15 1-002691 13-559 20 1-003590 13-647 30 1-005393 13-523 40 1-007201 13-499 50 1-009013 13-474 60 1-010831 13-460 70 1-012655 13-426 80 1-014482 13-401 90 1-016315 13-377 100 1-018153 13-363 Table X. Depression of Barometric Column due to Capillarity, (Pouillet.) Internal Diameter of Tube in mm. Depression in mm. Internal DiamoLer of Tube in mm. Depression in mm. 2 4-58 12 •26 3 2-90 13 •20 4 2-05 14 -16 6 1-51 15 •13 6 1-14 16 •10 7 •88 17 ■08 8 •68 18 ■06 9 ■53 19 ■05 •10 •42 20 -04 11 ■33 21 ■03 TABLES 273 Table XI. Elasticity and Tenacity. (Chiefly from Sir W. Thoinson's Mathematical and Physical Papers, vol. iii.) Young's Modulus. Tenacity. Substance. In Dynes per square In Grammes jjer cin. square cm. Iron (wire) l-826xl0'2 6-5 xl08 *Iroii (wire, soft) 1 27 ,, Steel ,, ... 1 846 ,, 9 9 , , pianoforte wire . 2 01 ,, ;i8 123 17 5 to\ 6 J' 3 *Manganese steel (hard wire) 1 648 ,, „ (soft „ ) 1 47 ,, 7 7 Steel (bar) 2 7 to 9- , Brass (cast) 63 ,, 1 27 , ,, (wire) 982 ,, 3 43 , Copper ,, annealed 1 031 ,, 3 20 , „ (hard) . 1 172 ,, 4 22 , Lead (sheet) 050 ,, 23 , „ (cast) 187 „ 22 Tin „ ... 409 „ ■12 Zinc (drawn) 856 ,, 1 58 , *German silver (wire) 1 354 ,, 6 Platinum ,, . 1 569 „ 3 50 , Silver (drawn) . 722 „ 2 96 Gold 829 ,, 2 75 , Bronze ,, . . 683 „ 2 52 , Glass 551 ,, 66 Slate 996 ,, 73 Oak . . . . 101 ,, 1 05 Ash . 111 ,, 1 20 Teak 0-166 ,, 1-05 * These are our own determinations. — W. F. B. PAKT I 274. PRACTICAL PHYSICS Table XII. Limits of Elasticity arid Breaking Stress hy Stretching. (Wertheim.) Giving the weight {p) required to stretch a wire 1 square mm. in section, so as to produce a permanent elonga- tion of -05 mm. per metre, i.e. ^^ per cent; also the breaking weight (P) for the same wire at a temperature of 15° C. Name of Condition of P P Substance. Substance. Grammes. Grammes. Lead . . / Drawn . . t Annealed . 250 2,070 200 1,800 Tin . . . / Drawn . . \ Annealed . 400 2,450 200 1,700 Gold . . Drawn . . \ Annealed . 13,500 27,200 3,000 10,080 / Drawn . . 11,000 29,000 Silver . . 1 Annealed . 2,500 16,020 J Drawn . . 12,000 40,300 Copper . . \ Annealed . 3,000 30,540 / Drawn . . \ Annealed . 26,000 34,100 Platinum . 14,500 23,500 Iron . . / Drawn . . \ Annealed . 32,500 61,100 5,000 46,880 Cast steel . Drawn . . Annealed . 65,600 5,000 83,800 65,700 TABLES 275 Table XIII. Torsional and Longitudinal Resiliences in cms. (From Sir William Thomson's Mathematical and Physical Papers, vol. iii. Substance. Torsional. Longitudinal. India-rubber band 120,000 Pianoforte steel wi re 130 to 1203 17,620 Platinoid , 271 to 1580 1,693 German silver , 168 514 Brass , 860 to 940 728 Delta metal , 760 to 1250 2,708 Phosphor bronze , 4,904 3,545 1,842 Silicium bronze , 3,166 Manganese ,, , 2,998 The torsional resiliences given in the above table show great differences for the same metal due to differences of temper. In like manner various specimens of brass show the following differences in elasticity ; — Young's modulus . . 9'48 to 10'44 "j Simple rigidity . . 3-53 to 3-90 I x 10" Volume elasticity . . 10-02 to 10-85 J The volume elasticity (p. 172) and compressibility of water at different temperatures are as follows : — Volume elasticity Compression for one'j megadyne per sq. - cm. . . . j 0° ir 18° 43° C. xlO'o xl0-» 2-02 4-96 2-11 4-73 2-20 4-55 2-29 4-36 276 PEACTICAL PHYSICS Table XIV. Moments of Inertia. M = mass of the body, I = length, a, b, c = length of the sides, if the body is of rectangular cross-section, R = external radius, r = internal radius. Body. Moment of Inertia. Direction of the Axis of Oscillation. Uniform thin rod 4 At end, perpendicular to length. >) j» «S Through middle, perpen- dicular to length. Rectangular lamina . »E Through centre, parallel to one side, and bisecting side a. )» )) "'^ Through centre, perpen- dicular to plane. Rectangular parallelo- \ piped . . . / *^ 12 Perpendicular to side con- tained by a and b. Circular plate . ." Through centre, perpen- dicular to plane of plate. „ ring . M 4 Through centre, perpen- dicular to plane of ring. Right cylinder . Mf The axis of figure. Hollow cylinder Through centre, perpen- dicular to axis of cylinder. Axis of figure. J) ») ^[l2+ 4 ) Through centre, perpen- dicular to axis of cylinder. Sphere |mr2 5 Any diameter. TABLES 277 Table XV. Rigidity Moduli. (In grammes per square cm.) Substance. ■IL. Substance. 71. Brass Iron (wrought) Iron (cast) Steel 350 X lO'' 785 ,, 542 „ 834 ,, Copper . German silver Platinoid Glass 456x10" 496 „ 476 „ 241 „ Table XVI. Scale of Hardness. H = 1. Talc (common laminated variety). H = 2. Gypsum (crystallised). H = 3. Calcspar (transparent). H = 4. Fluorspar (crystalline). H = 5. Apatite (transparent). H = 6. Felspar (cleavable). H = 7. Quartz (transparent). H = 8. Topaz (transparent). H = 9. Sapphire (cleavable). H = 1 0. Diamond. Bodies scratched by finger nail Bodies that scratch copper Polished white iron . Window glass Bodies scratched by a penknife Steel point or file Flint H = 2'5 or less. H = 3 or more. H=4-5. H = 5 to 5-5. H = less than 6. H = 6 to 7. H = 7. 278 PRACTICAL PHYSICS Table XVII. Coefficients and Angles of Friction : without lubricants (Morin). Substance and position fibres. On Point of Motion. In Motion. ■0- f- *• p- Oak on oak (parallel) . „ „ (crossmse) „ „ (endwise). Elm on oak (parallel) . „ ,, (crosswise) Wrought iron on oak " (parallel) Copper on oak (parallel) Wrought iron on oast ' iron Cast iron on cast iron 31° 50' 28° 20' 23° 20' 34° 40' 29° 40' 31° 50' 31° 50' 10° 45' 9° 5' 0-62 0-54 0-43 0-69 0-57 0-62 0-62 0-19 0-16 25° 40' 18° 45' 10° 45' 23° 20' 24° 15' 31° 50' 31° 50' 10° 10' 8° 30' 0-48 0-34 0-19 0-43 0-45 0'62 0-62 0-18 0-15 Table XVIII. Compressibility (Quincke). Compression in Milliontlis for one Atmosphere. Substance. Temp. rc. At 0° C. At r c. Mercury 2-95 1-87 15-00 Glycerine 25-24 25-10 19-00 Rape oil ... 48-02 68-18 17-80 Almond oil . 48-21 56-30 19-68 Olive oil ... 48-59 61-74 18-30 Water (see also p. 275) . 50-30 45-63 22-93 Bisulphide of carbon 53-92 63-78 17-00 Oil of turpentine . 58-17 77-93 18-56 Benzol .... 62-84 16-08 Alcohol .... 82-82 95-95 17-51 Ether .... 115-57 147-72 21-36 Petroleum 64-99 74-50 19-23 TABLES 279 Table XIX. Viscosity of Liquids. Temperature Coefficient of cent. Viscosity. Water 0° 0-01783 n ... 10° 0-01309 M ... 20° 0-0102 )» ... 50° 0-0056 80° 0-0036 Water with 94-00 % glycerine . 8-5° 7-444 80-31 1-022 64-05 0-222 49-75 0-093 Alcohol meth. . 10° 0-0069 ethyl. . 0-0154 Glycerine . 2-8° 42-180 M ... 26-5° 4-944 Mercury 17-2° 0-016 Table XX. Viscosity of Gases (between 1 5° and 20° Experiments. C). From Graham's Gas. Coefaciency of Viscosity. Transpiration Time, Oxygen = l. Air 0-000192 0-901 Oxygen .... 0-000212 1-000 Hydrogen .... 0-000093 0-437 Nitrogen .... 000184 0-875 Chlorine .... 0-000141 0-666 Marsh gas . 0-000120 0-551 Olefiant gas 0-000109 0-507 Carbon monoxide 0-000184 0-875 Carbon dioxide . 0-000160 0-727 Air at 0° C, recent observa- tions .... 0-000168 280 PRACTICAL PHYSICS Table XXI. Effusion and Diffusion of Gases (Graham). Air=l. Gas. Density. 1 Rate of Effusion. Diffusion Volume. ^/density. Hydrogen 0-0693 3-7998 3-613 3-83 Marsh gas 0-5590 1-3375 1-322 1-34 Carbonic oxide . 0-9678 1-0165 1-012 1-01 Nitrogen . 0-9713 1-0147 1-016 1-01 Oxygen . 1-1056 0-9510 0-950 0-95 Nitrous oxide . 1 -5270 0-8092 0-834 0-82 Carbonic dioxide 1-5290 0-8087 0-821 0-81 For table of coefficients of inter-diffusion of gases in sq. cms. per sec, see Maxwell's Heat, last ed., p. 342. Table XXII. Surface Tensio7is at 20° C. in grammes and dynes per Lineal Centimetre (Quincke). Substance. Surface in contact mth air. Grammes. Dynes. Surface Tension Water=l. Water 0-0826 81-0 1-000 Mercury 0-5504 540-0 6-667 Bisulphide of carbon . 0-0327 32-1 0-396 Chloroform . 0-0312 30-6 0-378 Alcohol 0-0260 25-5 0-315 Olive oil 0-0376 36-9 0-455 Turpentine . 0-0303 29-7 0-367 Petroleum . 0-0323 31-7 0-391 Hhrdrochlorio acid 0-0715 70-1 0-865 Sllution of hyposulphate of soda 0-0790 77-5 0-957 TABLES Table XXIII. Surface Tensions (Various observers). The values found by Quincke for water and mercury given in the last table are only applicable when special precautions are taken to avoid impurity and surface contamination. In fact, " It seems doubtful whether the tension of water is really so high as that recorded by Quincke. Observations upon very clean surfaces, in which the tension was determined from its eiFect upon the propagation of ripples, gave 0'074 grammes " = 72'6 dynes per cm. (Eayleigh). The following table gives the values of the surface tension, T, of different liquids found by other observers. Substance. Density. T. in dynes. Temp. Authority. Water 1-00 75-2 0°C. Brunner. ii )) 74-2 8 Desains. Mercury 13-66 453-2 15 ^j Sulphuric acid 1-85 62-1 14 Frankheini. Nitric acid 1-50 41-9 16 ,, Absolute alcohol 079 22-3 20 Wilhelmy. Amylic ,, 0-82 23-8 .J Ether 0-72 17-6 M Mendeleeff. Solution of soap 1 to 40 1-01 27-7 Mensbrugghe Solution of sajjonine 45-8 }) Water containing camphor 44-2 )» In connection with the behaviour and measurement of contaminated water surfaces, Miss Pockels' recent valuable and suggestive experiments on surface tension should be read ; see Nature, vol. xliii. p. 437, and vol. xlvi. p. 418. t ^ 282 PRACTICAL PHVSICS REDUCTION OF RESULTS— PROBABLE ERROR (i.) After a series of observations have been made in an experiment, numerical reductions have generally to be made before the final result can be obtained. Before beginning these numerical reductions the observations should be arranged in a tabular form. Tables of logarithms and other tables should be used when necessary to lessen the labour of calcu- lation. Whilst for most physical calculations four figure logarithms are sufficient, the student should be on his guard against trusting to these when accuracy is required in certain kinds of calculations, such as correcting the density of water for temperature where the first significant figure may occur in the fifth or sixth decimal place. The student ought, however, to avoid the tendency to over -refinement in his calculations, such as using many decimal places when the experiments do not warrant it, a good general rule is to go one decimal place beyond that which denotes the accuracy which he expects from his result. (ii.) The degree of accuracy attainable in an experiment varies very much with the precision of the instrument employed ; the errors due to the instrument alone are called constant errors, and it is very important that they should be found out and allowed for in the final result. Thus it would be useless to read the barometer to the hundredth of an inch if the barometer itself be unreliable, either through the Torricellian vacuum being imperfect or the instrument not quite vertical, or to read the cathetometer to a tenth of a millimeter if the scale be badly divided or the levelling imperfect. We should always, in the first place, ascertain what reliance can be placed in the instrument we are using either by comparison with a standard or other means. It is veryi desirable that the student should if possible check his EEDUCTION OP EESULTS PROBABLE EEEOE 283 results by another method of experiment which, even if less sensitive, will serve to reveal gross errors. Thus in finding the internal diameter of a capillary tube by the method of weighing a given length of mercury, the result may be checked by the diameter of the tube being measured directly by means of the reading microscope. (iii.) If in an experiment two instruments are employed which possess different degrees of accuracy, it would be use- less to read or observe to the extreme degree of accuracy of the most delicate one. Thus in an experiment involving the product of a weight into a temperature it would be waste of time and give a misleading sense of accuracy to weigh to a milligramme if the thermometer only read to half a degree. (iv.) It is also desirable to make each experiment under conditions which give the most favourable result. Thus in determining the specific gravity of a body, it is best to use as large masses as the balance will weigh, for although the sensitiveness of a balance is less with a heavy load, the loss in this respect is more than compensated by the gain derived owing to the smaller percentage error (see p. 38). (v.) When all the separate determinations of an experiment are entitled to an equal degree of confidence, the arithmetic mean of the whole gives the most probable value of the required result. If the separate determinations be now com- pared with the mean value, differences greater or less will be found to exist, and from these differences the probable error of a single observation, as well as the probable error of the result, may be deduced as follows : — If n = the number of observations, 81, 82, 83 etc. = the differences, S = 61^ + 82^ + . . . 8„^ or the suni of their squares, then the probable error of a single observation will be 284 PEACTICAL PHYSICS ± 0-6745 \/ ^ And the prohMe error of the result is n-\ ±0-6745 a/ -^^ * n{n- 1) (vi.) The graphic method (Expt. 15, p. 40) affords another means of detecting errors in a series of observations. By plotting the results on millimetre paper, and using a flexible rule to draw a smooth curve, small irregularities are at once seen, which indicate errors of experiment. Care mnst, how- ever, be taken not to carry the smoothing process too far. The results being shown in a curve, the intervening values not actually determined by experiment can now readily be ascertained by mere inspection ; in this way " interpolation " is best accomplished, otherwise recourse must be had to the more troublesome interpolation formulae. * See Airy's Theory of Errors of Observation, part 1, p. 24. A full discussion of the subject of mean and probable error will be found in the introduction of Kohlrausch's Physical Measurements. THE END Printed by R. & R- Clark, Edinburgh MESSRS, PERCIVAVS EDUCATIONAL WORKS Practical Physics. An Introduc- tory Handbook for the Physical Laboratory. In Three Paris. By W. F. Barrett and W. Brown. 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