mm Cornell University Library The original of tiiis 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/cu31924031363421 Cornell University Library arW3865 A laboratory manual of experiments in ph 3 1924 031 363 421 olln,anx A Laboratory Manual of Experiments in Physics •«• FOR THE STUDENTS OF THE SOPHOMORE YEAR IN THE UNI- VERSITY OF UTAH, by L. JV. Hartman, Ph. D. Associate~ProfessQr of Physics REVISED AND ENLARGED UNIVERSITY OF UTAH 1907 TorBUNE-REPORTER PRINTINQ CO., SALT LAKE PREFACE This course in Laboratory Physics is intended for the , students of the sophomore year in the University of Utah. The completion of the course in general physics for students of this grade will require: (1) one laboratory period of three or four hours per week throughout the year, during which time it is expected that a minimum of thirty selected experi- ments from the manual will be performed ; and (2) a three- hour course of lectures and recitations on the general theory. The student is supposed to have had a course in Elementary Physics, — Physics "a" of the University, or its equivalent. In the preparation of this manual, it has been the pur- pose of the writer to state the physical theory as briefly as possible and to make the directions jor the physical manip- ulation sufficiently complete and clear so that the student will have no trouble in performing the mechanical manipulation and taking the necessary observations. Detailed descriptions of apparatus and methods have purposely been omitted. In or- der to get a thorough grasp of the theory involved in the ex- periment, it is hoped that the student will avail himself of the numerous references cited at the beginning of each experi- ment. Wherever possible the graphical method has been util- ized and the interpretation of physical constants has been em- phasized. In preparing this material for publication, the writer wishes to state that he has made free use of the notes written by his predecessors in the department, and also of the numer- ous laboratory manuals and text-books of physics at his com- mand. A number of new experiments have been incorporated in the mimeographed set of notes formerly used here, and many of the old experiments have been entirely re-written. In each experiment taken from the old set of notes, some change or addition has been made. No claim, however, is made for originality in the arrangement of the experiments or of the material contained herein. Although numerous excellent manuals are at hand, no single volume was suited to the needs of this laboratory. Hence it would have been necessary for the student to pur- chase at considerable cost several different texts. This set of notes has been prepared, therefore, so that all the informa- tion and the suggestions necessary for the experimental man- ipulation in this laboratory may be obtained at a moderate price. This is the sole justification claimed by the writer for his work. For numerous helpful suggestions and kind criticisms, the writer wishes to acknowledge his indebtedness to Mr. G. C. Gilbert of the Department of English in the State University. For the preparation of the index and valuable assistance ren- dered in correcting the proof, the author wishes to thank his wife. TABLE OF CONTENTS General Snggestions to Students 6 List of Reference Books 8 MECHANICS. 1. The Use of the Vernier and Micrometer Calipers and the Diagonal Scale 11 2. The Traveling Microscppe — Calibration of a Screw 12 3. The Traveling Microscope — The Calibration of the Bore of a Glass Tube 14 4. The Comparator — The Determination of the Number of Centimeters in an Inch 16 5. The Micrometer Microscope^ — The Measurement of the Diameter of a Glass Tube 19 6. The Cathetometer 20 7. The Spherometer 21 8. Linear Acceleration 23 9. Angular Acceleration 26 10. Linear Harmonic Motion 29 11. Angular Harmonic Motion 30 12. Moment of Inertia 32 13. Moment of Inertia — Angular Acceleration 34 14. Atwood's Machine — Gaertner Pattern 35 15. The Atwood's Machine 37 16. The Reversible Pendulum — The Determination of Gravity. . 39 17. The Physical Pendulum — The Determination of Gravity. ... 41 18. The Determination of Gravity by Means of Kater's Pendulum 43 19. The Variation of the Periodic Time of the Physical Pendulum as the Position of the Knife Edges is Changed 44 20. The Coefficient of Friction 45 21. The Force Table 47 22. Equilibrium of Three Forces 48 23. Impact — Linear Momentum 49 24. Inelastic Impact 50 25. Elastic Impact • 50 26. Centrifugal Force 50 27. Moments and Center of Gravity 51 28. The Sensibility of a Balance 54 29. The Determination of the Density of a Solid 55 30. The Determination of Density by Means of the Specific Gravity Bottle 57 31. The Determination of the Density of a Liquid by Means of the Pyknometer 58 32. The Determination of Density by Means of the Nicholson Hydrometer 59 33. The Determination of the Density of Liquids by Means of Balanced Columns 60 34. The Coefficient ^of Restitution 62 35. Young's Modulus by Stretching 63 36. Young's Mt)dulus by Bending 64 J,T. The Coefficient of Rigidity— Twisting Method 65 38. The Coefficient of Rigidity of a Wire — Method of Oscilla- tions 67 39. Boyle's Law 68 SOUND. 40. Stationary Vibrations 73 41. The Sonometer — Law of Vibrating Strings ." 74 42. The Velocity of Sound in Metals— Kundt's Method 75 43. Vibrating Air .Columns 1(> 44. The Velocity of Sound in Air — Resonance 11 45. The Determination of the Absolute Frequency of Vibration of a Tuning Fork 79 HEAT. 46. The Calibration of a Mercury Thermometer 83 47. The Calibration of the Bore of a Thermometer. 85 48. The Heat of Fusion of Ice 86 49. The Heat of Vaporization of Water 88 50. The Heat of Vaporization of Water — Electrical Method 89 SL The Specific Heat of a Solid 90 52. The Determination of the Specific Heat of a Metal with the Joly Steam Calorimeter 91 53. The Radiation from a Warm Body 93 54. The Fusion Point of a Solid 94 55. Molecular Depression 95 56. The Determination of the Coefficient of Linear Expansion of a Meta;l 96 57. The Coefficient of Apparent Expansion of Mercury — The Weight Thermometer 99 58. The Determination of the Coefficient of Expansion of Mercury — Regnault's Method 101 59. The Pressure Coefficient of Air at Constant Volume — The Air Thermoijieter 102 60. The Coefficient of Expansion of Air at Constant Volume — The Jolly Air Thermometer 103 61. The Coefficient of Expansion of Air at Constant Pressure.. 106 62. The Maximum Vapor Pressure of a Liquid 107 63. The Mechanical Equivalent of Heat — Puluj's Method 109 ELECTRICITY AND MAGNETISM. 64. Electrostatic Induction 115 65. The Toepler-Holtz Machine 117 66. The Plotting of Magnetic Fields 120 67. The Determination of Magnetic Dip 121 68. The Comparison of the Horizontal Intensity of the Earth's Magnetic Field at Different Points 122 69. The Determination of the Ratio M:H by the Magneto- meter Method 123 70. The Theory of Shunts 126 71. The Measurement of Resistance by Means of the Wheat- stone Bridge 127 72. The Measurement of the Resistance of a Galvanometer — Thomson's Method 129 73. The Resistance of a Galvanometer — Method of Shunts.... 130 74. The Measurement of the Resistance of a Battery — Mance's Method 131 75. The Measurement of Current by Electrolysis 132 76. The Determination of the Mechanical Equivalent of Heat with the Electrocalorimeter 134 77. Equipotential Lines and Lines of Current Flow 137 LIGHT. 78. The Determination of the Index of Refraction by Means of the Microscope 141 79. The Focal Length of a Concave Mirror 142 80. The Focal Length of a Lens 143 81. The Measurement of the Angle of a Prism 146 82. The Determination of the Index of Refraction of a Prism 148 83. The Dispersion of a Prism as a Function of the Wave- length 150 84. The Measurement of the Wave-length of Light by Means of the Grating 151 85. The Magnifying Power of a Microscope 152 GENERAL SUGGESTIONS TO STUDENTS. The purpose of the course is to illustrate certain of the fundamental principles and laws of the science, and thus im- press them on the mind of the student; to teach skill and accu- racy in physical manipulation and method of observation ; and independence in thought and action. The student should there- fore read over the directions of the experiment before he comes to the laboratory, and thus familiarize himself with the ob- ject of the experiment and the means employed in the experi- mental manipulation. If unfamiliar principles are involved, he should avail himself of the numerous references cited at the beginning of each experiment in order that the import- ance of each step in the manipulation may be appreciated and its significance understood. Then each step will be intelli- gently taken and the experiment will prove of some value. All observations and data obtained in the process of the experiment, should be neatly recorded in a note-book provided for that purpose. Each observation taken should be inde- pendent of any that has preceded it; that is, the mind of the observer should be unprejudiced by any known facts concern- ing the experiment when he makes a given observation. With- in a week after the experiment has been completed, the report of the experiment should be written in a note-book devoted exclusively to this course. It should be the earnest purpose of each student to keep his note-book neat and orderly, concise, and explicit. The reports must be written in ink. Whenever possible, tabulations of data should be made; plainly label all headings and divisions, and columns in tabulations. Write all descriptions and data on the right-hand page of the note-book, reserving the left-hand for drawings of apparatus and dia- grams. In all cases accuracy of observation and care in manipu- lation should be exercised. After the data have been obtained and tabulated, endeavor to deduce the law that is fulfilled, or the theory that is confirmed by the experiment. In any case where a graphical representation of the results is possible, it should be utilized and the constants of the graph should be given a physical interpretation. For example, in the case of the straight line, the slope and the intercept should be given a physical interpretation. Whenever the mathematical prep- aration of the student will permit, the formulae used in the experiment should be derived. This will help the student to fix in mind the physical and mathematical principles involved in the experiment, and will make clear to the instructor that the student thoroughly understands the work that he has placed in his note-book. The report of the experiment should include the follow- ing: (1) The Number and Name of the Experiment. (These are given in the Manual.) (2) The Date and the Names of the Observers. (3) The Object of the Experiment. (4) The Complete List of Apparatus. (5) The Theory of the Experiment in the Student's Own Words. This must not be copied from the Manual. (6) The Manipulation of the Experiment in the Student's Own Words. This should be a statement in his own words of all that the student did and how it was done, together with a tabulation of the data and calculation of the results. A sample computation should always be included in full in the report. (7) The Summary should include a statement showing how the law has been proved or the theory verified. This should also include the interpretation of all curve constants. (8) All assigned problems should be included at the end of the experiment. (9 )For plotting curves use millimeter cross-section paper. On all such curves indicate the co-ordinates and use a suitable caption. Whether the student works alone or with a partner, the writing of the experiment and the discussion and treatment of the data should be done independently and alone. The note-book should be handed in each week with all the expe- riments written up to date and with all former experiments cor- rected, or new work may not be assigned. Credit for labora- tory work may be withheld until this is done. A minimum of 30 experiments and 100 hours of actual work in the laboratory is required for the completion of the course. A list of re- quired experiments will be furnished, upon the completion of which the student will be permitted to exercise his choice concerning the experiments remaining for him to perform. The apparatus for each experiment must be obtained at the beginning of the laboratory period and a receipt given for all the pieces received. In case a piece of apparatus is want- ing, or is defective or out of adjustment, it should be reported to the instructor at once, as the student will be held respon- sible for all damages to or breakage of apparatus. All ap- paratus must be returned before leaving the laboratory, un- less special permission to keep it out longer is obtained from the instructor in charge. On entering the laboratory for the first time, present the receipt for the laboratory deposit. REFERENCE BOOKS. The following is a list of books to which reference has been made in this manual. AMES, J. S.— Theory of Physics. 1897. AMES, J. S., and BLISS, W. J. A.— A Manual of Experi- ments in Physics. 1898. CARHART, H. S.— University Physics, 2 vols. 1896. CARHART, H. S., and PATTERSON, G. W.— Electrical Measurements. 1900. EDSER, EDWIN— Heat for Advanced Students. 1901. EDSER, EDWIN— Light for Students. 1904. FERRY, ERVIN S.— Practical Physics for Students of Science and Engineering, 2 vols. 1903. GANOT — Elementary Treatise on Physics, translated by E. Atkinson. 1899. GLAZEBROOK, R. T.. and SHAW, W. N.— Practical Phys- ics, 3rd edition. 1899. GRAY, ANDREW — Absolute Measurements in Electricity and Magnetism. 1888. GRAY, THOMAS— Smithsonian Physical Tables, 3rd edi- tion. 1904. HASTINGS, C. S., and BEACH, F. E.— General Physics. 1899. KOHLRAUSCH, F.— Praktische Phvsik, 9th German edi- tion. 1901. LOUDEN, W. J., and McLENNAN. J. C— Experimental Physics. 1895. MILLER, D. C— Laboratory Physics. 1903. MILLIKAN, R. A. — Mechanics, Molecular Physics and Heat. 1903. NICHOLS, E. L. — Laboratory Manual of Physics and Ap- plied Electricity, 2 vols. 1894. NICHOLS, E. L., and FRANKLIN, W S.— Elements of Physics, 3 vols. 1898. OSTWALD, W., and LUTHER, R.— Physico - Chemische Messungen, 2nd German* edition. 1902. PRESTON," THOMAS— Theory of Heat. 1894. PRESTON, THOMAS— Theory of Light, 2nd edition. 1895 STEWART, BALFOUR and GEE, HALDANE— Practical Physics, 3 vols. 1898. THOMPSON, S. P. — Elementary Lessons in Electricitv and ■ Magnetism. 1905. TORY, H. M., and PITCHER, F. H.— Manual of Labora- tory Physics. 1901. WATSON, W.— A Text-book of Physics. 1900 EXPERIMENTS IN MECHANICS 11 EXPERIMENT 1. The Use of the Vernier and Micrometer Calipers and the Diagonal Scale. Object. — To learn the method of the vernier, and to deter- mine the dimensions of several objects and the volume of a regular solid. Apparatus. — Vernier caliper, micrometer caliper, diagonal scale, pair of dividers, brass cylinders, and pieces of wire. References. — Ames and Bliss, pp. 30-42; Ferry, Vol. I, pp. 1-14; Ganot, p. 4; Louden and McLennan, pp. 3-7; Miller, p. IS; Stewart, and Gee, Vol. I, pp. 14 and 45; Watson, p. 16; Physics "a" Manual, Experiment No. 4. Theory. — The linear dimensions of an object can be meas- ured directly by the use of a suitable scale, but the volume of a cylinder must be computed after the linear dimensions have been measured. The volume of a cylinder is vrH, where r is the radius and 1 is the length of the cylinder. The volume has to be computed after r and 1 have been measured. Since r enters into this relation as the square and 1 as the first power, it is evident that the first quantity must be determined with the greater care. The theory of the vernier should be developed with the aid of some of the references cited above. Manipulation.^Study the vernier and make a drawing of itj explaining its theory as suggested above. Take one of the hollow cylinders and measure its internal and its external diameters. At least ten independent measurements of these diameters should be made at different points along the cylin- der. Record the readings in centimeters and in inches. Take the mean of all the internal diameters and all the external diameters, both in centimeters and in inches. Record the results in tabular form as follows : DIAMETERS OF CYLINDERS Internal External | Inches Cm. Inches Cm. Means 1 i 12 Now take the mean of the centimeter readings and the mean of the readings in inches, and reduce the inches to centimeters. Measure the length of the cylinder in centimeters at least four different times and take the mean value. Calculate the area of the external surface and the volume of the cylinder. In measuring the internal diameter of the cylinder, the reading of the calipers must be increased by the width of the jaws of the vernier. This value may be determined by means of a second pair of calipers. Why does one need to measure the diameter with the greater accuracy? Now study the micrometer caliper and make a drawing of it. Measure the diameter of each piece of wire at least five times and then find the mean value in each case. Compare the observed values with the values given in standard tables for the size of wire used, and see what the percentage error is in each case. Now with the dividers and the diagonal scale, measure the length of two lines and the sum of their lengths. For this part of the experiment, select the lines on the inch side of the vernier caliper. Measure the distance AB, BC, and then the distance AC. Compare the value of AC with the sum of the values AB plus BC. From the measurements, assuming the unit on the diagonal scale to be 1 cm. (or 2 cm.), calculate the number of centimeters in an inch. In using the vernier and micrometer caliper, the zero reading of the instrument should be determined several times before taking any measurements. What is meant by the "least count" of a vernier? What is the least count in inches of the vernier used? In centimeters? What are the sources of error in this experiment? EXPERIMENT 2. The Traveling Microscope Calibration ot a Screw. Object. — To determine the pitch of a screw and .to find the number of centimeters in an inch. Apparatus. — Traveling microscope, steel scale with centi- meter and inch divisions. References. — Ames and Bliss, p.49; Miller, p. 26; Nichols' Lab. Man., Vol. I, p. 31 ; Stewart and Gee, Vol. I, p. 35. Theory. — No screw is perfect. The distance between the threads of a screw cannot be made exactly one unit dis- tance apart, — even though this unit be the fractional part of i;r inch or a centimeter^ — nor can the pitch along the length of 13 the screw be made constant. In this experiment the student is expected to determine the average pitch of the screw along its entire length. The screw is mounted upon a suitable base and carries a nut along its length as it is turned. Attached to this nut is a microscope which can be focused upon the scales to be compared. The scale to be measured is firmly fastened on the iron base of the instrument by means of clamps and set-screws. Care should be exercised to see that the scale is parallel to the direction of motion of the microscope on the screw. Focus the microscope on a division of the scale so that the cross hairs bisect the division thus : As the microscope is moved forward from one scale-division to the next, the turns and fraction of a turn through which the circular scale on the drum of the screw is turned, should be counted. It is not safe to depend on the scale on the base of the instrument. These observations should be repeated three or four times, care being taken to avoid the back-lash of the screw. Having determined the number of turns of the screw required to move the cross-hairs through one division of the scale, the distance through which the cross-hairs of the microscope are moved when the screw is turned one revo- lution can be determined. This is the pitch of the screw. Manipulation. — Adjust the centimeter scale on the base of the traveling microscope so that the cross-hairs of the micro- scope move parallel to the length of the scale when the^ screw is turned. Set the cross-hairs on one of the centimeter divisions near the end of the scale. Call this No. 1. Focus the microscope carefully so that there is no parallax. Take the reading on the circular drum of the screw and record. Set the cross-hairs on the next centimeter division of the scale, counting the number of turns and fractions of a turn through which the screw has been turned. Care must be taken to see that the screw is always turned in the same direction and that one always approaches the scale division on the scale from the same direction. Thus "back lash" in the screw is avoided. Call this division No. 2. Continue this until No. 5 is reached. Then move to division No. 10, counting the turns and the fractions of a turn through which the microscope is moved, just as was done above. Now take divisions No. 11 to No. IS 14 inclusive. Record these observations in a column parallel to the first. Then record the differences No. 11 — No. 1, No. 12 —No. 2, etc., in terms of turns and fractions of a turn. Each difference is a determination of the number of turns in ten centimeters. Take the mean value. Repeat, moving the screw in the opposite direction. Use these observations as above and take the mean value. With the mean of these two values, calculate the value of one turn of the screw. Express the lesult in millimeters. Use this value in the following operation : Adjust the inch scale in position under the microscope as above and set the cross-hairs on the inch, or half-inch lines, as the screw is ad- vanced. Take differences each time between lines four inches apart. At least five different sets of differences should be recorded. Read both ways along the screw as above, and take the mean value. Determine the number of turns in one inch, and from this value find how many centimeters in one inch. What is meant by "back lash" in a screw? What is parallax? How does one correct for parallax? What are the sources of error? EXPERIMENT 3. The Traveling Microscope — ^The Calibration of the Bore of a Glass Tube. Apparatus. — Traveling microscope, millimeter scale, glass tube to be calibrated, mercury, small crucible, thermometer, set of weights, and balances. References. — Ames and Bliss, p. 49; Ferry, p. 141; Miller, pp. 26 and 160; Nichols' Lab. Man., Vol. I, p. 31 ; Stewart and Gee, Vol. I, p. 35, Experiment No. 2. Theory. — The bore of a piece of glass tubing has various values at different points along the length of the tube. If one places a known mass of mercury in the bore of the tube and measures the length of the column of mercury, the average cross-section of the tube can be computed for that position of the mercury column. The density of mercury for the par- ticular room temperature at which the experiment is per- formed can be obtained from suitable tables. Knowing the density and the mass of the mercury, and the length of the mercury column, it is obviously an easy matter to determine the bore of the tube at different points. Manipulation. — Make the adjustments indicated in the previous experiment. By means of ten different measurements 13 determine the value of the pitch of the screw of the traveling microscope. This should be expressed in millimeters. See that the glass tube is clean and dry. Place in the bore of the glass tube a small globule of mercury approximately one-half a centimeter long. By means of a small piece of rubber tubing attached to one end of the glass tube, bring the globule of mer- cury so that one end of the mercury column is just in contact \\ ith a scratch placed near one end of the piece of glass tubing. The glass tube having been clamped in position, the cross- hairs of the microscope are focused on the scratch on the glass tube. This should also be the position of the end of the mer- cury column. Take the reading on the graduated drum of the screws. Set the cross-hairs of the microscope on the other end of the mercury column and take the reading on the drum as before. Now determine the number of turns and fractions ol a turn that one has to move the microscope in order to bring the cross-hairs from one end of the mercury column to its opposite end. Proceed as in the previous experiment. Now rnoA'e the mercury column along the tube about one centimeter and determine its length as before. At least ten different settings along the length of the tube should be' taken. Fre- t,uent settings on the scratch on the glass should be made to see that the tube has remained in position. In two parallel columns tabulate the data so as to show the length of the mercury column and the distance of the center of the mercury column from the scratch in the glass for each position of the mercury column. After completing this operation, remove the mercury and weigh it. Note the room temperature. Knowing the mass of the mercury and its density at the room temperature, and the length of the mercury column at differ- ent positions along the tube, the cross-section in each position can be found. In another column parallel to that containing the lengths of the mercury column, record the diameters of the tube in the different positions concerned. With these latter values as ordinates, and with the distances in milli- meters from the scratch as abscissae, plot a curve. This will be a calibration curve of the glass tube. From the curve, what will be the diameter of the tube just four centimeters from the scratch? Indicate the sources of greatest error. What quan- tities does one need to measure with the greatest accuracy? Why? 16 EXPERIMENT 4. The Comparator — The Determination of the Number of Cen- timeters in an Inch. Object.-^To make a study of the comparator and to de- termine the number of centimeters in an inch. Apparatus. — Comparator, standard meter, standard yard, and thermometer. References. — Ames and BHss, p. 55 et seq. ; Miller, p. 23 et seq. ; Watson, p. 19; Fig. 1. Theory. — The object whose length is to be measured is placed on one of the two long platforms of the comparator and the standard bar on the other platform. These platforms are long iron bars supported by points resting on a bed. This bed can be moved at right angles to its length by means of a rack and pinion. Thus, either the standard or the unknown unit can be brought under the two microscopes. The micro- scopes themselves can be moved along a way parallel to the platforms. By means of screws the platforms can be leveled and moved parallel to their own length. If it is required to nieasure the distance between two points, the line joining these two points is adjusted parallel to the standard bar, the latter being parallel to the platforms. Tlie cross-hairs of one microscope are focused on one of the points and the cross- hairs of the second microscope upon the other point. The standard bar is then moved under the microscopes and with- out moving either microscope, the platform is moved parallel to itself until some line of the standard bar comes directly be- neath the cross-hairs of one of the microscopes. If the other microscope has a micrometer eye-piece, its cross-hairs may be moved until they coincide with the nearest line on the stand- ard bar. By means of the readings on the head of the micro- meter screw of the microscope, the distance through which the cross-hairs are moved can be determined. Care must be taken to avoid any error that may be introduced by the back- lash of the screw, and care should also be taken that a complete turn of the screw is not added to (or deducted from) the actual distance that the cross-hairs have been moved. The number of divisions on the standard bar between the cross-hairs of the two microscopes should now be noted. From these data, the distances between the two points can be determined as soon as the calibration of the screw is known. In one of the microscopes on the instrument used, the cross-hairs are fixed and should be set on a line as in the 17 figure of Experiment No. 2 (ab and cd being the cross- hairs). In the other microscope, two of the cross-hairs, ab and cd, are fixed as in Fig. 1, and,cd should be set parallel to the lines on the scale of the standard bar. Two other lines, ef and gh of Fig. 1, can be moved by means of the micrometer screw, the motion being in the direc- tion ab. In calibrating the screw, the line ef may first be brought into coincidence with the center of a hne on the scale by turning the drum of the screw continuously in one direction. The reading on the circular head of the screw should then be noted. Then turning the screw always in the same direction so as to avoid back-lash, bring the same cross-hair in coinci- dence with the center of the next line of the scale, counting the turns and the fractions of a turn through which the screw has been turned in passing from one line of the scale to the next. The whole number of turns of the screw can be deter- mined by means of the piece provided with saw-teeth, mounted in the principal focal plane of the eye-piece. In this piece there is one tooth for every turn of the screw, every fifth tooth be- ing deeper than its neighbors. Fractions of a turn are deter- n'iined in the usual manner, by reading the position of the divided head of the screw. This operation should now be repeated with the cross-hairs moving in the opposite direction. Then repeat on other divisions of the scale until at least ten different sets of readings have been taken. Thus the screw can be calibrated either in inches or in millimeters, depending upon which scale has been used. In this experiment it is desirable to calibrate the screw in millimeters. Manipulation. — ^Study the instrument and learn how to move the platforms along their length and at right angles to their length, how to level them, how to raise or lower the plat- forms, how to move the microscopes at right angles or paral- lel to the platforms, how to focus the microscopes and how to turn the cross-hairs in the field. Now place the standard meter bar on the platform nearest the observer, being very careful not to touch the scale, and make the necessary adjustments so that: (1) Both microscopes are in focus. (2) As one microscope is moved along the ways parallel to the platforms, it remains the same distance above the surface of the scale of the meter bar, i. e., so that the micro- scope remains in focus as it slides along. (3) As the microscope is moved along the ways parallel to the platforms, the cross-hair ab runs right along the center 18 of the scale. If this adjustment cannot be made exact all along the length of the way, make it so near the two ends. (4) The motion of the microscope across the scale of the standard is at right angles to the length of the scale. (5) The movable threads ef and gh are parallel to the' lines on the scale. (6) The motion of the bar as the platform is moved for- ward or backward is parallel to the lines on the scale. The screw of the micrometer should now be calibrated. The method given above may be followed, or the following method may be utilized : Set the thread gh in coincidence with one of the millimeter lines of the standard bar and note the reading of the micrometer screwhead. Continuing to turn the screw in the same direction, bring the cross-wire ef in coinci- dence with the same millimeter line, and read as before. Now^ continue to turn the screw in the same direction as before until the cross-wire gh is in coincidence with the next millimeter line of the scale and take the reading. Now bring the cross- wire ef in coincidence with the same line and take the reading. Knowing the number of turns and fractions of a turn through which each cross-line has been moved, one has two calibra- tions of the screw. Repeat these settings on at least five other millimeter divisions of the scale. Find the average value, in millimeters and decimal fractions of a millimeter, of one divi- sion of the micrometer screwhead. Now place thfe standard yard on the other platform and adjust as the standard meter was adjusted. Also, adjust so that the intersections of the cross-hairs in the microscope without the movable cross-hairs will be exactly on one of the millimeter divisions at that end of the bar, and at the same time be on the zero division of the standard yard when the latter is brought under the cross-hairs of the microscope. Now adjust the cross-hairs of the second microscope on an inch line near the end of the standard yard, say the three-foot line, and then by turning the micrometer screw measure the dis- tance from the nearest millimeter line on the standard meter to the cross-hairs. Make at least four separate measurements, using different millimeter divisions each time. Repeat these measurements on the two-foot and one-foot lines of the stand- ard yard. From the data thus obtained, compute the number of centimeters in an inch. Indicate the sources of error. If possible, make correction for the temperature at which the observations were made. Does this increase or decrease the value of the determination? Whv? 19 EXPERIMENT 5. The Micrometer Microscope — The Measurement of the Dia- meter of a Glass Tube. Object. — To determine the pitch of a screw and to find the internal diameter of a glass tube. Apparatus. — Micrometer microscope, stage micrometer, pieces of glass tubing. References. — Ames and Bliss, p. 55 ; Miller, p. 20 ; Nichols' Lab. Man., Vol. I, p. 35 ; Stewart and Gee, Vol. I, p. 35 ; Ex- periments No. 2 and No. 4; Fig. 1. Theory. — The microscope is provided with an eye-piece having movable and fixed cross-hairs mounted at the princi- pal focus of the eye-piece so that both the object and the cross- hairs are visible at the same time. The movable cross-hairs are attached to a framework so that they can be moved by a screiy. This screw is provided with a milled head for turn- ing and a divided circular drum for measuring fractions of a turn. The whole number of turns can be obtained by means •of a fine "comb" mounted in the eye-piece, each tooth of which i« distant from its neighbor by the pitch of the screw. The cross-hairs ab and cd. Fig. 1, are fixed, while ef and gh can be moved across the field by turning the screw. The stage micrometer consists of a plate of plane glass with fine lines one-tenth of a millimeter apart ruled parallel to each other upon its surface. Manipulation. — Focus the micro- scope on a line of the stage microm- eter, just as in Experiment No. 4 one focused on the lines of the standard meter. Note the position of the wires on the comb and the reading on the scale of the circular drum. Now move the cross-hairs to the next Hne and record the position again. Repeat this ten times, using lines on different parts of the stage micrometer. Com- pute the pitch of the screw and determine the value in milli- meters of one division on the circular drum. Now measure the diameter of the bore of the several pieces of glass tubing in terms of millimeters and in terms of turns of the screw. At least five dififerent determinations of the diameter of the bore •of each tube should be made and the mean value taken. Re- f;^.i. 20 cord all the readings. What is meant by the calibration of the screw? Does the calibration of the pitch of the screw depend on the magnification of the microscope? Why is it necessary tc calibrate a screw? Name some of the defects of a screw. EXPERIMENT 6. The Cathetometer. Object. — To learn the use of the cathetometer, and tQ de- termine the number of centimeters in an inch through the measurement of the height of a barometer column. Apparatus. — Cathetometer, barometer, level and thermo- meter. References. — Ames and Bliss, p. 58 ; Kohlrausch, p. 76 ; Louden and McLennan, p. 8; Miller, p. 19; Millikan, pp. 107- 112; Nichols' Lab. Man., Vol. 1, p. 29; Stewart and Gee, Vol. I, p. 27 ; Watson, p. 20. Theory. — The cathetometer is an instrument used to meas- ure vertical differences in level at some distance from the ob- server. The scale is attached to a vertical support, and if the cross-hairs of the instrument are focused on any point of a distant object and then upon another point at a different level, the difference between the two readings on the vertical scale gives the height of the one point above the other. Manipulation. — Study the different adjustments of the cathetometer and the vernier on the scale. By means of a plumb-bob or a level, adjust the upright supporting the tele- scope until the column is exactly vertical. Then adjust the telescope until it is horizontal, as will be indicated by the level pttached to it. Bring the cross-hairs of the telescope into focus, and then focus the instrument upon a distant object so that there will be no parallax. Suspend the barometer at a suitable level so that it will be vertical and at a distance of several feet from the cathetometer. Now adjust the barometer and set the vernier. Take the reading of the barometer in inches. If the barometer has two scales use only the inch scale. Now set the intersection of the cross-hairs on the upper meniscus of the mercury in the barometer, and take the read- ing. Then set on the lower meniscus and take the reading again. The difference between these two readings will give the height of the mercury column in centimeters. Repeat this manipulation and all of these observations ten times and find the number of centimeters in an inch for each determination. Take the mean value. Find the greatest percentage of deviation of the observations from the mean. Ob- 21 serve the temperature of the room and correct the read- ings to zero C. From the data thus obtained, compute the number of centimeters in an inch. Indicate the sources of error. If no correction for temperature were made, would the value be too large or too small? Why? EXPERIMENT 7. The Spherometer. Object. — To measure the radius of curvature of a lens or ■X mirror. Apparatus. — Spherometer, lens, glass plate, dividers, and diagonal scale. References. — Ames and Bliss, p. 43 ; Louden and' McLen- nan, p. 10; Miller, p. 19; Nichols' Lab. Man., Vol. I, pp. 15-26; Stewart and Gee, Vol. I, p. 12; Ganot, p. 5; Watson, p. 459; Fig. 2. Theory. — The spherometer is an instrument having three legs with sharp pointed feet at the corners of an equilateral triangle, and also having a vertical screw with a large head di- vided into degrees, which passes down through the instrument so as to form the center of the triangle mentioned above. By means of this head, small advances of the screw can be read. Attached to the top of this screw-head is a lever to. indicate when the lower end of the screw makes contact with an object placed beneath it. When the instrument is on a perfect!}^ plane surface, the reading of the scale is called its "zero" reading. In general this will not be the scale-division marked zero. If any object is placed on the plane surface and the central point made to touch its upper surface while the three feet rest on the plane, the difference between this reading and the "zero" reading gives the thickness of the object. If the spherometer is placed on the surface of a lens, the difference between the reading then 22 obtained and the "zero" is the height of the surface of the lens above the center of the triangle formed by the feet of the in- strument. Let this height be called h. This is evidently a measure of the curvature of the lens. In Fig. 2, let P be the conical point of the screw, F one of the feet of the spherometer, and O, the center of the sphere of which the lens forms a part. In the figure, FB is perpendicular to OP, therefore PB=h. Let FB=a. Now OP=OF is the radius of curvature of the lens which we are to determine. Call this R. Then OB=R — h. The triangle FBO is a right triangle, therefore, (OF)^=(OB)=+(FB) = or, R==(R— h)^+a= a= h Therefore R= ) 2h 2 If the distance between the feet of the spherometer be called 1, it can be shown that the formula for the radius of curvature takes the form 1- h R=- + - 6h 2 Compute the value of a, using the distance between the feet and the fact that one has an equilateral triangle, as the basis from which to start. What is the percentage error in the re- sult, taking the mean measured value of a as the correct result? AVhat quantity is it necessary to measure with the greatest accuracy? Why? Manipulation. — Study the instrument- carefully. Now place the instrument on a piece of plane glass and find its "zero." To do this, turn the screw until its lower point touches the plane surface and raises the pointer on the lever at the top of the screw to the zero point of its scale. The screw may be turned either by means of the small milled head, or by taking liold of the upright support on which is the scale for the lever, but in no case should the student take hold of the screw itself or of the circular disk with its graduated scale. Read the zero to thousandths of a millimeter. Ten readings on different parts of the plane surface should be taken. Now measure the thickness of a piece of plane glass. Place the piece of glass on the plane surface under the conical point of the screw and take the reading again. Take five readings on one side and five on the other. Take the mean of these readings. Find the thick- 23 iiess of the piece of glass. What is the greatest percentage variation of the results from the mean? Now measure the thickness of the glass plate with a pair of calipers. Take ten readings oii different parts of the glass plate and take the mean. Do the two measurements of the thickness of the glass plate agree? If not, why not? Which is the larger? Why should this be so? Now place the spherometer on one side of the lens, noting which side, and take five readings with the instrument, as on the glass plate above. Use the mean of the readings thus ob- tained, and taking the "zero" found above, compute the value of h. Repeat, using the other side of the lens. Then place the spherometer on a sheet of paper and make an impression of the three feet and the central screw point on the paper. By means of a pair of dividers and a diagonal scale, measure the distances from the central point and the .three feet respectively. Then measure the distances between the three feet. Find the means of these and take the means thus obtained as the value of a and 1 respectively. Calculate the radius of curvature for both sides of the lens, using both formulae given above. If h is very small, --can be neglected. What will be the percentage error introduced here by neglecting h? Do not neglect it, however, in case of the lens that is used here. The value for the "probable error ' is given by the form- ula : — Ch =aV r( h— h.)' -f (h- h,)- + (h— h,)- + - (h— h„)^] n(n-l) J Where h is the mean value of all the separate values of h, and the values with subscripts are the separate values of h, and n is the number of the separate determinations of h. In this case n is five for each side of the lens. Now assume the probable error in a to be 0.1 mm. Find the probable error in h and R. The probable error in R is expressed thus : — i e„ = V (erf en Define probable error. EXPERIMENT 8. Linear Acceleration. Object. — To determine the laws of falling bodies, and to test the equations of uniformly accelerated motion of transla- tion. 24 Apparatus. — Linear acceleration apparatus (Gaertner pattern), glass plate, "Bon Ami," meter-stick, pair of dividers, diagonal scale, and plumb-bob, or level. References. — Ames and Bliss, p. 70 ; Carhart, Vol. 2, p. 16 ; Miller, p. 74; Millikan, p. 1 ; Nichols' Lab. Man., Vol. I, p. 55; Fig. 3. Theory. — The velocity of a body starting from a given point with an initial velocity, v^, and moving for a time, t, with uniformly accelerated motion, is given by the equation : — (1) v=v„+at where a is the acceleration with which the motion of the body is being accelerated. The distance, d, traversed in time, t, is given by the equation : — (2) d=v„t+J.^at= In the apparatus used in this experiment, the prong of a vi- brating tuning fork which slides down between vertical ways, carries a stylus that presses lightly against the surface of a glass plate. The surface of the glass plate is covered with a thin coat of "Bon Ami" and the stylus, as it falls, leaves a trace of its path on the glass plate. At the top of the apparatus the fork is held in place by means of a clamp, which also keeps the prongs slightly spread. Turning the handle of this clamp releases the fork, which falls unimpeded with the prongs in vibration, until it is stopped by dash pots at the bottom of the apparatus. The stylus thus makes a wavy trace on the glass. If Ai, Aj, A,, Ai, Fig. 3, are successive wave-crests, the dis- tances Aj-Ao, A2-A3, Ag-Aj, are wave-lengths that have been traversed in equal intervals of time, viz., during one complete vibration of the fork. Taking the time of one complete vibra- tion of the fork as the unit of time, we can find the average velocity during this interval of time. (A2-A3) — (Ai-A„) is the acceleration during the second interval, (A.,-A4) — (A,-A3),the acceleration during the third interval, etc. These differences should all be the same if the acceleration is constant, as it should be in the case of a falling body. If this is found to be constant, it proves that equation (1) is true, where a is the con- stant of acceleration. This will give the value of the acceleration in terms of the period of vibration of 'the fork as the unit of time. In order to change to centimeters per second per sec- ond, the frequency of the fork and the manner in which the numerical vahie of the acceleration varies with the magnitude of the unit of time, must be known. The frequency of the fork is stamped upon it. The law ol vibration is given by the defining equation — (2). An ex- perimental determination of this law is a verification of equa- tion (2). The following is a simple test of this equation, which is independent of the value of a. Let dj, do, dj, be the distances traversed by the fork in time t,2t,3t, etc., respective- ly, gravity being the accelerating force. According to the law of freely falling bodies, since the distances d^, do, dg, etc., are all measured from the same starting point, the following relations hold : d,=v„t+l/2gt= d,=2v„t+4/2 gt^ d3=3v„t-f 9/2 gt= etc., etc., etc., where y^ is the initial velocity of the fork at the time when the observations were begun. Subtracting each of the above equations from the next succeeding, the successive differences represent the distances passed over in successive intervals, or the average velocities acquired during any particular in- terval of time. Representing these average velocities by Vj, v„, V3, etc., we have d— d„=Vi=Vot+l/2gt= d,— d,=v,=Vot+3/2 gt== d3—d2=V3=Vot -1-5/2 gt^ etc., etc., etc.. In like manner subtracting the successive values of v^, Vj, V3, etc., we get the acceleration acting upon the fork in terms of centimeters per period per period. Representing this by a, we have V2— Vi--=a=gt2 V3— V2=a=gt= etc., etc., etc., or, the acceleration is proportional to the square of the unit of time. Thus, if successive values of a are found for any given interval of time, the value of gravity can easily be de- termined. This operation is to be performed with the data obtained. A second simple test can be carried out according to the method indicated in the succeeding experiment on angu- lar acceleration. This should also be done by the student, as it affords an independent test of the accuracy of his measure- ments. 26 Manipulation. — Adjust the instrument so that the ways are vertical. Then remove the plate and cover it with a uni- form thin layer of "Bon Ami." Replace the plate and adjust the stylus so that when the fork falls a clear, fine trace is left vn the prepared surface. Clamp the fork at the top and release it so that it will produce a wavy trace on the plate as it falls, in this way obtain three good traces. Then remove the plate and mark with a pin or a sharp pencil each tenth vibration. Then with a meter stick, measure the distances A^ A^, A^ A3, Aj Aj, etc. Tabulate the total distances, their successive dif- ferences which represent the average velocities, v, and the suc- cessive differences of the latter which represent the accelera- tions, a. The latter values should be almost constant. Find the mean value of the accelerations and the greatest percentage deviation of any one value from this mean. Now record in another table the values of v and their differences, the accel- eration, a, when the unit of time is the period of ten vibrations ; also when the period is fifteen vibrations, and when the period is twenty vibrations. By comparison of the mean values of the accelerations for different units of time, deduce the law that connects the numerical value of the acceleration with the lengfth of the unit of time. From this value and the known frequency of the fork, determine the value of g, the acceleration of a freely falling body expressed in centimeters per sec. per sec. If the value of a does not appear constant along the whole length of the plate, what reason can be assigned for the variation ? Using the value found for g, calculate : — (a) The veloc- ity with which a body must be thrown upward so that it will rise 100 meters; (b) the time that will elapse before it will reach the ground again; (c) the velocity it will possess 50 meters from the ground going up and also coming down. EXPERIMENT 9. Angular Acceleration. Object. — To study the motion of a body moving with uni- form angular acceleration, and to determine the angular veloc- ity and acceleration of the body. Apparatus. — Angular acceleration apparatus (consisting of an iron wheel and axle, and a tuning fork with a stylus mounted upon it), cord, camphor, strip of paper, diagonal scale, pair of dividers, set of weights, and pair of outside calipers. 27 References. — Ames, p. 28; Ames and Bliss, p. 79: Carhart, Vol. II, p. 85 ; Fig. 3. Theory. — The angular velocity of rotation of a body at any instant, when its angular acceleration is constant, is given by the equation : — (1) (D=(o„-|-at where w is the angular velocity at any time, t, after the angu- lar velocity was ^3 — ^2' etc., are the average angular velocities in these intervals, given in terms of one vibration as the unit of time. Call these ojj, o>.,, etc. Then the differences u^ — Wj. 0)3 — and a. Then take ten vibrations as the unit of time, measuring the successive values of w for ten steps at a time instead of five. Then do the same for fif- teen vibrations at a step. Tabulate the values of ^Mia^) iCI+M^r^+J^M.a^) (F) To find I, take off both cylinders and find T as above. Now it was seen that 'r-=A^n :K EHminating K we get : — Ti^:T==(I+M,r^+i4Mia=) :I (G) Substitute this value of I and the values of K, T, M, and r in equations (1), (2), and (3), and see if they are satisfied. EXPERIMENT 12. Moment of Inertia. Object. — To determine the moment of inertia of a regular solid about a given axis. Apparatus. — Rectangular bar with wire suspension, ring ct rectangular cross-section, two cylinders, set of weights, platform balance, stop-watch, steel scale, vernier caliper, pair of dividers, and diagonal scale. References. — Carhart, Vol. II, p. 84; Ferry, Vol. I, p. 91 ; Miller, p. 104; Millikan, p. 78 et seq. ; Stewart and Gee, Vol. I, p. 243; Watson, p. 93. ^"W-Q^ "i-^- 33 Theory. — Let the bar (see Fig. 4.) be suspended from a support by means of a wire, and made to oscillate about an axis passing through the center. It will then oscillate with angular harmonic motion. If its moment of inertia, I, is not known, the period, T, obtained by vibrating the bar alone, and then the period, T^, obtained when a known moment of inertia, I^, has been added to that of the bar, can be observed and then I can be obtained by elimination. In the first case we have the relation : — (1) T==4,r^I:K and in the second case we have the relation : — (2) T,^=4,r=(I+I,) :K Eliminating either I or K we get (3) I=IiT=:(Ti=— T^), and K=47r=I:(T,=— T=) The moments of inertia of the solids used in this experiment are as follows : The moment of inertia of a uniform cylinder of radius r and mass M when suspended from a point in its ■ geometrical axis is represented by I„=i^Mr^ For a hollow cylinder having an inner radius, r, and an outer radius, r,, the moment of inertia, I,, is represented by L=>^M(r,^+r^) P"or a rectangular bar of length 1 and width a, the moment of inertia, I , is represented by the expression I = ^M (a= + 1=) If the moment of inertia of any body about an axis through the center of gravity is I,,, the moment of inertia about a paral- lel axis at a distance x from the former axis will be expressed "by the relation : — I,=I„+Mx= Manipulation. — Measure the distance x. Suspend the tar and start it vibrating. Avoid a sidewise swing. Deter- -mine T by taking the time of 50 vibrations with the stop- watch. Place the two cylinders just over the circles drawn •on the bar. Determine Tj as before. Calculate Ij, the moment -of inertia of the two cylinders about the axis of suspension. Find the value of I and of K for the bar. Place the ring on the bar and determine the new period Tj. Compute the moment of inertia of the ring, I2, and substitute in (3) and •solve again for the values of I and K for the bar. Take 34 the mean of the two values of I thus found. Now calculate I fronE the equation for the moment of inertia of the bar. It should be- noted that in the second set of relations for the values of I and K, the quantities Ij and T^ take the place of the quantities. Ij and Ti in the first solution for I and K. How does the com- puted value of I agree with the experimental value? What; is the percentage error in the observed value? How could, one determine the moment of inertia of an irregular body? Tabulate data as follows: TTMK TN SECONDS FOB 100 VIBEATIONS No. With Bar With cylinders on bar With ring on bar 1 2 3 564 sec. 560 sec. 556 sec. Averaee 560 sec. Period 5.60 sec. DIMENSIONS, MASSES and MOMENTS of INERTIA of APPARATUS Object Length Radius (width) Mass MOMENTS OF INERTIA Computed Observed Ring I,= 1= Cylinder Io= 1= Il= 1= Bar 1= EXPERIMENT 13. Moment of Inertia. Object. — To determine the moment of inertia of a body by means of angular acceleration. Apparatus. — ^Angular acceleration apparatus, gum cam- phor, matches, pair of tongs, weight, set of weights, cord. References. — Carhart, Vol. II, p. 84 et seq. ; Millikan, pp. 78-83 ; Stewart and Gee, Vol. I, p. 242 ; Watson, p. 93. • Theory. — If a force is applied to a body so as to produce rotation, the angular acceleration will be expressed by the relation, o=Fr-^I, where I is the moment of inertia and Fr is- the moment of force acting on the body. The moment of in- 35 ■ertia can also be obtained by measurement of the mass and •dimensions. For a circular disk the formula would be J/aMr^. The values of I thus obtained should be equal. Manipulation. — With the burning camphor smoke the side •of the metal disk that does not have the graduations upofi it. Wrap a fine cord three or four times about the circumference •of the large disk, not around the small shaft. To the free end -of this cord attach sufficient mass to just equalize the friction of the ball bearings and of the stylus as it bears against "the face of the disk. Connect a battery to the terminals of the tuning fork circuit, and start the fork to vibrating. Add 100 grams to the mass attached to the cord and suddenly release the disk, at the same time moving 1;he fork forward along the track so as to prevent overlapping of the traces. As soon as the weight, m, touches the floor, stop the disk, remove it from the frame, and carefully mark off groups of 50 wave lengths as in Experiment No. 8. Now replace the disk in the frame, set the cross-hairs of the low-power microscope upon the marks that have just been made upon the blackened surface of the disk, and read the magnitude of the angles between successive settings with the liigh-power microscope on the graduated side of the disk. In this way read the position of ten different series of 50 waves •each. Make two more traces and repeat. Make m equal to _200 and 300 grams respectively for these last two traces. Find a. The force F which produces the rotation is the tension •on the cord. The force acting upon the mass, m, is mg, but part of this is expended in giving the mass, m, an acceleration, a. Therefore mg^F-i-ma. Hence F=m (g— a) =:m (g— ro) .Measure the radius, r, of the disk and find ]=--Fr^r-a. Weigh the disk and show that I=>^Mr== liow do they agree? EXPERIMENT 14. Atwood's Machine — Gaertner Pattern. Object. — To verify the laws of the motion of a body under the action of a constant force. 36 Apparatus. — Gaertner Linear Acceleration Machine with attachments and weights, glass plate, Bon Ami, set of weights, meter stick or diagonal scale, and pair of dividers. References. — ^Millikan's ^Mechanics, Molecular Physics and Heat, pp. 17-18; Experiment No. 8. Theory. — If a body is acted upon by a constant force, it will move with uniformly accelerated motion of translation, the acceleration being proportional to the force and inversely proportional to the mass of the body moved. In the case of falling bodies the force varies as the mass of the body, and hence the acceleration of all falling bodies (at the same point) IP the same. Observations of freely falling bodies, however, do not enable us to determine the relation between force, mass and acceleration. Thus if two masses, m, and m^, are dropped downward from a given position, the fact that they fall shows that they are acted upon by certain forces which can be designated by i^ and f,. Since they both fall at the same rate, these bodies may be said to be acted upon by a constant acceleration which will be called g.' From Newton's Second Law of Motion, it follows that fi :f,=mig :m2g=mi -.m. If, now, each of these masses fall and drag with it a mass, m, which would otherwise remain at rest, the same forces, f^ and fj, produce motion of the combined masses. For example, if a cord having a mass, m, attached to both ends be hung over a frictionless pulley and the mass, m^ be placed on either side, the moving force will be fj, the mass moved will be (mi-|-2m) and the acceleration imparted to the system will be a measurable quantity, ai. If the masses on the pulley be restored to their original position and the mass, m,, be placed on one side, the moving force acting on the masses, (m2-|-2m) will be fa, and the imparted acceleration will be a^. As above, it follows from the Second Law that fi :f2=(mi-|-2m)ai :(mj,-f 2m)a2 from which it follows that mi:m2=(mi4-2m)ai:(mo+2m)a2. This is one of the relations that is to be verified by experi- ment. If m, mi, and z^, or m^ and a^ are determined, the value of g can also be found, since fi=ra,g=(mi-t-2m)aj, and f2=m2g=(m2+2m)a. 37 Manipulation. — Level the instrument as in Experiment 8. Weigh the fork and frame and replace them in position. At- tach the frictionless pulley to one of the standards of the machine so that it will allow the cord attached to the weight- holder to pass over freely when the free end of the cord is attached to the fork and frame. Adjust the mass, m, so that the fork, when it is given a slight downward motion, will continue to move with uniform velocity, but without accelera- tion. This will eliminate the friction of the pulley and the friction between the fork and frame and the standards. If the weight, m, on the weight-holder just balances the fork and frame, the removal of any weight, mi, will cause the frame and fork to fall with an acceleration which can be measured according to the method of Experiment No. 8. The moving force in dynes will be mig, and the mass moved will be (m — nil) plus the mass of the fork and frame. Now remove a mass, nil, (say 200-400 grams), from m and obtain two traces as in Experiment No. 8. Repeat, when m has been diminished by a second mass, approximately the same as m^. The moving force in the first case is mjg, in the second case, mjg. Meas- ure the acceleration in both cases as Avas done in Experiment No. 8, taking the time of 20 vibrations of the fork as the unit of time. Now determine the mass of m, m^ and m^. Tabulate all observations and data in suitable form and record the mean values of the accelerations for each trace, and take the average of the two accelerations found by using m^. Call this a^. Do the same with the acceleration obtained with m^ and call this a^. Determine the value of a^ and z^ in cm. per sec. per sec. also. If now the mass of the fork and frame is denoted by M, we have the relations mig=[A'I-|-(m — mjjai, and mag— [M-(-(m — mjja, from which it follows that fi:f2=mi:nl2=[M+(m— mj]a, :[M+(m— m2)]a, IS this relation verified? What is the percentage error? EXPERIMENT 15. The Atwood's Machine. Object. — The determination of gravity by means of the Atwood's Machine. Apparatus. — Weights, rider, meter stick, and set of weights. 38 References. — Ames and Bliss, p. 98 ; Carhart, Vol. I, p. 12 et seq. ; Ganot, p. 65 ; Hastings and Beach, p. 28; Miller, p. 77 ; Nichols' Lab. Man., Vol. I, p. S3; Nichols and Franklin, Vol. I, p. 40. Theory. — ^The Atwood's Machine consists essentially of a vertical standard about two meters high, on the top of which is mpunted a pulley in such a way as to make the friction of the bearings as small as possible. Over the pulley passes a light cord to which weights are attached. If equal weights are suspended from both sides of the pulley, the wheel will remain at rest, but if a small additional weight be placed on one side, the heavier weight will fall with uniformly accel- erated motion. The force of gravity acting on this small addi- tional mass is utilized to set the two weights on both sides of the pulley in motion. The resulting acceleration is conse- quently much less than if the small mass alone fell. By suit- ably varying the masses on the pulley the resulting motion can be made as slow or fast as desired. The apparatus can thus be used to determine the value of the acceleration of gravity. For measuring time the machine is provided with an electromagnet and sounder, which are connected in circuit with a second's pendulum. By means of the electromagnet placed at the top of the vertical standard, the weights can be released at the beginning of a second. An adjustable bracket on the standard permits one to remove the rider from the weights at the end of any second, while the weight itself con- tinues in its motion. If the mass of the rider is m, the force producing motion of the weights is mg. This force equals the total mass moved multiplied by the acceleration acquired. If the masses on both sides of the pulley be 2w and M, the equivalent mass of the jjulley, we have mg=(m-|-2w-fM)a where a is the acceleration acquired by the moving masses. As soon as we have determined 2w, m, and M, and the value of a, the value of g is known. The values of m and 2w are found by weighing, and the value of a is computed from the observed measured distances fallen by the rider in a given time. The value of M can be eliminated from two equations and the value of g then found, or M can be found graphically, accord- ing to the following method : — Manipulation. — Adjust the weights on the cord passing over the pulley of the machine so that if one gives either a slight downward motion, the system will continue to move uniformly. This can be done by placing tin-foil between the weights so 39 as to overcome the friction. Place the rider in position and determine the distance fallen through in one, two, three, etc., seconds respectively. Remove a weight from each side and repeat the process. Continue this procedure until all the small weights are removed. Determine the masses on the cord for each observation. As the distance fallen in a given time, with given masses on the cord, take the mean of four independent settings of the platform. From the mean values, compute the acceleration in each case. Now plot a curve upon cross-section paper, using total masses hung on the pulley as abscissae, and the reciprocals of acceleration as ordinates. The curve should be a straight line. What is its equation ? The curve should pass a.*-, nearly as possible through all the points plotted on the paper. It will have an intercept on the x-axis Which will be the equivalent mass of the pviUey. Why is this so? What is the meaning of the slope of the line? Find the value of the acceleration of gravity from the curve. How does its value compare with the mean of the separate determinations of g? Tabulate results as in the following table, where x is the distance traversed in time t and a is the acceleration with which it moves. Msss in gms. 100 gms. 140 ems. 180 gms. 220 gms. 260 gms. Time in sec. X a X a X a X a X a 1 sec. 2 sec. 3 sec. 4 sec. Mean accelerations Reciprocals of accelerations Value of g EXPERIMENT 16. The Reversible Pendulum — The Determination of Gravity. Object. — The determination of the acceleration of gravity by means of the reversible pendulum. Apparatus. — Reversible pendulum, seconds clock, relay, comparator, standard meter, meter stick. 40 References. — Carhart, Vol. II, p. 84; Ganot, p. 69; Miller, p. 84; Millikan, p. 93; Stewart and Gee, Vol. I, pp. 247-255; Watson, p. 131 ; Fig. 9. C) t:]e f;.,.*^- 1 Theory. — In the pendulum used in this experiment, two heavy bobs, A and B, and a slider, c (see Fig. 9), are movable along the rod. Suspending the pendulum from the knife-edge, a, will give to it one period, T. Suspending it from the other knife-edge, b, will give to it the period T^. By adjusting A and B, and finally c, these two periods may be made equal. It is, however, a very tedious process to make them equal. It is more satisfactory to adjust until the periods are nearly the same. Let the distance from the knife-edge, a, to the center of gravity be called h ; from the edg«, b, to the center of gravity be called hj. Then T==4H-=k=-^gh, and T^^=4ir-k^--^gh^ where k is the radius of gyration about a, and k^ is the radius of gyration about b. Therefore Show that k=— ki2=h2— hi^ 4,r= T=+Ti= T=_T^= g 2(h+hJ 2(h-h,) where h-j-hi is equal to the distance between the knife- edges and the quantity h — h^ may be found by balancing the pendulum on a knife-edge. The distances from the center of gravity and between a and b should be measured with the comparator. 41 Manipulation. — ^Two pendulums are said to swing in coin- cidence when both pass their equiHbrium positions in the same direction and at the same instant. If their periods are not the same, they will gradually get "out of step" until one has gained a complete vibration over the other. They will then be in com- plete coincidence again. Note the time by the clock at the beginning and at the end of the coincidence. Take the mean as the time of coincidence. Allow the pendulum to swing until a second coincidence is obtained as above. The time between the two coincidences should be about ten minutes. Now re- \erse the pendulum, set it into vibration when suspended from the other knife-edge, and take the time between coincidences as before. The times between coincidences should be nearly the same for the two knife-edges. The difference should be less than thirty seconds. If it is not, adjust A, B, and c until it is so. J\Iake five different determinations of both T and T^. Balance the pendulum on a knife-edge, and measure h and h^ with a meter stick, and h-(-h^ with the comparator. Deter- mine h-|-hi several times. Substitute these values in the equation and determine the vailue for the acceleration of gravity. EXPERIMENT 17. The Physical Pendulum — The Determination of Gravity. Object. — To determine the value of the acceleration of gravity by means of the physical pendulum. Apparatus. — Physical pendulum, screw driver, meter stick and watch. References. — Nichols' Lab. Man., Vol. I, p. 67. Theory. — The physical pendulum consists essentially of a- cylindrical bar of metal suspended from a support by means of a pair of knife edges so that it can swing about an axis at right angles to its geometrical axis. If the pendulum be displaced from its equilibrium posi- tion through an angle, the moment of force acting on the pen- dulum will be represented by the expression MgR sin p, where p is the angle of displacement. If now the angle is so small that sin p can be replaced by the angle expressed in radians, the moment of force will be proportional to the displacement and the pendulum will therefore have simple harmonic motion. The work done in displacing the pendulum through a small angle, p, will then be the average moment of force mul- tiplied by the angular displacement, or the potential energy 42 of the pendulum at its highest point will be represented by this -quantity, that is Potential Energy=Ei=>4MgR/>= From the law of the conservation of energy, the potential ■energy of the pendulum at the highest point of its vibration is equal to its kinetic energy at the lowest point of its vibra- tion, provided the energy dissipated can be assumed to be neg- Jigible. Assuming this, the energy of the pendulum at its lowest point is entirely kinetic energy of rotation, and can be represented by 'Since the pendvilum has simple harmonic motion we have o)=27rp :T Therefore E,=2,r=p=I:'P Since Ei=E,, we have MgRp^=47r^P^l:T\ or g=4,r^I:MRT= Where M is the mass of the pendulum, R is the distance from the knife-edges to the center of gravity of the bar, T is the period of vibration, and I is its moment of inertia with respect to its axis of suspension. But the moment of inertia, I, is I=M[^-hL-|-h''-|-|^] Where L is the total length of the bar, h is the distance from the knife-edges to the top of the pendulum and a is the radius of the bar. Therefore, 4^ TV RT^ [f -hL+h^+ f] Manipulation. — Firmly adust the knife-edges a few centi- meters from the top of the pendulum and then place the pen- dulum in position in the bracket. Displace the lower end through a small angle. Note and record the time (the hour, minute and second) of the 1st, 101st, 201st. . . .701st vibration. Repekt these observations again. Measure the length of the pendulum and the distance of the knife-edges from the lower ■end. Now adjust the knife-edges a few centimeters from the cejiter of the pendulum and repeat the observations taken above. Record and tabulate the data in suitable form. . 43 Now subtract the time of the 1st vibration from the time- ot the 701st vibration, the time of the 101st vibration from the time of the 601st vibration, the time of the 201st vibration from the time of the 501st vibration, the time of the 301st vibration from the time of the 401st vibration. If one then add all these time differences together and divide by 1600, he will have the time of one complete vibration. Carry out the results to the thousandths of a second. Substitute the proper values in the equation given above and determine the value- of g for each position of the knife-edges. Find the mean of the two values. Why does one not need the mass of the pen- dulum? What quantities should be determined most accu- rately? Why? What will be the efifect on g of an error of one per cent, in T? EXPERIMENT 18. The Determination of g by Means of Kater's Pendulum. Object. — The determination of the value of the accelera- tion of gravity by means of a simple form of Kater's pendulum. Apparatus. — Kater's pendulum, meter stick, watch or stop watch, simple pendulum. References. — Experiment No. 27 and references of the same. Theory. — Consult Nichols' Lab. Man., Vol. 1, p. 69. Manipulation. — About ten or twenty centimeters from one end of the pendulum, fasten one pair of knife-edges. Deter- mine the approximate period of vibration of the pendulum when it is suspended from this pair of knife-edges. This can be done by timing twenty vibrations with the stop-watch. Now suspend the simple pendulum from the pendulum sup- port and adjust its length until it vibrates in unison with the Kater's pendulum. This locates approximately the center of oscillation of the pendulum. The length of the simple pendu- lum is approximately the distance from the center of suspen- sion of the bar to the center of oscillation. Now set the second pair of knife-edges at a distance from the first pair equal to- the length of the simple pendulum. Suspend the bar from this second pair of knife-edges and take- the period of vibra- tion as above. The period should be the same as above. If the two periods differ appreciably, the knife-edges can be shifted slightly until the time of vibration of the pendulum from both sets of knife-edges is the same. Carefully take the time of vibration with the pendulum suspended from each pair of knife-edges. At least 500 vibrations should be timed in each 44 case. The method of the previous experiment is suggested ioT obtaining the period of vibration with a watch. Now meas- ure carefully the distance between the knife-edges and com- pute g according to the method indicated in the reference cited above. The formula g=4^(R+RJ :T= may be used.where R is the distance from the center of grav- ity to one pair of knife-edges, Ri is the di.stance from the cen- ter of gravity to the second pair of knife-edges, and T is the period of vibration. EXPERIMENT 19. The Variation of the Periodic Time of the Physical Pendu- lum as the Position of the Knife-edges is Changed. Object. — To determine graphically the relation between the periodic time of a physical pendulum arid the distance between the position of the knife-edges and the center of gravity of the pendulum. Apparatus. — Physical pendulum, meter stick, stop watch, and screw driver. References. — Nichols' Lab. Man., Vol. 1, p. 72. Theory. — From the manipulation in Experiment No. 17, it will be seen that the period of vibration of the pendulum depends upon the distance of the knife-edges from the center of mass of the pendulum. It will be the purpose of the present experiment to study this and to determine the distance from the center of mass that the knife-edges must be placed in order to secure the minimum periodic time. Manipulation. — Adjust the knife-edges near the upper end of the pendulum. Suspend the pendulum by its knife- edges from the bracket and set it swinging through a small arc. With the stop-watch take the time of 100 vibrations. As a check repeat for another 100 vibrations. Now place the knife-edges four or five centimeters nearer the center of the pendulum and repeat the above operation for two sets of 100 vibrations each. Again set the, knife-edges four or five centi- meters nearer the center of the pendulum and take the time of vibration as before. Repeat this procedure until the knife- edges are at the center of mass of the pendulum. From the data thus obtained find the periodic time of vibration of the pendulum for each position of the knife-edges. With these values as ordinates and the distances of the knife-edges from the center of the pendulum as abscissae, plot, on cross-sec- tion paper, a curve to suitable scale. From the curve show at what distance from the center of the pendulum the knife- 45 edges should be placed in order to have the minimum per- iodic time (period of vibration). For any two observed val- ues of the time of vibration of the pendulum, compute the value of gravity according to the method indicated in Ex- periment No. 1^. EXPERIMENT 20. The Coefficient of Friction. Object.— To determine the coefficient of friction between two different surfaces. Apparatus. — Smooth iron plate, wooden block with screw- eye, cord, weights, level, two weight-holders, set of weights, pulley. References. — Ames and Bliss, p. 146; Ferry, Vol. 1, p. 84; Hastings and Beach, p. 77; Miller, p. 112; Nichols and Frank- lin, Vol. 1, p. 101 ; Nichols' Lab. Man., Vol. 1, p. 44; Watson, p. 111. Theory. — In order to make a mass move over a horizon- tal surface a force parallel to the plane must be applied to the body that is moved. The applied forte will depend upon the nature of the two surfaces in contact and upon the mag- nitude of the normal force acting upon the mass. The mag- nitude of'the force required to start the body will be greater than the force required to keep the body moving with a uni- form velocity after it has once been started in motion. The ratio of the tangential, or moving force, to the weight moved, or the normal force acting upon the horizontal surface is called the coefficient of friction, or F '*= N where /u is the coefficient of friction, F is the moving force and N is the normal force acting upon the surface. This con- stant is called the coefficient of static friction when F is the force required to start the body in motion ; and the coefficient of kinetic friction, when F is the force required to keep the body moving with uniform velocity when once the body has been set in motion. It will be the purpose of this experiment to determine the coefficient of kinetic and static friction be- tween iron and a block of wood. Manipulation. — Carefully adjust the iron plate on a table so that its upper surface is horizontal, and place the wooden block upon it. At one end of the plate fasten the pulley to the table so that a cord fastened to the screw eye of the block and to the weight-holder will be horizontal when passed over 46 the pulley. Weigh the weight-holder. Place weights o£ 4 Kg, or 5 Kg. upon the block. Add sufficient weights to the weight-holder until the block begins to move. Repeat this with loads increased by 4 or 5 Kgs. until a load of 40 or 50 Kgs. is attained. Record the results in each case. Now repeat all this procedure adding sufficient weights to the weight-holder each time so that when the block is started in motion it will continue to move with a uniform velocity. Record these results. From the data thus obtained, the co- efficient of static friction and the coefficient of kinetic friction can be determined as soon as correction has been made for the friction of the pulley. To correct for the friction of the pulley, attach a weight- holder to the end of the cord which was attached to the screw- eye and hang the cord with its attached weight-holders over the pulley. Add one kilogram to each weight-holder. Then add small additional weights to one side so as to give uni- form velocity when the masses are started in motion. Re- peat, adding the small masses to the opposite side of the pulley. Record the results and take the mean. The masses of the weight-holders should be known. Add another kilo- gram to each side of the pulley and repeat this procedure until masses of 5 or 6 Kg. are suspended from either side of the pulley. This will afford sufficient data to correct for the friction of the pulley. "With the data last obtained plot a curve with the aver- age moving forces as ordinates and loads on the pulley as abscissae. What is the curve? What is its equation? What is its slope? Determine the coefficient of friction of the pulley. It will be noted that the force required to produce motion of the masses on the iron block included the force required to overcome the friction between the two surfaces and also the force required to overcome the friction of the pulley. It will also be noted that the friction of the pulley is dependent upon the normal pressure upon the pulley Knowing the coefficient of friction of the pulley, correct the moving forces in the first two cases so as to eliminate the friction of the pul- ley. For each case plot a curve with these corrected moving^ forces as ordinates and loads on the woo3en block as abscissae. What is its equation? What is its slope? ^^^hat is its inter- cept? Does it have an intercept? Find the coefficient of friction between the wood and the iron surfaces. What is the physical meaning oi //.? 47 EXPERIMENT 21. The Force Table. Object. — To verify the laws of equilibrium of three or more forces acting at a point. Apparatus. — The "Force Table" and attachments, set of weights. References. — Ganot, p. 18 ; Millikan, pp. 21-26 ; Watson, p. 80. Theory. — If several forces acting at a point are in equi- librium, their geometrical sum must be zero. The sum of the components in the x-direction must also be zero, or expressed mathematically, we have the relations : — X,+X,+X3r=0, and Y^+Y,+Y3=0 or, fiCosSi-|-f2Cos^,-j-f3Cos^3=0 and iiSm6i-\-i.^sin9^-\-igSmdg=0 From this we have that f.sin^, -l-f^sin^, tan6', ^^3^^ -- fjCOS^i-j-faCOS^a also f3==fr+f.^+2f,f,cos(e— e,) Manipulation. — Set two of the pulleys at the angles 30 degrees and 150 degrees. Calculate the angle and the magni- tude of the third force when the masses have the values given in the following table. On removing the central pin, be sure that the balance is not due to friction. Displace the juncture, and tap the stand to see that the position obtained is the true equilibrium position. If not, change the force or the angle, or both, until you get the proper values. Record the experimental and the theoretical results as in the table below. Repeat with the angles and the forces indicated in the table. Calculate the values by the equations, and also solve all by construction. Put a paper underneath the cords and draw the lines. Take f, and f^ as given, and solve geometrically for f^. 48 TABLE. ^1 e. M. M, Ma (comp) M3 (obs) h (comp) ^3 (obs) 45° 240° 100 g 150 g 60° 180° 200 g 100 g 30° 330° 200 g 50 g 30° 150° 100 g 200 g 0° 90° 150 g 200g 0° 90° 100 g 100 g Complete the above table and give all of the graphical solutions. EXPERIMENT 22. Equilibrium of Three Forces. Object. — To verify the law of the parallelogram of forces for the addition of tvvro forces. Apparatus. — Two pulleys, cord, frame attached to the wall, stick, set of weights.- References. — ^Ames, p. 68 : Ames and Bliss, p. 122 ; Ganot, p. 20; Miller, p,49"; Fig. 6. Theory. — If three forces acting at a point are in equilib- rium, their geomterical sum must be zero. If two are at right angles, then (see Fig. 6). — c cos a=a, c sin o=b, tan 0= — b a and c-=a--|-b" Manipulation. — Arrange the appa- ratus as shown in Fig 6. Make M^ equal to 1 Kg. Have the stick, PN, horizontal and add enough weights to M, to pull the stick away. Re- peat until the exact weight is ob- tained. Measure the lengths PC and PN and find sin a. In the note- book, draw . the direction of the forces, and by their length represent their magnitude to scale. Do they balance? Change the position of pulley A and repeat the above three times. Tabulate the values of PA, PB, PC, cos a, sin a, PCcosa PCsin a. Compare the results with the law above. "1. 1 Fig 6 49 EXPERIMENT 23. Iftipact — Linear Momentum. Object. — To verify the laws of linear momentvim in the case of colliding bodies. Apparatus. — Impact apparatus, two lead balls, two ivory balls, cord, and a set of weights. References. — Ames, p. 34; Ames and Bliss, p. 90; Ganot, p. 41; Miller, p. 51; Millikan, pp. 62-63; Watson, p. 96; Fig. 7. Theory. — A mass, m, moving with a velocity, v, collides with a mass, M, moving with a velocity, V, and after impact the masses possess velocities v^ and V^, respectively. The equation expressing this condition follows : — mv+MV=mVi+MVi This equation is to be verified. The velocity at A, (see Fig. 7), of a ball falling from D to A will be the same as if it fell along the arc from B to A, i. e., v-^=2g(DA). (AB)= (AB)= Since (DA) = therefore v'''-=2g.- (AE) (AE) If the ball is dropped from another position, P, then the veloc- ity Vj will be expressed by the relation (AP)^ V2'=2g. (AE) if the arc is net too large. The relation of the velocities is then v^ (AB) = v,= (AP) = Manipulation. — Adjust the lead balls so that they hang freely and just touch each other in the line of their centers. Note the reading of both m and M when at rest. Draw the large ball back about half way along the scale and note the reading on the scale. Release it and determine the reading at the end of the swing for both balls after collision. Repeat five times, starting from the same point. Tabulate results. De- duce the lengths of the arcs passed over and substitute in the 50 equation above, r Determine the mass of each ball. Is the first equation satisfied? Now draw back the small ball in a similar man- ner and repeat five times for two dif- ferent starting points. Adjust the ivory balls and repeat the above. Take five observations with the large ball and five with the small ball for only one starting posi- tion. EXPERIMENT 24. Inelastic Impact. Object. — To study the laws of inelastic impact. Apparatus. — Gaertner impact apparatus, steel cylinder and steel sphere. References. — Millikan, pp. 52-56 ; Ganot, p. 42 ; Watson, p. 951. Theory and Manipulation. — Consult Millikan. EXPERIMENT 25. Elastic Impact. Object. — To study the laws of elastic impact. Apparatus. — Gaertner impact apparatus, two hardened steel spheres, set of weights. References. — Millikan, pp. 62-63. Theory and Manipulation. — Consult Millikan. EXPERIMENT 26. Centrifugal Force. Object. — ^To verify the law of centrifugal force. Apparatus. — Whirling table and attachments for centrif- ugal force, set of weights, stop-watch, steel scale, and a pair of dividers. References. — Ames, p. 40 ; Ames and Bliss, p. 105 ; Ganot, p. 38 ; Millikan, p. 100 ; Fig. 8. Theory. — In order to make a mass, m, move along the arc of a circle of radius, r, with an angular velocity, w, it is neces- 51 sary to exert a force upon it. This force is directed toward the center of the circle and is equal to mrw^ dynes. Prove. Two masses, m^ and m^ (see Fig. 8), slide freely upon a hor- izontal rod, cd, which is whirled about the axis a until the mass M is just lifted. Then miriO)--(-m„r20)-=weight of M in dynes. Manipulation. — Determine the masses m^, m^ and M, and the radii of the paths which m^ and m^ must take in order to lift M. Whirl the rod, gradually increasing the speed until M is just lifted. Determine a by timing a given number of turns when turned at a constant rate. Repeat five times and cal- culate the centrifugal force and compare with the force in dynes acting on M. Repeat, us- ing different masses for m^ and m,. Repeat, using the same -^ (T, M If. =^ F^.. values of mi and m^, but a new value for r. EXPERIMENT 27. Moments and Centers of Gravity. Object. — To verify the law of moments and the formula for the center of mass. Apparaitus. — Support with knife-edges and meter bar, pans, set of weights, thread, meter stick, and pair of plumb- bobs. References. — Ames and Bliss,pp. 118-140; Carhart, Vol. II, p. 71; Ganot, pp. 21 and 56; Miller, p. 54; Millikan, pp. 29-39; Watson, pp. 75 and 124; Fig. 5. Theory. — A body is in equilibrium under the action of parallel forces (1) if the geometric sum of all the forces is zero, and (2) if the sum of all the moments about any axis is zero. In this case the mathematical expression takes the form : miXi-l-m^x^-l-maXa-l-etc. mi-f-nia+ms+etc. where m^Xi, m^x^, mjX^, etc., are the separate moments, m^, mj, mg, etc., are the separate masses and X is the center of mass of the system. Positive moments are taken clockwise; negative moments are taken counter-clockwise. X= 52 Manipulation. — (A) Clamp the bar at its center of grav- ity and suspend it from the adjustable knife-edges. Put the empty pans at such points as to balance. Write the equation, treating the masses of the pans as unknown. Put 10 gms. on one pan and balance again. Write the equation and compute the masses of the pans. Weigh them and check the results. (B) Place two different masses on the pans and balance. l£ the sum of the moments equal to zero? Do this for five different masses and distances. Tabulate results as in the following table : — -(-moments are clockwise. — moments are counter-clockwise. No. mi mrfpand) Xi m. mj-l-pan(2) Xa -|- momeiit — mount %dii. 1 20 g 5g 2 20g 10 g 3 25g 10 g 4 100 g 60 g 5 200 g 150 g In this table, as in the above equation, m and x are the mass and force arm respectively and the subscripts designate the individual quantities. (C) Support the bar from the fixed knife-edge, and ad- just the other edge so that the bar is in equilibrium in any position, i. e., so that the fixed knife-edge and the center of gravity coincide. Add masses to both pans (which are now placed at 10 and 90 cm. respectively) until a balance is ob- tained with the bar at an angle of 30 degrees with the horizon. To get the "force-arm," use a plumb-bob, cd, as in Fig. 5. Consider the mass of the knife-edge as concentrated at a. Write the equation of moments. Repeat with the bar at a dif- ferent angle. (D) The sensibility of a balance is defined as the angular displacement of the beam produced by the addition of any arbitrary small mass. To determine this, first bring the edge c down so that all three edges can be placed in one line. Place the pans near the ends of the beam (not in the grooves). Adjust to balance. Clamp a meter stick vertically behind the end of the beam. Add 1 gm. to one pan and note the deflection. Repeat several times. Add 200 gms. to each pan and repeat. Now raise the edge c about three cm. above 53 the edge of the beam, and repeat the process. Compare the sensibilities and state conclusion. Tabulate data as in the following table : — SENSIBILITY TABLE. Pans empty Pans loaded with 200 gms. each No. Zero reading Eeading with 1 gm. added Sensi- bility No. Zero reading Reading with 1 gm. added Sensi- bility 1 2 3 4 1 2 3 4 (E) Double weighings. — Clamp the bar in the middle, but put the pans at unequal distances from the point of sup- port. Balance with a counterpoise weight, t'ut a mass of 100 gms. on one pan and weigh. Place it on the other pan and weigh. Call these separate values Wi and Wj. If W be the true weight, we have W==V(WiW2). If the arms are nearly equal, W^J^ (Wj+Ws). Test both equations. First make x^ and x, nearly equal with the given mass and then make them quite different. Repeat with 200 gms. Tabulate as follows : — Mass Xl Wi X3 W2 100 g ICO 200 200 (F) Clamp the knife-edges about 20 cm. from the middle. Attach four or more masses by loops of thread from different points along the bar. Consider the mass of the bar concen- trated at its center of inertia. Substitute these values in the equation written above and see if the observed value of x, the position of the knife-edges, sat- isfies the value which you get by substituting in this equa- tion. 54 EXPERIMENT 28. The Sensibility of a Balance. Object. — ^To determine the sensibility of a chemical bal- ance and to find how it varies with the load. Apparatus. — Chemical balance, set of weights. References. — Ames, pp. 84, 208; Ames and Bliss, p. 151; Carhart, Vol. II, p. 75 ; Ganot, p. 59; Miller, p. 55 ; Millikan, pp. 38, 1 14 : Stewart and Gee, Vol. I, p. 63 et seq. ; Watson, p. 106 ; Experiment No. 27. Theory. — The deflections of the balance beams are read by means of a pointer passing in front of a graduated scale. The sensibility or sensitiveness is the deflection produced by the addition of one milligram to one of the pans. If the knife- edges are in the same straight line, the sensibility is indepen- dent of the load on the pans. If the middle knife-edge is higher, the sensibility decreases as the load increases. In weighing, always determine the "zero-point" of the balance with the pans empty. With the pans empty, gently lower the support that keeps the pans in position and allow the beam to rest on the knife-edges. With the case closed, set the beam to vibrating through about five divisions. Read the end of the swings, first on the left side, then on the right, then on the left, and so on, taking an odd number of consecutive readings. If the balance is slow, take sets of three on one side and two on the other: if the balance is fairly rapid, take sets of five on one side and four on the other. Average the readings on the right and those on the left. The means of the two averages will be the "zero-point.'' Thus Zero. 9.95 Always lift the beam from the knife-edges when touch- ing the pans or moving objects on them. Place the object to be weighed on one pan and the weights to balance (at nearly the same "zero-point") on the other. Take the "zero" reading again. Is the object heavier than the weight or lighter? Adda small weight, m (usually 5 mg.) to one side of the balance and determine the "zero" again. Let these masses and positions on the scale be represented thus : Left. Right. Mean Left. Mean Right 1.5 18.2 2.0 17.6 2.0 17.9 2.5 55 Mass Zero o Xo M Xi M+-m X2 Then the sensibiUty, s, is represented by the equation: — X, — x. The weight that must be added or subtracted to shift the zero from point x^ to point x^ isi(xo — Xj) or m(x„— xj Xj Xj Should this be added to the weights, or subtracted? If the arms of the balance are not equal, weigh the object in both pans. Then as in the previous experiment W=V(WiW2) If the arms are nearly equal in length, one may t4 (wi+w^) In still more accurate weighing, one must correct for the buoy- ancy of the air displaced. Manipulation. — See that the balance is level and that the pointer comes to rest near the center of the graduated scale, i. e., near 10. Find out how to raise and lower the beams and release the pans. Always handle the weights with the forceps, and do not jar the beam on the knife-edges. Determine the zero with no load on the pans. Repeat three times and take the mean. Make m equal to 5 mgs. Find the sensibility for loads of gms., 10 gms., 20 gms., to 80 gms. Novy plot a curve with loads as abscissae and sensibilities as ordinates. From the curve- determine at what load a maximum sensibil- ity is obtained. EXPERIMENT 29. The Determination of the Density of a Solid. Object. — ^To determine the density of a regular solid by finding its weight in vacuo and its volume, and to compare that value with its specific gravity as found by weighing the body in air and then in water. 56 Apparatus. — A metallic cylinder, or a sphere, chemical balance, box of weights, calipers, thread, thermometer, glass beaker, distilled water. References. — ^Ames, p. 117; Ames and Bliss, pp. 151, 189; Carhart, Vol. II, p. 110; Ganot, p. 108; Miller, p. 118; Millikan, p. 114; Nichols' Lab. Man., Vol. I, p. 34; Experiment No. 28. Theory. — The density of a solid in c. g. s. units is defined as its mass in grams divided by its volume in cubic centime- ters. The specific gravity of the body is equal to the ratio of the mass of the body to the mass of an equal volume of water. Since the body loses in weight when immersed in water an amount equal to the weight of the water displaced by the body, hence c- -n r- -^ Weight of the body (in vacuo) Specific Gravity = 5_ . £_J^ i Loss in weight of the body in water If water whose density is exactly unity be taken, the dens- ity of the solid as defined by the first definition will be ex- actly the same as the specific gravity defined by the second relation above. Before making the weighings, the previous experiment on the sensibility of the balance should be read carefully. All weighings must be made by the method of vibrations. Manipulation. — Carefully weigh the body, first on one pan of the balance and then on the other, and take the mean of the two values obtained as the correct weight in air. In weighing, first find the "zero-point" on the scale with no weights on either pan. Then place the body on one pan and the weights on the other until the pointer has nearly the same point of rest as before. (Caution. — Do not place the weights on the pan or in any way jar the balance while the beam is resting on the knife-edges.) Then determine the point of rest again. Add 5 or 10 mg. to one pan and again find the point of rest. Find the sensibility of the balance and compute the correct weight of the body. After having weighed the body in both pans and having ta- ken the mean, it is necessary to reduce to the weight in vacuo, i. e., to take account of the buoyancy of the displaced air upon the weight of the body and upon the weights themselves. First determine the volume of the body by means of the calipers. This should be accurately done. Let the volume in cubic cen- timeters be called V. Then the volume of air displaced by the body is also equal to V. Read the barometer and a thermom- eter placed in the balance case. Let the reading of the barom- 57 eter in millimeters be h, and of the thermometer t degrees C. Then the density of the air is expressed by the relation ; — h 273 D= X X 0.001293 760 273+t ]\lultiplying this density by V, we have the mass in grams of the air displaced by the solid. This quantity should be added to the mass as found by weighing. In order to find the buoyancy force on the weights, divide their mass by the density of brass, if they are made of brass. This will give the volume of air dis- placed by the weights. Multiply this volume by the density of the air, as found above, and the mass of air displaced by the weights will be obtained. This mass should be subtracted from the weight of the body as found by weighing. Why? Having applied these corrections for the buoyancy of the air upon the body and upon the weights, the true weight of the body in vacuo is obtained. Call this m. Divide this by the volume of the body as found by the calipers, and the density of the body is obtained. Record all the observations and data. Now suspend the body from the hook above one of the scale pans of the balance, and allow it to hang freely in an empty beaker placed on a bridge over the scale pan so as to allow the latter to swing freely. Weigh the body again. Let the weight in grams be m' Weigh only on one pan and cor- rect for the buoyancy of the air displaced by the body, but not for that displaced by the weights. Why? Then fill the beaker v/ith distilled water, noting its temperature, and allow the body to hang freely in the water. Do not allow any air bubbles to cling to the body. Weigh again (without any corrections this time). Call the weight in grams m". Then m' — m" is the mass of the displaced water. From tables at the end of the text find the density of pure water at the observed temperature. Dividing this density into the mass of the displaced water will give the volume of the body. Dividing the volume thus obtained into the mass, m, as found in the first part of the experiment, will give the specific gravity of the body. How does this agree with the value obtained for the density of the body? EXPERIMENT 30. The Determination of Density by means of the Specific Grav- ity Bottle. Object. — To determine the density of a solid and of a liquid by means of the specific gravity bottle. 58 Apparatus. — Chemical balance, specific gravity bottle, shot or pieces of copper wire, an unknown liquid, distilled wa- ter, set of weights. References. — Ganot, p. 110; Miller, p. 124; Nichols' Lab. Man., Vol. I, pp. 80-84; Watson, p. 174. Theory. — The density of a solid or liquid in c. g. s. units is defined as its mass in grams divided by its volume in cubic centimeters. Manipulation. — Carefully clean and dry the specific grav- ity bottle. Weigh the bottle empty. Add a certain mass of the solid and weigh again. Then fill the bottle full of distilled water, insert the cork so that the capillary of the cork is just full, and wipe all moisture from the outside of the bottle. Weigh again. The difference between the first two weighings will give the mass of the solid taken. The difference between the last two weighings will give the weight of the bottle filled with water less the weight of the water displaced by the solid. Now empty the bottle and refill with distilled water. Weigh again. The volume of the solid taken can now be determined and also its density. Compute the density of the solid. All weighings should be made by the method of vibrations as in the two previous experiments. Clean and dry the bottle again and fill it with the liquid whose density is sought. Weigh. From the weighings made, compute the density of the liquid. Corrections for the dens- ity of the distilled water at the temperature at which the observations were made, should be introduced. EXPERIMENT 31. The Density of a Liquid by Means of a Pyknometer. Object. — To determine the density of a liquid with the pyknometer. Apparatus. — Balance, set of weights, beaker, distilled water, two unknown solutions, pyknometer and thermometer. References.— Ferry, Vol. 1, p. 70; Ganot, p. 122; Kohl- rausch, p. 59; Miller, p. 124; Ostwald-Luther. p. 141; Stew- art and Gee, Vol. 1" p. 157; Wiedemann and Ebert, p. 63; and Watson, p. 176. Theory.— In the c. g. s. system the densitv of a liquid is determined by dividing its mass in grams by its volume in cubic centimeters. In this experiment the volume of the liquid is obtained by determining the mass of a given volume of the unknown liquid and comparing this with the mass of the same volume of water, the density of the latter being cor- rccted for temperature. The ratio of the two will give the density. Manipulation. — Carefully clean and dry the pyknometer. Weigh it empty, using the method of weighing by vibrations and correcting for the buoyancy of the air as indicated in Experiments Nos. 28 and 29. Now draw distilled water of known temperature into the pyknometer untill the water passes the fixed line on the capillary bore of the instrument. With a piece of filter paper absorb water until the meniscus of the water column reaches this fixed line. Weigh the pyknom- eter and its contents as above. Now empty the bulb and wash out the interior several times with some of the unknown solution. Fill and weigh again as above. Remove this so- lution and wash out two or three times with distilled water and lastly with the second unknown solution; then fill as be- fore with this last solution. Weigh as above. Now correcting for the density of water at the observed temperature, deter- mine the density of the unknown solutions. What would be the effect of using as the unknown, a liquid that was volatile? Indicate the sources of error. EXPERIMENT 32. The Determination of Density by Means of the Nicholson Hydrometer. Object. — To determine the density of a coin and of a liquid by means of the Nicholson hydrometer. Apparatus. — Nicholson hydrometer, set of weights, com- mon salt, glass cylinder, and a coin. References. — Ames and Bliss,p.l93; Ganot, p. 108; Miller, p. 151 ; Nichols' Lab. Man., Vol. I, p. 85; Watson, p. 180. Theory. — The principle on which this experiment is based is that the specific gravity of a substance is defined as its weight in air divided by its loss in weight when immersed in pure water. In the manipulation of this experiment one must bear in mind that the weight of the volume of a liquid displaced by any body immersed in it, is equal to the loss in weight of the immersed body. With these facts in mind the theory of this experiment becomes very simple. Manipulation. — Determine the weight, W. that must be added to the pan of the hydrometer in order to bring the pointer in contact with the surface of the water. Now place the coin on the pan and find what weight, w, must be placed on the pan in order to produce the same result as before. Now 60 place the coin in the basket in the water and determine the weight, w', that will be required to sink the hydrometer as before. The ratio of the weight of the coin in air to its loss of weight in water is its specific gravity, or W — w Specific Gravity = — To determine the specific gravity of a solution, the mass of the hydrometer must be known. Let this be denoted by M. Now make up a sufficient quantity of a 25 per cent, solution of com- mon salt. If W be the weight that will be required to bring the pointer in contact with the surface of the solution, then we have M+W Specific Gravitv = ,.- , ,,r ^ - M-f-W If desired, corrections can be made for the density of the water at the room temperature. Determine the density of a larger coin, using the salt solution in place of the water. EXPERIMENT 33. The Density of Liquids by Means of Balanced Columns. Object. — To determine the density of liquids by means of balanced liquid columns. Apparatus. — (a) Two pieces of glass tubing about one meter long and one centimeter in diameter mounted upon an upright stand or support and connected at the upper end to the two arms of a glass Y, rubber tubing, beakers, two un- known solutions, suction pump, meter stick, level, and ther- mometer; (b) a glass U-tube provided with long arms, mer- cury, distilled water, cathetometer, level, and thermometer. References. — Ames, p. 116; Ames and Bliss, p. 183 et seq. ; Ganot, p. 98; Miller, p. 129: Nichols' Lab. Man., Vol. 1, p. 92; Stewart and Gee, Vol. 1, p. 142; Watson, p. 174; and Wiedemann and Ebert, p. 76. Theory. — The liquid pressure at the base of a liquid col- umn is measured by the product of the density of the liquid, into its height. This will give the pressure in grams per unit area. If the pressure in dynes is desired, this product must be multiplied by the value of gravity. If the downward pressures exerted by two liquid columns are in equilibrium, it follows that the densities are inversely proportional to the two heights. Or, if the densities of these two liquids are represented by 8 and S^, and their heights by h and h,, then 8 :8,=hi :h 61 Hence if one density is known, the second density can be determined after the two heights have been measured, This principle can be used to determine the densities of liquids : — (a) that mix or react on coming in contact; (b) that do not mix on coming in contact. Manipulation. — (a) By means of short pieces of rubber tubing connect the two arms of the glass Y to the two pieces of glass tubing. Mount these against the wall in a vertical position. Place the lower end of one tube in a beaker of dis- tilled water and that of the other tube in a beaker containing the unknown solution whose density is sought. Carefully note the temperature of each. By means of a piece of rubber- tubing connect the upper end of the glass Y to the pump. Take a few strokes on the pump and close the rubber tube with a pinch cock. Care must be exercised that air does not' leak into the apparatus while the measurements are being taken. With a meter stick or a cathetometer measure the height of each liquid column above the level of the liquid in the beaker. Repeat this for ten different levels. Again ob- serve the temperature of each solution and take the average of the two temperature readings. Compute the density of the unknown solution for each case. Tabulate results in suitable form. With tubes of the cross-section indicated the error due to capillarity can be neglected. Or, correction for capil- larity can be eliminated as follows : If h and h', and h^ and h^', be respectively two successive readings in each tube, then one can write the relation 8 :8i=h'i-h, :h'-h in which the error due to capillarity is entirely eliminated. Now thoroughly wash out both tubes v/ith distilled water and repeat as above, using a second imknown solution. (b) In the second part of the experiment mount the U-tube in some convenient position so that the arms will be vertical. Pour mercury into the tube so that there will be several centimeters in each arm. Now pour distilled water into one arm until a marked difference in level between the upper ends of the twa columns is obtained. Having washed the thermometer in distilled water, take the temperature of the mercury and the water in the two arms. Now measure the height of the upper meniscus of each column above the level of the common boundary of the two liquids. Repeat taking ten different sets of observations. Again take the tem- perature of both the mercury and the distilled water. Take the mean of the two temperatures. Taking the density of water at the observed temperature from the tables, determine that of mercury. Capillarity can be eliminated by the method suggested above. Compare the measured value of the density 62, of mercury with the vakie given in the table at the end of the manual. What are the sources of error? What will be the effect upon the results if either tube has an irregular bore? If the two tubes were not parallel? If one were of smaller cross-section than the other? EXPERIMENT 34. The Coefficient of Restitution. Object. — To find the coefficient of restitution and the per- centage loss of mechanical energy on impact. Apparatus. — Clamp stand, iron or marble base, level, meter stick, plumb-bob, glass, ivory, and metal balls, impact apparatus, cord. References. — Millikan, pp. 61-63. Theory. — According to the law of conservation of momen- tum, when two bodies impinge miUi-|-m2U2=in,Vi-|-m2V3 and the coefficient of restitution may be defined according to the equation Velocity of separation _ Vg — Vi Velocity of approach Ui — u^ where mj and m^ are the masses of the two bodies and u^ and Uj and \\ and Vj are the velocities of each before and after impact, respectively. The loss of kinetic energy is evidently expressed by the equation L=(J^miUf+>^msui) — (3^m,Vi-t-}^m2vi) The fractional loss, 1, is this last expression divided by the initial energy. Show that if m^ is at rest, l=(l-e2)— Hi— . m 1 — 111 „ If mj is very large, this becomes, l=(l-e=) Manipulation. — (A) Arrange the stand and marble plate so that ivory or glass balls can be held in the clamp until the screw is turned to release them. In falling, they should pass through the ring, strike the marble plate beneath and rebound, retracing the path through which they fall. Fasten the clamp and drop the ivory ball first. Slip the ring up or down the 63 \ertical support until the ball, when rebouncUng, becomes just visible above the top of the ring. Let hi be the height of re- bound, in each case measured from the marble plate to the bottom of the ball. Take five different values of h^ and find s and 1 for each. Use balls of different material. Try a small It ad ball. Tabulate results in suitable form. (B) P.y means of the impact apparatus and by the meth- od described in the reference, find e and 1 for two sets of balls, making at least 10 different trials for each position. Use two different starting points for each set of balls. Tabulate your observations and results for both A and B. EXPERIMENT 35. Young's Modulus by Stretching. Object. — To verify Hooke's Law and to determine Young's Modulus. Apparatus. — Young's Modulus apparatus (Gaertner pat- tirn), optical lever, telescope and scale with clamp stand, meter stick, set of weights, micrometer calipers, and weights (1-10 Kg.). References. — Ames, p. 107; Ames and Bliss, p. 163 ; Ganot, p. 79: Glazebrook and Shaw, p. 141 ; Miller p. 93; Millikan, p. 65 et seq.: Nichols' Lab. Man., Vol. 1, p. 74; Stewart and Gee, Vol. 1, p. 176; Watson, p. 202.' Theory. — According to Hooke's Law, when a force dis- torts a body in some way, the stress is proportional to the strain so long as we do not exceed the elastic limit. Young's JNTodulus is then defined by the equation: — longitudinal stress FL longitudinal strain ae where E is Young's Modulus, F is the force in dynes produc- ing an elongation, e, in a wire of length, L, and of cross-sec- tion, a. Manipulation. — Add enough weights to the weight-holder to stretch the wire straight. Set the single foot of the mirror frame onto the brass collar attached to the wire. Set the pair of feet into the groove in the iron base. Now set up the tele- scope and scale so as to permit one to read the scale when the wire is stretched. Have the scale vertical, using a plumb-bob if necessary. With the naked eye find the image of the scale and set up the telescope between the eye and the mirror. When the reflected image passes into the telescope one will be able to read the .scale provided the eye-piece is properly focused. 64 As the wire is stretched the mirror is moved through a small angle, and the scale reading in the telescope is changed by twice that angle since the angle of incidence is equal to the angle of reflection. Let b equal the perpendicular distance from the singje foot on the mirror frame to the line connecting the pair of feet, let e equal the elongation of the wire, let d equal the distance from the mirror to the scale, and let c equal the change in the scale reading. Show that be Take the reading of the scale when the initial load is at- tached. Add another weight and read again. Record the force (in dynes) added and the difference in the telescope read- ings. Do this for five different added weights. Wait a minute cr more after adding each weight before taking the reading. Take the weights off in the reverse order and observe the readings as before. Repeat the whole process twice, and observe whether the ratio of the force added to the elongation is a constant. In each case find the elongation due to a weight of, say, 1.0 Kg. Measure the length of the wire and its cross-section. Calculate Young's ^Modulus. On the same sheet of cross- section paper plot a curve for each length of wire used, with total forces in grams as abscissae and total elongations in centimeters as ordinates. ^^'hat is the curve? What is the meaning of its slope? Obtain E from the curve. What does the curve prove? Repeat for three different lengths of wire. (Express force in dynes, lengths in centimeters, areas in square centimeters.) Compare the observed value of the modulus with that given in some standard table. Problem. — Using the observed value of the modulus, how much would a rod of the same material 5 meters long, and of 50 sq. mm. cross-section, be stretched by a force of 100 million dynes? What is the physical meaning of Young's Modulus? EXPERIMENT 36. Young's Modulus by Bending. Object. — To verify Hooke's Law and to determine Young's Modulus. 65 Apparatus. — Bending apparatus, electric bell, storage bat- tery, vernier calipers, meter stick, set of weights, weights (500 gm.-5 Kg.), several rods of rectangular cross-section. References.— Ames, p. 107; Ganot, p. 83; Kohlrausch, p. 128; Miller, p. 98; Stewart and Gee, Vol. I, p. 179; Watson, p. 23; Experiment No. 35. Theory. — Young's Modulus by bending may be deter- mined by the aid of the following equation : — For a rectangular rod — E = 4a'be For a circular rod — E=FL^-4(37rr*)e where E is Young's Modulus, F is the force applied in dynes, L is the length of the rod between the knife-edges, b is the breadth and a is the depth of the rectangular rod, and r is the radius of the circular rod, and e is the depression or bending of the rod in centimeters. Manipulation. — Place the rod on the knife-edges, having first placed the rectangular collar at about the middle of the rod. Adjust so that the rnicrometer will touch the platinum contact of the collar. Connect the bell, battery, and apparatus in series. Take the readings when the bell just begins to ring. Attach a weight to the weight-holder and take the reading again. Do this for five or more different weights. Each time take off the weights added and see that the bar comes back nearly to its initial position. Wait a minute each time before taking the reading. Record the forces added and elongations produced. Is Hooke's Law obeyed ? Now add some large weights which will not permanently with the hypsometer. When a constant temperature has been obtained, pour the shot into a known mass of cold water (as many degrees below the room temperature as the final temperature will be above the room temperature), and, keeping the water thoroughly stirred, take its temperature until there is no longer any rise in temperature of the water in the calorimeter. A known mass of the metal should be taken each time. The temperature of the steam should be determined as in the previous experiment. Repeat the experiment until three concordant values of the specific heat are obtained. Define specific heat of a metal, water equivalent of a calorimeter, calorie, thermal capacity of a body. EXPERIMENT 52. The Determination of the Specific Heat of a Metal with The Joly Steam Calorimeter. Object. — To determine the specific heat of a solid with the Joly steam calorimeter. Apparatus. — Joly steam calorimeter apparatus, set of 92 weights, Bunsen burner, hypsometer, rubber tubing, ther- mometer and barometer. References. — Edser, p. 156; Ferrv Vol. 1, p. 221; Ganot, p. 451 ; Preston, p. 236; Watson, p. 249. Theory. — When a mass of cold metal is placed in an at- mosphere of steam, the temperature of the metal will rise and steam will be condensed upon the metal until the latter at- tains the temperaure of the steam. The amount of steam con- densed will be a measure of the specific heat of the metal taken, since the heat given up by the steam will numerically equal the quantity of steam condensed multiplied by the heat of vaporization of steam. The quantity of heat absorbed will be equal to the mass of the metal multiplied by its spec- ific heat and by its rise in temperature. These two quanti- ties are equal to each other. Knowing the heat of vaporiza- tion of steam, the specific heat of the metal can be readily determined. The Joly steam calorimeter consists of a steam chamber mounted on an adjustable stand beneath one of the arms of a sensitive balance. This chamber encloses a small scale pan which is suspended from the arm of the balance by means of a fine wire that passes down through a small opening in the top of the steam chamber. In this form of apparatus care has been taken to prevent steam from condensing on the wire leading into the steam chamber by having the suspension wire pass into the chamber through a tube that is surrounded by a steam jacket. Thus errors in weighing are avoided. With the scale pan empty, steam is passed into the steam chamber and the condensed steam is weighed. The chamber is then cooled to room temperature and the condensed mois- ture carefully removed. The specimen whose specific heat is sought is then placed upon the scale pan and carefully weighed. Steam is again passed into the chamber and the condensed steam is carefully weighed. Knowing the mass of the specimen and the mass of steam condensed upon it, which is the diflference between the two observed masses of con- densed steam, the specific heat of the metal can be com- puted as follows : Let the difference between these two mass- es be represented by m and the heat of vaporization of the steam by h, the mass of the specimen by M, its specific heat by s, the room temperature by t and the temperature of the steam by t^. Then equating the heat lost by the steam con- densed upon the metal to the heat absorbed by the metal, we have mh =Ms(t, — t). Manipulation. — Determine the boiling point of water from the barometric reading. Weigh the solid whose specific 93 heat is to be determined. Having assembled the apparatus, see that the scale pan is freely suspended. Briskly pass steam from the hypsometer into the steam chamber. Note the tem- perature of the steam. When the condensation on the scale pan of the steam chamber has ceased, the supply may be re- duced so as not to disturb the weighing of the scale pan and its condensed steam. Thus the temperature of the steam chamber will be maintained constant. The weighing should be done by adding weights to the scale pan of the balance above the steam chamber. Without disturbing the weights on this scale pan, bring the balance to rest and disconnect the hj'^psometer from the steam chamber. Allow the appar- atus to cool to room temperature and carefully remove all the condensed steam from the scale pan in the steam cham- ber. Add to the scale pan of the balance weights to equal the mass of the specimen. Again briskly pass steam into the chamber for several minutes until the specimen has reached the temperature of the steam, when the supply may be re- duced and the weighing performed as above. The difference between the two masses of condensed steam will give the mass of steam condensed by the specimen in passing from room temperature to the temperature of the steam. Com- pute the specific heat of the metal according to the formula indicated above. Repeat until four or five satisfactory values have been obtained. EXPERIMENT 53. The Radiation from a Warm Body. Object. — To determine the rate of radiation of heat from a warm body and to study the law of radiation of heat from bodies of different materials. Apparatus. — A blackened calorimeter, a polished calori- meter, Bunsen burner, two tenth-degree thermometers, vessel for heating water, tripod, set of weights, platform balance, insulating supports. References. — Ames and Bliss, p. 311 ; Carhart, Vol. II, pp. 117, 124; Edser, p. 127; Ferry, Vol. I, p. 197; Ganot, p. 413; Miller, p. 175; Nichols' Lab. Man., Vol. I, p. 119; Nichols and Franklin, Vol. I, p. 208; Preston, p. 444 et seq. ; Watson, p. 301 et seq. Theory — In the case of a body partly surrounded by a gas, such as air, the cooling is due in part to convection cur- rents, and in part to radiation. In such a case Newton assumed that the rate of cooling, i. e., the number of calories of heat giv- en off in unit time is proportional to the difference of temper- 94 ature between the body and the surrounding medium. This law holds only for small differences of temperature, e. g., such as are met in ordinary work in calorimetry where the difference in temperature does not exceed 20 or 30 degrees. Manipulation. — Fill the two calorimeters with hot water, scy at a temperature of 85 degrees C, and note the tempera- ture of the water and of the surrounding air in the room ever}' minute. The calorimeters should be placed on insulating sup- ports and the water should be kept thoroughly stirred in both calorimeters. With a little care both can be observed simul- taneously. Begfin the observations when the water has reached a temperature of 75 degrees C. Continue the observations until 30 degrees C. is attained by the water in each calorimeter. Now plot curves starting from the same point as the zero of time, showing the relation between temperatures as ordinates, and times as abscissae. These are called cooling curves. What can one conclude from these curves as regards the rate of cool- ing from the two calorimeters? Determine the water equivalent of the calorimeters as in Experiment No. 48. Determine the quantity of heat lost from each calorimeter while the temperature of the calorimeters fell from 70 degrees C. to 60 degrees C. Weigh each. Let t equal the number of seconds. Let Q equal the quantity of heat lost during t seconds. Let d equal the average difference of temperature between the calorimeter and the surrounding air. Find the value of Q^-dt for each calorimeter for ten dif- ferent stages in the process of radiation. Is it a constant or does it vary? If so, what is the relation between the temper- ature and the quantity, O? What is the meaning of the slope of the curves? EXPERIMENT 54. The Fusion Point of a Solid. Object. — ^To determine the fusion point of a solid. Apparatus. — Thermometer, test-tube, stirrer, pieces of glass tubing, Bunsen burner, beaker, gauze, tripod and paraffin. References. — ^Ames, p. 226; Ames and Bliss, p. 291 ; Edser, pp. 147, 166; Ferry. Vol. I. p. 228: Ganot, p. 323; Preston, pp. 270, 286; Watson, p. 244. Theory. — ^All crystals and most pure substances keep their temperature the same at the melting-point until all the sub- 95 stance is fused. This law holds true only when the mixture ci liquid and solid is well stirred. When waxes and certain other bodies begin to melt they become "past}" — like plumb- er's solder — and the temperature changes until the entire process is complete. If in the liquid state, on cooling they do not begin to solid- ify at the temperature at which they began to melt. The average of these two temperatures, however, is constant. It is called the fusion-point. Manipulation. — Take one of the short glass tubes and seal it at one end. While still warm, introduce small bits of paraffin and allow the whole to cool. Arrange the thermometer and stirrer as if preparing to determine the freezing point of ice, but since the fusion point is above the normal temperature of the room, raise the temperature of the water in the beaker. Take the tube containing the paraffin and attach it, by means of rubber bands or thread, to the thermometer so that the center of mass of melted paraffin will be about the same height as the center of mass of the mercury in the thermometer. Be sure to get the thermometer into the axis of the test-tube and have the upper part of the glass tubing above the surface ol" the water. Plunge the test-tube into the water in the beaker, which should now be gently warmed. Take the temperature when the paraffin first begins to melt, reading to one-tenth of a degree. When it is all melted, remove the flame and allow the water to cool. When the paraffin begins to solidify, take the temperature again. The mean of the two points will be the fusion-point. Again heat the water in the beaker and make a second c'etermination of the fusion-point. Repeat this six or eight times. If the same result is not obtained each time, repeat the experiment with a fresh specimen of paraffin. Take the mean of the separate values as the fusion-point of the solid. How does the observed value agree with the value given in ?ny of the standard tables? EXPERIMENT 55. Molecular Depression. Object. — To determine the lowering of the freezing point by dissolving salt or sugar in water. Apparatus. — Test-tube, stopper, stirrer, thermometer' divided into tenth-degrees, crushed ice or snow, salt, sugar, beaker, set of weights, and balance. 96 References. — Ames, p. 230; Edser, p. 168; Ferry, Vol. I, p. 226; Ganot, p. 327; Kohlrausch, p. 166; Ostwalcl-Luther, p. 288; Watson, p. 268; Wiedemann and Ebert, p. 175. Theory. — When pure water is cooled under ordinary con- ditions it freezes at zero degrees C. If the water is rendered im- pure by the presence of dissolved substances it freezes at a low- er temperature. The solid which then separates out is pure ice and not a solid solution like the liquid solution. The molecular depression or molecular lowering is the extent to which the freezing point is depressed by dissolving in 100 grams of the solvent a gram molecule of any substance (i. e., a weight in grams equal to the molecular weight of the substance). For the same solvent this is found, under certain conditions, to be a constant. For water the molecular depression is 18.5 degrees, for acetic acid about 39, and for benzine about 49. Manipulation. — Make a solution of cane sugar in water, using 2.5 g. of sugar to 100 g. of distilled water. Determine the freezing point of this solution as you determined the freez- ing point of pure water in Experiment 46. Alake another solu- tion having twice as much sugar per 100 g. of water. Deter- mine the freezing point again. Make a solution three times as strong and determine the freezing point as before. Make a solution four times as strong and determine the freezing point. Let m equal the molecular weight of dissolved substance. Let 1 equal the observed depression of the freezing point. Let g equal the grams of substance in 100 grams of solvent. Let D equal the molecular depression. Then g Make a solution of 3 grams of common salt in 100 g. of water. Determine the freezing point and molecular depression. Salt solutions do not give a depression of 18.5, since in electro- lytes a part of the solute is dissociated into ions. EXPERIMENT 56. The Determination of the Coefficient of Linear Expansion of a Metal. Object. — To determine the coefficient of linear expansion of a metal rod. Apparatus. — Coefficient of expansion apparatus with opti- cal lever, two thermometers, boiler, Bunsen burner, rubber tubing, telescope and scale, pair of dividers and diagonal scale. 97 References.— Ames, p. 204 ; Ames and Bliss, 265 ; Carhart, Vol. II, p. 28; Edser, p. 39; Ferry, Vol. I, p. 157; Hastings and Beach, p. 172; Miller, p. 153; Millikan, p. 219; Nichols and Franklin, \'ol. I, p. 153; Preston, p. 98; Tory and Pitcher, p. 105 ; Watson, p. 214. Theory. — Most bodies undergo a change of shape or dimensions when subjected to a change of temperature. For example, when a rod, of length lo, at zero degrees, undergoes a change of temperature of t^ degrees, its length will change to the value 1. The change in length due to thermal expansion will be expressed by the relation 1 — l,,, and this change in length will be found to be proportional to the rise in tempera- ture and to the length at zero degrees. The proportionality factor is called the coefficient of linear expansion, and it may be expressed thus : — lo(ti-to) The coefficient, k, may be defined as the increase in length per unit length per degree rise in temperature, the length at ^■ero degrees being taken as the unit length. In making the experimental determination of k, th'e increase in length of the metal rod is the quantity that must be determined with the greatest accuracy, because it is a very small quantity. This change in length can be determined with a micrometer, or by means of some form of lever which will greatly magnify the increase in length. In this case it will be determined by means of an optical lever. In the form of apparatus that is used in this experiment, the actual rod is supported at one end by means of a Y-knife- edge, while the other end is supported on the knife-edge of the optical lever. This optical lever is composed of a stirrup- shaped piece of steel, on which are ground two parallel knife- edges. The stirrup is supported on one knife-edge, while the metal rod rests on the other knife-edge. Fastened to this stirrup is a mirror and a pair of counterpoises. When the length of the rod changes, the lever is tilted and the angular movement can be measured by means of the telescope and scale. The length of the rod between the knife-edge of the Y and the knife-edge of the optical lever should be measured with a meter stick at the beginning of the experiment, and the change in length due to the given change in temperature can be ob- tained from the measurement of the angle of deflection of the 98 mirror. If the telescope and scale are placed at a distance, R, in front of the mirror and the reading on the scale is noted at the lower temperature, as the temperature of\the rod is raised the mirror will be tilted and the reading of the cross-hairs of the telescope and scale will be changed. Suppose that the change in reading is s. Then s :R equals tan 28 where 8 is the angle of rotation of the mirror. If the distance between the two knife-edges of the mirror be denoted by d, and the elonga- tion of the rod by e, then we have the relation : — e:d=tan 8, or, e^(l — Ij,) ^d tan 8. But this is equal to k l,, (t — t,,). The coefficient, k, can there- fore be determined. The relations involving the angular meas- urements should be derived. Manipulation. — By means of ^a rubber tube connect the boiler to one of the side arms of the steam-jacket surrounding the metal rod. By means of the vernier calipers, or by means of a pair of dividers and a diagonal scale, determine the dis- tance between the knife-edges of the optical lever. Place the mirror in an upright, position on the apparatus and place the rod in position. Measure the distance from the one point of support of the rod to the other point of support of the rod. Now place the telescope and vertical scale at a distance of one meter in front of the mirror, and focus the cross-hairs of the telescope on the scale. The scale-reading should be one that is near the middle of the scale. Note and record the tempera- ture of the two thermometers that have been inserted in the two rubber corks of the expansion apparatus. Pass steam from the boiler into the steam jacket for a time until the temperature of the two thermometers ceases to rise. Note and record the temperature of the interior of the steam jacket and the scale reading of the telescope. The difference between the readings on the scale is the quantity s. Now allow the apparatus to cool down to about the temperature of the room and repeat. From these data, compute the value of e, as indicated above. Now determine the value of k for both cases and take the mean of the two values. Indicate the sources of greatest error. What quantities need to be measured with the greatest accuracy? Whv? 99 EXPERIMENT 57. The Coefficient of Apparent Expansion of Mercury. — The Weight Thermometer. Object. — The determination of the cubical coefHcient of expansion of mercury. Apparatus. — Glass bulb with stem, mercury, mercury tray, glass tube with cork stopper, Bunsen burner, tripod, thermometer, stirrer, crushed ice, or snow, crhcible, set of weights, and hypsometer. References. — Ames, p. 210; Ames and Bliss, p. 274; Car- hart, Vol. II, p. 30; Edser, p. 66; Ferry, Vol. I, p. 164; Ganot, p. 509; Hastings and Beach, p. 169; Miller, p. 155; Millikan, p. 213 et seq. : Preston, p. 175 ; Tory and Pitcher, p. 100; Watson, 1... 219. Theory. — In this experiment the mass of a given volume of mercury at zero degrees centigrade is determined. Know- ing the density of mercury at that temperature, the voltime can easily be determined. This volume then is the volume of the containing vessel at that temperature. If the mass of mer- cury completely fills the containing vessel at zero degrees C., and if the cubical coefficient of expansion of mercury is greater than the cubical coefficient of expansion of the containing vessel, then as the latter with its contained mercury is heated to some temperature, t, a portion of the mercury will be ex- pelled from the vessel, which in this case is of, glass. Knowing the cubical coefficient of expansion of glass; the volume of the glass tube at temperature, t, can be determined and therefore the volume of the contained mercury. All the data necessary for determining the coefficient of expansion of mercury then is at hand. This coefficient may be defined as the change in volume per unit volume per degree rise in temperature, the unit volume at zero degrees C. being taken as the unit. For temperatures ranging from zero to 100 degrees C., the coefficient of expansion of glass has been determined. The cubical coefficient of expansion of glass may be taken as 0.000025 and the linear coefficient as 0.0000083. Mercury con- fined in a glass vessel has an apparent expansion which is too small by an amount equal to the coefficient of expansion of glass. . Manipulation. — Thoroughly clean and dry the glass bulb, and then weigh it. Fit a short piece of glass tubing into a rub- ber stopper, and bore the hole through the stopper so that the stem of the glass bulb will fit snugly. Pour some mercury into 100 the glass tube and heat the bulb gently so as to drive out some of the air from the interior. Allow it to cool and some mercury will be drawn into the bulb. Repeat the process until the bulb is partly filled with mercury. The mercury should be put in little by little and heated to drive out any air bubbles. When the bulb is full pack ice ©r snow around it and fill the stem completely. Weigh a small crucible or beaker. When the temperature of the glass bulb and the enclosed mercury has attained the temperature of the ice or snow, remove the glass tube and stopper used in filling the bulb. Scrape off the mercury even with the top of the stem. Remove the bulb from the ice; as it warms and globules of mercury form at the top, scrape them off into the beaker or crucible. When no more globules ap- pear to come out, put the bulb into the hypsometer over boiling water. The water in the hypsometer should be gradually brought up to the boiling point so that no mercury is lost. As the mercury expands, collect the globules in the beaker or crucible. When the temperature of the mercury attains that of the boiling water, the globules will cease to appear. Note the reading of the barometer and calculate the boiling point of the water. Weigh the mercury that has been forced out of the bulb. Call this weight m. Weigh the bulb with the remaining mercury. Repeat this procedure once more* Let M equal the weight of mercury in the bulb at t degrees C. Let do equal the density of mercury at zero degrees C. Let d equal the density of mercury at t degrees C. Let k equal the coefficient of expansion of mercury. Let b equal the coefficient of expansion of glass (0.000025). Let Vj, equal the volume of the bulb at degrees C, equal the volume of the mercury at degrees C. Let V equal the volume of the bulb at t degrees C. Then we have the relations : — V-V„ M M+n. V=V„ (1+bt). Therefore b= Now V=— , and y„= Vo t. d do and do= d (l+kt), and therefore Vo=(M-|-m)-H[d (1+kt)]. M M+m d d (1+kt) M+m .t 101 Substituting these values in the equation above, the relation is obtained : — m^ m Therefore k = b dH |-| — M/ Mt t d(l+kt) This relation gives the value of k, the coefficient of expansion of mercury, in terms of known quantities. Substitute the observed values and solve for k. EXPERIMENT 58. The Determination of the Coefficient of Absolute Expansion of Mercury. — Regnault's Method. Object. — To determine the coefficient of expansion of mercury according to Regnault's method. Apparatus. — Regnault's Expansion of Mercury apparatus, three thermometers, air pump, Bun"sen burner, rubber tubing, crushed ice or snow^, or cold running water. References. — Ames, p. 210; Carhart, Vol. II, p. 31 ; Edser, p. 71 ; Ferry, Vol. I, p. 162 ; Ganot, p. 308 ; Hastings and Beach, p. 175; Millikan, p. 218; Preston, p. 168; Watson, p. 221. Theory. — In this experiment the determination of the co- efficient of expansion of the liquid is independent of the shape, size, or expansion of the containing vessel. The apparatus consists essentially of a double U-tube with long outer arms. The liquid to be studied is poured in the outer arms, but the two portions of the liquid are prevented from meeting by the presence of an air column in the inverted U of the middle branch. One-half of the apparatus is maintained at some tem- perature, t, while the other half is kept at the temperature of melting ice. By compressing the air in the middle portion of the tube, the liquid can be forced up to considerable heights in the outer tubes. In the form of apparatus used in this laboratory, the two outer arms are joined by a cross-tube, and the mercury is forced up by the pressure of the air in the inverted U-tube until the mercury just fills the horizontal cross-tube. If now h and h,,, respectively, be the heights of the mercury levels in the two outer columns above the free surfaces in the inner columns, and if the densitites be d and d„, then since both columns have the same pressure on their outer and inner ends, it follows thaji 102 If V and V,,, respectively, be the volumes of one gram of mercury in the two branches, then vd = Vodtj. Therefore Tho = Vq h- Hence it follows directly from definition that the coefficient of expansion, « h-h„ ^^ hot - All the quantities in this expression can be measured, hence the coefficient can be determined. Manipulation. — Very carefully pump air into the reservoir ])etween the two side arms of the apparatus until the mercury flows across in the horizontal cross-arm at the top of the apparatus. The air pressure should be so adjusted that the mercury stands at about the axis of this connecting tube. Pass ice water through the water jacket of one arm and steam tJirough that of the other. As soon as steady conditions have been attained, take the temperatures. With the cathetometer measure the difference in level of the two mercury columns in the air reservoir at the base of the apparatus. This will give the value of (h^-h,,). A\^ith the meter stick or the catheto- meter measure the height of the column in the cooler tube. This will give h,,. Substitute these values in the equation above and solve for the coefficient of expansion. Repeat this operation twice more. If it is found necessary to introduce corrections with the particular form of apparatus used here, the student is referred to Ferrv. EXPERIMENT 59. The Pressure Coefficient of Air at Constant Volume. — The Air Thermometer. Object. — To determine the average pressure coefficient of air between zero and 100 degrees C. at constant volume. (The Air Thermometer.) Apparatus. — ^Air thermometer, hypsometer, beaker, ice or snow, IGO-degree thermometer, cathetometer, clamp stands, mercury. References. — Ames, p. 212 et seq. ; Ames and Bliss, p. 278 ; Carhart, Vol. II, p. 39; Edser, p. 100; Ganot, p. 318; Hastings and Beach, pp. 164, 182; Kohlrausch, p. 151 ; Louden and ^Ic- 103 Lennan, p. 146; Miller, p. 166; Millikan, p. 123; Nichols and Franklin, Vol. I, p. 146 : Preston, pp. 129, 187 ; Tory and Pitch- er, p. 106; Watson, p. 229; Experiment No. 6. Theory. — If the volume of a gas be kept constant, the pressure, P, at any temperature, t degrees C, is given by the equation : — ■ . P=Po(l+;8t) Mil where Po is the pressure at zero degrees C, and /8 is the pres- sure coefficient. It has been found that p equals l-;-273 or 0.003665 for all gases. (Gay Lussac's Law.) If the volume is allowed to vary and the pressure kept constant, the volume, V, at any temperature, t degrees C, is given by the equation : — v=v;(i+;8t) where \'o is the volume at zero degrees C, and /i is the coeffi- cient of cubical expansion of the gas. It has been found that this ;8 is also the same for all gases. (Charles' Law.) This value of j8 is the same as that in the former equation. The object of this experiment is to determine /3 in the first equa- tion. Manipulation. — The volume of air in the air thermometer may be kept constant by bringing the mercury in the closed arm always up to the same mark. The pressure can be changed by raising or lowering the open arm. When the gas in the bulb cools the pressure is reduced and the mercury in the closed arm will rise. Heat the bulb by immersing it in the steam rising from boiling water. Bring the mercury column up to the mark and measure the difference in height of the mercury columns. To this must be added the height of the barometer. Lower the mercury at once to prevent its being drawn into the bulb. Plunge the bulb into ice and again bring the mercury column up to the mark. Measure the pressure as before. From these two values of P calculate /S. Repeat the experiment. EXPERIMENT 60. The Coefficient of Expansion of Air at Constant Volume. — The Jolly Air Thermometer. Object. — The determination of the coefficient of expansion of a gas at constant volume by means of the air thermometer. Apparatus. — ^Jolly air thermometer, hypsometer, thermom- eter, rubber tubing, Bunsen burner and beaker. 104 References. — Ames, p. 212 et seq. ; Ames and Bliss, p. 278 : Carhart, Vol. I, p. 39; Edser, p. 109; Ferry, Vol. I, p. 176: Ganot, p. 39; Hastings and Beach, pp. 164, 180; Miller, p. 168 Nichols and Franklin, Vol. I, p. 39; Preston, pp. 129, 187; Watson, p. 229. Theory. — ^When a gas within an enclosure of constant volume is heated, its pressure increases. If the given" volume of gas is heated from zero to one degree C, the ratio of the in- crease in pressure to the initial pressure, is called the pressure coefficient of air at constant volume, or, if the coefficient be represented by p, the pressure at zero by P,,, and the pressure at temperature t degrees by P, the following relation exists : p p Pot It can be shown that this is also the coefficient of expansion of sir at constant volume. The apparatus employed in this experiment is the Jolly Air Thermometer. This consists of a glass bulb filled with dry air connected to an open manometer tube filled with mer- cury. This glass bulb, in turn, is enclosed within another vessel so as to be surrounded with crushed ice or snow, or with water at any desired temperature. Immediately beneath the bulb and the envelpping vessel is a tube containing an ■index of colored enamel. The volume of the gas is kept con- stant by adjusting a metal plunger at the base of the instru- ment until the surface of the mercury in the tube is in contact with the pointer. The pressure of the gas in the bulb is ob- tained by measuring the difference in level of the two mercury columns, and adding this to (or subtracting it from) the baro- metric pressure, which should be determined at the beginning and end of the experiment. Since the part of the tube containing the pointer is not at the same temperature as the gas in the bulb, and since the volume of the glass bulb does not remain constant with change of temperature, the equation in its simple form as it is written above, cannot be applied directly, but corrections for these changes must be introduced. The modified equation, devel- oped from Ferry, is given below. Let po be the pressure in the bulb at degrees C, or T„ absolute. Let p be the pressure in the bulb at t degrees C, or T ab .solute. Let \„ be the volume of the bulb at degrees C. 105 Let V be the volume of the bulb at t degrees C. Let M„ be the mass of the gas in the bulb at degrees C. Let M be the mass of the gas in the bulb at t degrees C. Let V, be the volume of the exposed part of the bulb when the bulb is at degrees C. Let Vo be the volume of the exposed part of the bulb when the bulb is at t degrees C. Let nio be the mass of the gas in the exposed part of the bulb when the bulb is at degrees C, and let m be the mass of the gas in the exposed part of the bulb when the bulb is at t degrees C. If the temperature of the exposed part of the bulb is assumed to be the same as that of the room, the error thus introduced will be negligible. If this temperature be denoted by tj degrees C, or T„ absolute, it follows that v„ will be approx- imately the same as v^. From the gas laws, therefore, it fol- lows that PoVo = RMoTo p V = EM T PoVi = RnioTa p V, =Rm Tj Since the mass of gas remains constant throughout if we substitute the values of M„, M, nio and m from the above equations, we get : — PqVo I Po^l = P V 4. P Vl To T2 T ^ Ta h' we represent the coefficient of expansion of the glass by b, we have — , V =Vo(l+bt) Substituting this value in the last equation above, we have — ry (.+ ^. ^ = p C^oa±bt) , V, ) Dividing through by v„ and denoting the ratio v, :v„ by the constant k, we have the relation : — 106 Solving for p, neglecting terms that contain powers of j8 or b higher than the first degree, we obtain : — „ _ p (1+k+bt) - po (1+k) '^ ~ Po (t+ts+kt) — p(t2+kt) Manipulation. — In carrying out the experimental manipu- lation, first obtain the value of the ratio Vj :v^ from the instructor, or else determine it yourself. This can be done by first finding the volume of the bulb when it is full and then v.'hen it is filled with water up to the proper points. The bulb should now be filled with dry air that has been passed over calcium chloride. The air should be forced out and then drawn into the bulb several times until the bulb is thoroughly dry. After this process has been carried out several times, the small glass capillary tube can be sealed off. Now fill the vessel i^urrounding the glass bulb with crushed ice or snow. Bring the mercury column so that it is just in contact with the col- ored glass index. This can be done by adjusting the plunger at the base of the apparatus. Note the difference in height between the level of the mercury in the two columns. This plus (or subtracted from) the barometric pressure will give the value of p„. Now add cold water to the ice and carefully heat it by passing in steam from the hypsometer. Care must be taken not to crack the glass of the bulb in this operation. Steam should be slowly introduced into the ice water until the temperature of the water in the bath is about 30 degrees C, £0 degrees C, 70 degrees C, and 90 degrees C, reading each time the difference in level of the two mercury columns and taking the barometric pressure. Keep the mercury surface on the one side in contact with the index on that side. In this way find the average pressure coefficient for 30 degrees C, 50 <:legrees C, 70 degrees C, and 90 degrees C. EXPERIMENT 61. The Coefficient of Expansion of Air at Constant Pressure. Object. — ^To determine the average coefficient of expan- sion of air at constant pressure. Apparatus. — Glass bulb with stop-cock, boiler, glass tank, ice or snow, thermometer, set of weights, balance. References. — Ames, p. 212 ; Ames and Bliss, p. 278 ; Car- hart, Vol. 11^ p. 39: Edser, pp. 96, 104; Ganot, p. 314 et seq. : Hastings and Beach, pp. 164, 182; Kohlrausch, p. 71 ; Louden 107 and McLennan, p. 149 ; Miller, p. 166 ; Millikan, p. 123 ; Nichols and Franklin, Vol. I, p. 146: Preston, pp. 129, 187; Tory and Pitcher, p. 113; Watson, p. 229. Theory. — At constant pressure the volume, V, of any gas at a temperature of t degrees C. is given by the equation : — V=Vo(l+;8t) where \'(, is the volume at zero degrees C, and p is the cubical coefficient of expansion of the gas. The object of this experi- ment is to determine the value of p. Manipulation. — First see that the bulb is dry and filled with dry air. Then weigh the glass bulb with stop-cock. Then place in the boiler, and with stop-cock open bring the bulb to the temperature of the steam. Note the barometric pressure and calculate the boiling point. Close the stop-cock and re- move the bulb from the boiler. Pack ice and snow around it and place the end of the tube under the surface of water which has a temperature of about degrees. Opening the stop-cock, make the levels of the water inside and outside the same by lowering the bulb into the water. Close the stop-cock, re- move, dry and weigh again. Then entirely fill the bulb with water at a known temperature, t, and weigh again. From the three weighings calculate the vajue of V-:-V„. Substitute this value and that of t in the equation and determine the value of /3. Repeat the experiment with another dry bulb and make a second determination of /? and compare with the former de- termination. EXPERIMENT 62. The Maximum Vapor Pressure of a Liquid. Object. — ^To determine the maximum vapor pressure of a liquid at temperatures between zero and 100 degrees C. Apparatus. — Hypsometer, rubber tubing, Bunsen burner, thermometer, beaker, and vapor pressure apparatus. References.— Ames, pp. 233, 236; Ames and Bliss, p. 306; Edser, p. 220 et seq. ; Ferry, Vol. I, p. 180; Ganot, p. 337; Hastings and Beach, p. 213; Millikan, p. 152 et seq.; Preston, p. 317 et seq. Theory. — If a liquid is introduced within a vacuum or an enclosed space, vapor will be given oflf and a pressure will be exerted upon the surface of the liquid and the walls of the enclosing vessel. This pressure will vary with the mass of liquid that evaporates and the temperature. For a given tem- perature the vapor pressure will be a maximum when the en- 108 closure containing the vapor is saturated. As mentioned above, this value of the maximum vapor pressure will vary with the temperature. In this experiment a small quantity of the liquid to be tested is introduced into the space above the mercury of a barometer column. The vapor pressure is determined by ob- serving the change in height of the mercury column that results when the liquid is thus introduced into the tube. The appara- tus used consists of a barometer tube, which is expanded at its upper end into a small closed bulb, and at its lower end is con- nected to an open manometer. Connected to the latter is an iron well containing a plunger. This well also contains mer- cury. By means of the plunger the mercury in the apparatus can always be brought up to a fixed point so that its surface will be in contact with the colored glass pointer attached to the interior of the tube. Surrounding this tube is a vessel into which one can introduce water at any desired temperature, and thus the vapor pressure of the liquid can be studied at any temperature between zero and 100 degrees C. The liquid introduced above the mercury column is water. In the bulb, therefore there will be mercury and water vapor. At temperatures below 100 degrees C. the vapor pressure of I'lercury is so small that it can be neglected, or if it is desired, to make correction for it, the value of the pressure of mercury vapor at the given temperature can be taken from tables and correction made for it. The maximum vapor pressure of water, therefore, will be the atmospheric pressure less the difference in level of the mercury in the two columns on either side of the apparatus. Manipulation. — Observe the barometric pressure at the beginning and at the end of the experiment. Introduce water into the vessel surrounding the bulb at the top of the barometer tube, and pass steam into the water until it attains a tempera- ture of about 40 degrees C. Care should be taken that the steam does not come against the glass walls of the bulb, or other parts of the apparatus, because the sudden expansion ot the glass may crack it. Now adjust the level of the mercury column so that the surface of the mercury is just in contact with the colored index. Keep the water in the enclosing vessel thoroughly stirred, and when the temperature becomes con- stant, note if the surface of the mercury is in contact with the colored index. If so, read the difference in level of the two mercury columns. This is done by setting the cross-wire of the movable slider on the top of the meniscus of the mercury column in the open tube, and then reading the difference in 109 level of the cross-wire and the end of the index. To this must he added the number of millimeters of mercurj^ equivalent to the water on the top of the mercury in the barometer tube. Compute the maximum vapor pressure of water for the tem- perature observed. Repeat these observations for every five or ten degrees up to 90 degrees C. With the values thus ob- tained, plot a curve with temperatures as abscissae and corre- sponding vapor pressures as ordinates. On the same sheet, plot a similar curve with the values taken from a table in some standard text-book. What is the percentage error at 40 degrees C? At 50 degrees C. ? At 70 degrees C? at 80 degrees C? At 90 degrees C. ? EXPERIMENT 63. The Mechanical Equivalent of Heat. — Puluj's Method. Object. — To determine the value of the mechanical equiv- aleht of heat. Apparatus. — Mechanical equivalent of heat apparatus, 50- degree thermometer, set of weights, stop-watch. References. — Edser, p. 274 ; Ferry, Vol. I, p. 270 ; Miller, p. 194; Preston, p. 575 et seq. ; Experiment Nos. 48, 53 and 76. Theory. — When mechanical energy is changed into heat, the amount of energy that disappears bears a constant ratio to the quantity of heat that is developed. This constant is called the mechanical equivalent of heat. It may be defined as the number of units of work, expressed' in ergs, that will be required to produce one calorie. The numerical value of this constant is 4,18x10'^ ergs. In the method described below the heat is produced by friction between two metallic cones, one of which fits into the other. These cones are supported by insulating rings in a cup mounted at the top of a revolving axle. The outer cone will rotate while the inner one is prevented from rotating by a weight attached to the circumference of a wooden disk, which is fastened to the inner cone. A thermometer passes through the disk into water contained in the inner cone. A stirrer may also be used to keep the temperature of the water uniform. At the base of the standard supporting the disk is a speed-counter which registers the number of rev- olutions given to the outer cone when the hand wheel is turned. If it is necessary, weights may be placed upon the disk in order to secure sufficient friction between the cones. The speed of rotation of the disk is so adjusted that the energy no represented by the friction between the cones will be sufficient to keep the mass, M. so that the cord attached Jo it will be tangent to the circumference of the disk. The force, Mg, will then be acting on the circumference of the disk. If it is neces- sary to turn the disk n times in order to keep the mass in this position, the result will be the same as though the disk were revolved n times and the cone were caused to remain at rest. Thus, to cause the disk to turn n times against the force Mg would require the expenditure of 27rrnMg ergs of work, r being the radius of the disk. This is the amount of work that is transformed into heat during the process of the experiment. Some of the heat thus generated is absorbed by the water in the cone and by the cones themselves; some of the heat is con- ducted away by the supports, and part is radiated off into the surroupding air. If desired, the friction can be taken into account according to the method indicated in Ferry, and the results corrected accordingly. If w be the water equivalent of the cones and the ther- mometer, the energy relations can be expressed thus : — The work done is equal to 27rrnMg ergs and the heat de- \eloped is expressed by the relation : — H=w(t,-t,)-|-E, where t^ is the initial temperature and t, is the final tempera- ture, and R is the heat that is radiated. The latter is deter- mined according to the method utilized in Experiment No. 53. We have therefore, 2wrn]\Ig=Jw(t2— tj)-fR, where J is the mechanical equivalent of heat. Manipulation. — Weigh the brass cones and thermometer. Determine the water equivalent of the cones and stirrer and thermometer as in Elxperiment No. 48. Repeat this deter- mination until three consistent results are obtained. The radiation correction can be determined by the methods indicated in Ferry, Vol. I, pp. 197-199, or according to the method indicated in Experiment No. 76. Now fasten a mass of 100-200 grams to the string attached to the wooden disk, and note the reading of the speed-counter. Weigh out a mass of 25-50 grams of cold water, approximately ten degrees colder than the room temperature, and pour it into the metallic cup. After so doing, let the water stand for at least ten minutes, stirring meanwhile and taking its temperature every mirjute. At the end of this time begin to turn the hand wheel so as to keep the weight in a position such that the string attached Ill to it will be tangent to the circumference of the disk. This will require some little care and patience as the mass must be kept suspended by friction alone. One observer should now stir the water and keep a record of the time and the tempera- ture. Observations of the temperature should be taken at the end of every minute. It will be well to start with the water at a temperature approximately ten degrees below the room tem- perature and at the end have the final temperature as many degrees above the room temperature. When this temperature has been attained, stop turning the hand wheel but continue to take the temperature observations for ten minutes after the turning has ceased. A record of the time at which the water attained the room temperature should also be made. From the first part of your observations, determine the rate at which the temperature of the water increased be- fore the turning began. Knowing this, find the gain in temperature of the water during that part of the experiment before the room temperature was attained. This will give the heat absorbed due to the radiation from the room. From the final observations determine the rate of cooling and then find the loss in temperature during tnat part of the experiment after the room temperature had been- attained. This will be the loss due to radiation from the water. The corrected temperature change, through which the v\'ater has passed can now be determined. Having determined this change in temperature and the thesrrnal capacity of the calorimeter and its accessories, the heat developed can be found. Read the speed-counter and determine the work ex- pended in turning the disk through the number of revolutions indicated. Compute the mechanical equivalent of heat. The temperature observations of this experiment can be treated in the same manner as the temperature observations in Experiment No. 76 are treated. This latter method is sug- gested to the student as the simpler method. EXPERIMENTS IN ELECTRICITY AND MAGNETISM 115 EXPERIMENT 64. Electrostatic Induction. Object. — To study electrostatic induction. Apparktus. — Influence machine, Leyden jar, electroscope, liollow insulated sphere, insulated cylindrical conductor with hemispherical ends, proof plane, vulcanite and glass rods, flannel and silk cloth, suspended pith ball, and copper wire. References. — Ames, p. 269 et seq. ; Ames and Bliss, pp. 325- 331 ; Carhart, Vol. II, p. 171 et seq. ; Ganot, p. 733 et seq. ; Hast- ings and Beach, p. 312 et seq.; Nichols' Lab. Man., Vol. 1, pp. 122-130; Nichols and Franklin, Vol. II, p. 99 et seq.; Watson, p. 625 et seq. Theory. — On charging a body or system of bodies, equal quantities of positive and negative electricity are developed. For example, if one rubs a glass rod with a piece of silk, a positive charge is said to be developed on the surface of the glass and an equal quantity of negative charge is developed on the silk. The energy of electrification is believed to exist m the insulating medium surrounding the two bodies which ciirry the two equal charges of positive and negative electricity. That is, the insulating medium is said to be under a strain. If a charged sphere, for example, be brought in the imme- diate neighborhood of an uncharged body, there will be sep- arated on the latter two charges of electricity of opposite sign ; one charge of the same sign as that on the sphere being repelled to the side of the uncharged body farthest from the sphere, and a charge of the opposite sign on the side nearest to the sphere. If the body which was originally uncharged be con- nected to earth, the repelled charge will go to earth and the remaining charge is then said to be a "bound charge." The charge that passes to earth is said to be a "free charge." To completely discharge a conductor and cause the field that surrounds it to disappear, it will be necessary that these two equal charges of opposite sign, just mentioned, be made to unite. Such a body may then be called a neutral body. Positive and negative electricity always exist at the positive and negative ends of electrical lines of force. The po- tential of a body containing a "free" positive charge is said to be positive as regards the potential of the earth, and the poten- tial of a body carrying a "free" negative charge is said to be negative as regards the potential of the earth. The potential of the earth is arbitrarily taken as zero. The potential of a body is positive when a positive charge passes to earth upon 116 grounding the body ; its potential is negative when a negative charge passes to earth upon grounding the body; it is said to be at zero potential when no charge passes to earth on ground- ing. The distinction between the charge that may be on a body and its potential should be clearly kept in mind. A body may be positively charged and still be at zero potential, or it may even have an excess of negative electricit}' on it, and be at positive potential. The character of a charge on a body may be tested by means of the proof-plane and the electroscope. In using the electroscope it must be remembered that it is the first motion of the leaves, as the charged proof-plane is brought near it, that is to be noted. If the proof-plane has on it a considerable charge whose sign is opposite to that on the leaves of the electroscope, it will cause them to collapse, and afterwards to diverge as the proof-plane is brought quite near the electro- scope. Manipulation. — By means of a wire, connect the hollow sphere to the outer coating of a Leyden jar. Charge the Leyden jar by connecting the inner coating momentarily to one terminal of the friction machine when the latter is in a charged condition. Now place the cylindrical conductor in the neighborhood of the charged sphere. With the aid of the proof-plane and the electroscope study the character of the charge on the different parts of the conductor. By means of a suspended pith-ball, try to get some idea of the direction of the lines of force constituting the electric field about the con- c'uctor. Note the difection in which the pith-ball tends to move. Record results. Keep the conductor insulated all the time during this procedure and exercise care not to accidentally ground it, for so doing will give misleading results. Now re- move the insulated conductor to a distance and test again as above. Record the results of the observations. Again bring the conductor near the charged sphere and ground, and test with the proof-plane and electroscope as before. Now bring the conductor a little nearer the sphere and test again with the proof-plane and electroscope. Is any change found? Ground the conductor and then remove it, still insulated, to a distance from the charged sphere. Test with proof-plane and electroscope again. Care must be exercised to keep the con- ductor from being grounded while handling it and also to keep a charge from passing from the charged sphere to the conduc- tor during the performance of the experiment, otherwise the results of the observations will be vitiated. Repeat the obser- 117 vations several times and record all the restilts in suitable tabulated form. With a charged pith-ball determine the direction of the lines of force in a field surrounding (1) a charged sphere, (2) a cylindrical conductor with hemispherical ends, (3) the sphere and the conductor when charged with like charges, (4) the sphere and the conductor when charged with unlike charges, (5) the charged sphere and the, grounded conductor. Draw figures illustrating each case and show a number of the equipotential surfaces. Define : Unit electric charge, unit electric field of force, electric difference of potential, electric potential at a point, equipotential surfaces, electric capacity. Prove that lines of force and equipotential surfaces are mutually perpendicular, if the charge on the insulated sphere were positive, what would be the potential of the conductor in each of the cases studied ? What determines the electrical potential of a body in any given case? Draw a vertical section of the two conductors showing the lines of force and the equipotential surfaces, (1) when both are insulated and near each other, (2) when the cylindrical conductor is grounded. Explain the charging of the electroscope. Describe fully the various steps. Describe fully the experimental work so that it can be easily followed. EXPERIMENT 65. The Toepler-Holtz Machine. Object. — To study an induction electrical machine and to become familiar with the principles and the operation of such a machine. Apparatus. — A Toepler-Holtz machine, an electroscope, a rubber rod, a piece of flannel cloth, a proof-plane, a pith-ball covered with tin foil, centimeter rule, and Leyden jars. References. — Ames and Bliss, p. 332; Carhart, Vol. II, p. 180; Ganot, p. 766; Hastings and Beach, p. 333; Nichols' Lab. Man., Vol. I, p. 122 et seq. ; Nichols and Franklin, Vol. II, p. 160 ; Thompson, p. 59 ; Watson, p. 666 ; Experiment No. 64. Theory. — A charged body placed in proximity to an un- charged body will induce two equal and opposite charges on the uncharged body. If another uncharged body be brought in contact with the latter, it will receive a charge of the same sign as that of the charging body; or if a body bearing points on 118 its surface be brought in the immediate neighborhood of the second body, the pointed portions will receive a charge of the same sign as that on the charging body, i. e., a discharge of the repelled electricity will pass to the points of the body. If now a series of positively charged bodies is caused to rotate past the points of an uncharged body, the lat- ter will receive a charge of electricity from each positively charged body so long as the potential of the moving body is above that of the body receiving the charge. By means of con- densers, or Leyden jars, these charges of electricity received by the pointed conductor can be stored until there, is sufficient charge stored up to break down the dielectric of the condenser. In the case of the Toepler-Holtz machine, we have, in brief, conditions somewhat as suggested above. Primarily there are two upright discs made of some insulating material, mounted on a suitable frame so that one is stationary and the other is capable of rotation about an axis. The stationary disc has on the back two sheets of paper, symmetrically placed as regards the axis. Underneath these sheets of paper are smaller sheets of tin-foil which serve as "inductors." The rotating plate of the machine carries a number of small metallic but- tons placed near the periphery of the plate. . These serve as carriers and collectors of electric charges. In front of these two plates are arranged a set of metallic points in the form of a comb, which receives charges from the collectors on the ro- tating disc. Each comb is attached to a horizontal conducting frame-work, that connects in front to one terminal of a Leyden jar. On this frame is a pair of handles which carry a metallic ball at one end and an insulated handle at the other. These latter serve to discharge the Leyden jars. In front of the rotary disc is a metallic cross-bar placed at an angle of about 45 degrees with the horizontal. This cross-bar serves to carry repelled charges from the metallic carriers on one side of the machine to the carriers on the opposite side of the machine. From the inductors on the rear plate, an arm provided with a brush extends around in front of the rotary plate and makes contact at the proper point with the rotating carriers, and thus the potential of the inductors is maintained. In starting to work with the machine, one must assume that one of the inductors is at a higher potential than the other, or else that a positive charge is given to one of the inductors. Assuming this, as one of the carriers on the front plate comes in the neighborhood of this positively charged inductor, a negative charge is bound on the inner side of the carrier and a positive charge is repelled to the outer side of the carrier. As ll'J this carrier comes under the horizontal comb mentioned above, the combs being mounted in front of the inductors, the posi- tive charge in part is discharged to the combs and the negative charge remains bound on the carrier. Passing on, the carrier comes in contact with the brush of the cross-bar, and the re- maining portion of the separated positive charge is taken off and neutralized by the negative charge that is received at the same instant at the other end of the cross-bar. The next stage finds the carrier with an excess of free negative charge, and as it passes on in front of the inductor on the opposite side of the machine, it gives up a part of its negative charge to the inductor on the rear and thus lowers the potential of that inductor. This transfer of negative charge is made through the contact of the carrier with the little brush attached to the inductor-arm, as mentioned above. The process, as described, is then repeated in the next half revolution of the disc with the sign of the charge reversed. If the knobs in front of the machine are apart, the charges that are removed from the carriers by the combs are stored in the Leyden jars until sufficient difference of potential is developed to break down the air-gap separating the two coatings of the Leyden jars, and then a spark leaps across from the terminal of one Leyden jar to that of the other jar. This process is repeated indefi- nitely. Manipulation. — Set the machine in rotation. With a piece of flannel or cat's fur, gently rub one of the inductors on the rear plate of the machine until the machine "picks up." As soon as a spark passes from one terminal of the machine to the other, the machine will de- velop electrical charges. Run the machine a few minutes until it is fully charged. Pull the poles a few centimeters apart. Then stop the machine, and with the electroscope and proof- plane test the character of the charges on each part of the machine. By means of a diagram show the charge on the dif- ferent parts of tlie machine. Repeat several times so as to be sure that no error has been made. Does the machine become reversed? Explain. Observe the difference in the discharge when the Leyden jars are on the machine and when they are removed. What is the purpose of the jars? Determine the maximum length of spark that the machine will develop when it is running steadily. Remove the cross- bar and repeat with the machine running at the same rate as before. Reverse the direction of rotation of the disc and de- termine under what conditions the machine will now work. 120 Now take the machine in the dark room, and with the ter- minals in contact and cross-bar in place, run the machine steadily. Describe what is observed. With the terminals apart and the cross-bar in position, run the machine. Remove the cross-bar and repeat the same steps again. Describe in full all that is observed. Try to reverse the polarity of the machine and observe the effect as seen in the dark-room. Explain the operation of the machine. Explain the func- tion of the cross-bar and of the Leyden jars. Show how the machine becomes fully charged when one inductor is given a small charge and the machine is run steadily. EXPERIMENT 66. The Plotting of Magnetic Fields. Object. — To plot the magnetic field of a magnet under dif- ferent conditions. Apparatus. — Small compass or magnetic needle, board divided into squares, bar magnet, piece of soft iron or horse- shoe magnet, cross-section paper. References. — Ames, p. 247; Ames and Bliss, p. 344; Car- hart, Vol. II, p. 310 et seq. ; Ganot, p. 703 ; Hastings and Beach, p 361 ; Nichols' Lab Man., Vol. I, p. 141 ; Nichols and Franklin, Vol. II, p. 27; Thompson, p. 109; Tory and Pitcher, p. 127; Watson, p. 498 et seq. Theory. — Surrounding a magnet is a region in which a magnetic needle will be acted upon by magnetic forces, caus- ing the needle to assume at each point in the region the direction of the maenetic force at that point. It is the object of this experiment to study with a small compass the magnetic field surrounding a bar magnet when the magnet alone is in the region, and secondly when a piece of soft iron is in the neighborhood of the magnet. Manipulation. — Place the board divided into squares so that one edge coincides approximately with the earth's mag- netic meridian. In the center of this board place the bar mag- net so as to make an angle with the lines on the bpard. On each alternate intersection of the lines forming the squares, place the small compass. When the needle has come to rest, indicate on the piece of cross-section paper the position of the magnet and the position of the needle. Repeat this until you have moved the magnet over the whole board. Now place the piece of soft iron or the horse-shoe near the bar magnet. 121 but not in contact with it, and repeat as above. After this has been completed, draw in the lines of magnetic force and locate the neutral points. Observe the distorting effect of th« earth's magnetic field. What is meant by a magnetic field? A mag- netic line of force? What is a magnetic equipotential surface? EXPERIMENT 67. The Determination of Magnetic Dip. Object. — To determine the dip or inclination of the earth's magnetic field at a given place. Apparatus. — ^Dip circle, small compass, large magnet. References. — Ames and Bliss, p. 349; Carhart, Vol. II, p. 275; Ganot, p. 694; Hastings and Beach, p. 363; Louden and McLennan, p. 190 ; Stewart and Gee, Vol. II, p. 275 ; Thomp- son, p. 138; Watson, p. 611. Theory. — Consult references. Manipulation. — Place the dip circle at the point where the angle of dip is desired. Level it carefully. Place the circle in a horizontal position and move about a vertical axis until the needle points in the direction of the horizontal axis. Then being careful not to disturb the adjustment about the vertical axis, turn the circle into the vertical plane with the graduated face of the circle facing east. Read both ends of the needle, recording the separate values and the mean. This eliminates errors due to the eccentricity of the circle. Repeat this four times. Now turn the circle completely over so as to face the west. Repeat all that was done with the circle facing east. Reverse the needle in its bearings and repeat all that has been done thus far. Then with a bar magnet reverse the polarity of the needle and repeat the whole experiment. Take means of corresponding sets all the way through, i. e., means of each set with the circle facing east and with the circle facing west, and then take the average of these means. Then take the means likewise with the polarity of the needle reversed. The final mean, which will be the value of the angle of dip at the partic- ular place where the observations are taken, will then be the average of the two mean values obtained with reversed polar- ity. Record the results in tabular form. 122 EXPERIMENT 68. The Comparison of the Horizontal Intensity of the Earth's Magnetic Field at Different Points. Object. — To compare the value of H, the horizontal in- tensity of the earth's magnetic field, at different points. Apparatus. — Vibration box, small bar magnet, stop-watch or clock, micrometer calipers, centimeter scale, set of weights iin4(A-fD) /* = sinj4 (A) 149 where A is the angle of the prism and D is the angle of mini- mum deviation of the prism. The student will derive this expression and also show that the angle of deviation is a mini- mum when the angle of incidence is equal to the angle of emergence. Manipulation. — Adjust the spectrometer as in the previous experiment. If the angle of the prism has not been found, determine it as in the previous experiment. .Place the prism on the prism table of the spectrometer. Having made the slit vertical and narrow, place the sodium flame in front of the slit and find the refracted ray by looking through the prism with the unaided eye. Set the telescope in position and focus on the refracted image of the slit. Turn the prism stand and see which way the image in the telescope moves. Follow it with the telescope. Turn the prism so as to decrease the angle of deviation. At length a point will be reached where a motion of the prism in either direction will cause the image to move in the same direction. Set the cross-hairs of the telescope at the turning point of the image. The telescope is now set for the angle of minimum deviation. Take the reading by both ver- niers. Swing the telescope on the other side of the circle, move the prism around and again determine the position of minimum deviation. Read both verniers again. The angle through which the telescope is swung is either twice the angle of mini- mum deviation, D, or else (180° — D). Make three separate determinations and take the mean. Substitute the values of D and- A in the equation ami calculate the value of /<,, the index of refraction of the prism. A simpler and more direct way of determining the angle of minimum deviation is as follows: Set on the position of minimum deviation as above. Take the reading on both ver- niers. Then remove the prism and set the telescope on tiie image of the slit as the ray of light passes in a straight line through the collimator and telescope. Then read both verniers again. The difiference between these two readings will rive the angle of minimum deviation direct. Determine the angle of minimum deviation by both methods and find the value of the index of refraction. Repeat this second method at least twice, and take the mean of the results. Define index of refraction. What is its physical meaning? How will it vary with the wave-length? Will it vary at all with the wave-length? Why does one use the sodium flame in this experiment? Would a lithium flame serve as well? 150 EXPERIMENT 83. The Dispersion of a Prism as a Function of the Wave-length. Object. — To calibrate a glass prism and to obtain its dis- persion curve. Apparatus. — Spectrometer, prism, Bunsen burner, small induction coil, vacuum tubes containing H, N, O, etc., bottles containing Li, K, Na, Sr, Ca, and Ba salts with platinum wires for same to be used in producing flame spectra. References. — ^Ames, pp. 455-467; Carhart, Vol. I, p. 293; Edser, pp. 83, 330; Ganot, p. 569; Hastings and Beach, pp. 704-710; Nichols and Franklin, Vol. Ill, p. 76; and Watson, pp. 514-518. Theory. — When a beam of white light passes through a prism it is deviated from a straight line, and upon emergence from the prism it is found to be separated into component wave-lengths forming a spectrum. It is evident therefore that the different wave-lengths are deviated differently according to their position in the spectrum. It will be found that the short wave-lengths in the violet end of the spectrum are de- viated most, while the long wave-lengths in the red end of the spectrum are deviated least. It is the purpose of this experiment to measure the angular deviation of wave-lengths in different parts of the sepctrum. Manipulation. — Adjust the spectrometer as in Experi- ment No. 82, so that minimum deviation for the sodium flame will be obtained. Obtain the angle of minimum deviation for this wave length as directed in the experiment cited. Ad- just the prism on the stand so as to again secure the angle of minimum deviation for the sodium wave-length. Being care- ful not to disturb the prism after it has been once adjusted, obtain the deviation for the wave-lengths emitted by the sources named above. This can be done by setting the cross- wires of the telescope upon the vertical image of the slit in each case and then reading the angular position of the tele- scope. After all these observations have been completed, re- move the prism and set the telescope upon the image of the slit. Read the angular position of the telescope. From the data thus obtained the angular deviation for the different wave-lengths can be obtained. Having done this, tabulate the results with wave-lengths in one column, angular devia- tions in the second column and indices of refraction in the third column. On a sheet of cross-section paper plot a curve with wave-lengths as abscissae and angular deviations as or- dinates. On the same sheet plot a second curve with indices of refraction as ordinates and the same values as above for 151 abscissae. The former is a calibration curve of the prism and the latter shows approximately the variation of the index of refraction with the wave-length. EXPERIMENT 84. Measurement of the Wave-Length of Light by Means of the Grating. Object. — To determine the wave-length of sodium and lithium light by means of a plane grating. Apparatus. — A horizontal track provided with two verti- cal supports, one of which has a narrow slit and attachments for two meter sticks, a grating ruled on a piece of plane glass, three meter sticks, Bunsen burner, sodium and lithium salts and platinum wires. References. — Ames, p. 479; Ames and Bliss, p. 477; Car- hart, Vol. I, p. 303; Edser, p. 448; Ganot, p. 651; Hastings and Beach, p. 650; Kohlrausch, p. 257; Nichols and Franklin, Vol. Ill, p. 94; Nichols' Lab. Man., Vol. I, p. 273; Preston, p. 226 et seq; Watson, p. 531. Theory — A plane grating consists essentially of a piece of polished plane glass having ruled upon its surface a series of equidistant parallel lines or scratches. These lines are ruled by means of a diamond point attached to the screw of a divid- ing engine. A plane wave of monochromatic light incident normally upon such a grating will be transmitted or reflected in several different directions determined by the equation dsin^=nA where d is the grating space or the distance between the lines of the grating, A. is the wave-length of the incident light and n is any inte ger corresponding to the order of the spectrum and d is the angle between the normal to the grating and the normal to the wave-front of one of the transmitted or reflected waves. Accordingly, there will be angles 6^, 6^, 6^, etc., cor- responding to A, 2\, 3A, etc., in the formula above. Manipulation. — The horizontal track should be set up with one of the vertical supports at one end. This support should be provided with a narrow slit and with clamps or brackets so as to hold the two meter sticks horizontal at right angles to the length of the track, and just below the level of the slit. At the other end of the track should be placed the second support which is provided with a slot or pocket in which is placed the grating. Place the sodium flame behind the slit and let the beam of light from the flame fall upon the grating. If the grating is 132 placed in the slot of the second support so that the rulings arc vertical, several images of the illuminated slit will be seen symmetrically placed upon either side along the length of the two meter sticks, provided these have been properly adjusted. The distances between these images will depend upon the dis- tance between the grating and the slit and upon the distance between the rulings of the grating. If white light had been used in place of the monochromatic light, what would have been the difference in the result that is obtained? Try it and see. To do this, simply place the fingers over the air supply at the base of the Bunsen burner and note the result. Note and re- cord the distance between the grating and the slit, and the dis- tances between several succesive images and the slit, noting in each case the order of the image. Now move the grating a little nearer the slit and repeat these observation as above. From these data the tangent of the angle through which the light is bent by diffraction can be readily determined and hence the sine of the angle of the diffraction can easily be found. Knowing the grating space, which will be given by the instructor, and the order of the spectrum, the wave-length of sodium light can be determined. As a final value, the mean of all the separate determinations should be taken. With a lithium bead and a clean Bunsen flame repeat the observa- tions, and from the data thus obtained compute in a similar manner the wave-length of the lithium flame. At least five different settings of the grating should be made in each case. What will be the effect of having a grating ruled with a large number of lines per cm? What is the difference between a spectrum produced by a grating and a spectrum produced b}- a prism? What is a normal spectrum? EXPERIMENT 85. The Magnifying power of a Microscope. Object. — To determine the magnifying power of a micro- scope. Apparatus.- — Microscope, stage-micrometer, eye-microm- eter, pair of dividers, and diagonal scale. References. — Ames, p. 450; Ames and Bliss, p. 457; Edser, p 203; Ganot, p. 594; Hastings and Beach, p. 631 et seq. ; Kohlrausch, p. 184; Louden and McLennan, p. 58; Miller, p. 221 : Nichols' Lab. Man., Vol. I, p. 265 ; Nichols and Franklin, 153 \'ol. Ill, p. 67 ; Tory and Pitcher, p. 77 ; Watson, p. 489 et seq. ; Fig. 16. Fig 16. Theory.^ — A converging lens system will produce a real image, c'd', of an object (or an image) cd. The magnifying power is the ratio of the length of c'd' to the length of cd. If .-I be the distance from the lens to the object and b be the dis- tance from the lens to the image, the magnifying power is equal to the ratio of b to a. In the compound microscope such a real image of the object is formed and this image is further magni- fied by the eye-piece. If K be the magnifying power of the eye-piece, and M be the total magnifying power of the micro- scope, then Kb M = — a As the tube is changed in length by raising or lowering the eye-piece, both b and a change, and the magnifying power varies as the ratio of b to a varies. Manipulation. — With the eye-piece tube pushed entirely in, put the stage-micrometer on the stage and focus on it. Bring the lines which are one-tenth millimeter apart into the field. If the eye-micrometer is present, determine by means of it the act- ual distance between these tenth millimeter lines. The ratio of the distance between two adjacent lines to one-tenth millimeter will be the magnifying power. If there is no eye-micrometer, place a piece of paper on the side of the stage and look with one eye through the microscope at the lines on the stage-mi- crometer and with the other outside the microscope at the paper. The two will appear superposed and the lines of the micrometer will seem to be on the paper. With a pencil outline six or more lines. With dividers and scale measure the distance between these and then determine the magnify- ing power. Now measure the distance from the object to the objective and from the objective to the image. The image is formed in the focal plane of the eye-piece. The position of this plane, which is at one of the diaphragms, can be deter- 154 mined by putting a fine hair into the tube and seeing whether it is in focus when the eye-piece is in place. Move the tube with the eye-piece up an inch or more and determine the magnifying power again. Measure the distances a and b again. Repeat for different positions until the tube is pulled out to its greatest length. Record the magnifying power and the values of a, b, and K for the different positions. Is K a constant? Make a diagram of a compound microscope showing how the image of an object is formed and how it is viewed by the eye-piece. TABLES OF PHYSICAL CONSTANTS 157 ELASTICITY CONSTANTS OF SOLIDS IN DYNES PER SQUARE CENTIMETiER. young's rigidity elastic substance modulus modulus limit Aluminum 6.5 2.4 — 3.3 4.5—11.0 Brass 7.7— 9.3 3.1—3.6 Copper 8.5—11.6 3.2—4.5 3.0— 7.0 Glass 4.0— 6.0 1.7—2.4 2.3 Iron, cast 6.0—10.8 2.6—4.2 7.0 Iron, wrought 19.3—20.9 7.7—8.5 20.0 Lead 1.7 2.0 Nickel 20.0 Steel 20.1—21.6 8.0—8.8 33—40 Silver 7.0 2.6—3.0 3.0—11.0 Tin 4.5 0.2 Wood 0.7—1.5 .05'-.1.3 1.5—2.4 Zinc 9.0 0.15—0.5 Platinum 15.0 6.1—6.6 14.0—26.0 DENSITIES OF VARIOUS SUBSTANCES. SOLIDS. Aluminum 2.7 Brass 8.2— 8.7 Copper 8.5— 8.9 Cork 0.20 German Silver.. 8.3 — 8.77 Glass, common.. 2.4 — 2.6 Glass, flint 3.0— 5.9 Gold 19.3 Ice 0.917 Iridium 21.8—22.4 Iron 7.1— 7.8 Ivory . . . Lead . . . . Nickel . . Platinum Silver . . . Tin 1.9 11.3 8.8 21.4 10.5 7.3 Wood 0.35— 1.2 LIQUIDS. Alcohol 0.789 Carbon Disulfid 1.264 Chloroform 1.489 Ether 0.715 Glycerin 1.23 Petroleum 0.88 Sulfuric Acid . . . Nitric Acid Turpentine .... Sea Water Mercury at 0°C. Pure Water at 4°C . 1.832 . 1.522 . 0.87 . 1.026 .13.595 . 1.000 GASES. Air 0.001293 Carbon Dioxide. .0.001965 Chlorine 0.003091 Hydrogen 0.00008987 Nitrogen 0.001251 Oxygen 0.001429 158 VELOCITY OF SOUND IN METERS PER SECOND IN DIFFERENT MEDIA. SOLIDS. Meters Meters Sec. Aluminum 5104 Brass 3500 Copper 3560 German Silver 3700 Glass 5026 Iron 5127 Iron, soft 5000 Ivory 3013 Sec. Lead 1228 Nickel 4700 Steel 4990 Steel, soft 5000 Silver 2610 Tin 2500 Zinc 3700 LIQUIDS. Meters Meters Sec. Alcohol 1160 Petroleum 1395 Sec. Sea Water 1400 Water 1435 GASES. Meters Sec. Air 331.8 Carbon Dioxide 261.6 Carbon Monoxide . . . 337.1 Hydrogen 1286.0 Meters Sec. Illuminating Gas 490.4 Nitrogen Oxygen 317.2 159 DENSITY OF WATER AND MERCURY AT DIFFER- ENT TEMPERATURES CENTIGRADE. Temper- ature. 1 2 3 4 5 () 7 8 9 10 12 14 16 18 20 22 24 26 28 30 100 Water. 099987 jO.99993 0.99997 0.99999 1.00000 0.99999 0.99997 0.99993 0.99988 0.99981 0.99973 0.99953 0.99927 0.99897 0.99862 0.99823 0.99780 0.99732 0.99681 0.99626 0.99567 0.95886 Mercury. 13.596 13.593 13.590 13.588 13.586 13.583 13.580 13.578 13.575 13.573 13.571 13.567 13.562 13.557 13.552 13.547 13.542 13.537 13.532 13.528 13.523 13.352 BOILING POINT OF WATER AT DIFFERENT BAROMETRIC HEIGHTS. Barometer Height Boiling Point. .5.50111111 9t.20 .->(;o !)l.S(i 570 92.15 .580 92.62 590 93.08 600 93.53 610 93.97 620 94.40 630 94.80 640 95.27 650 95.70 Barometer Height, 660 (mt 680 690 700 710 720 730 740 750 760 770 780 790 800 Boiling Point. 96.10 !>(>.."i2 96.92 97.32 97.71 98.11 98.50 98.88 99.25 99.63 100.00 100.37 100.73 101.09 101.44 160 VAPOR PRESSURE OF WATER BETWEEN— 10° C AND 100 °C. T represents the temperature in degrees centigrade and P the pres- sure in millimeters of mercury. T. P. T. P. T. P. -10 1.9 42 61.0 -S 2.3 44 67..S -u 2.S 4(i 75.1 —I 3.3 48 83.2 _o o.S 50 92.0 !)l) 4.(! 52 lOl.G !)1 •546.2 2 5.3 54 112.0 92 567.1 4 6.1 56 123.3 93 588.T 6 7.0 58 135.6 94 611.0 8 8.0 60 148.9 95 634.0 10 9.2 62 163.3 96 657.7 12 10.5 64 178.8 97 682.1 14 12.0 66 195.7 98 707.3 16 13.6 68 213.8 99 733.2 18 15.5 70 233.3 100 760.0 20 17.5 • 72 254.3 22 19.8 74 276.9 24 22.4 76 301.6 26 25.3 78 327.6 28 28.4 80 355.4 30 31.8 82 385.2 32- 35.3 84 417.0 34 39.5 86 451.0 36 44.2 88 487.3 38 49.3 40 54.9 161 HEAT CONSTANTS OF COMMON LIQUIDS AND SOLIDS. Substance c o i'i ;3l K a. o 13 O O Melting Point Degrees C. a ■§■2 a a '4-1 N I c m Alu Bra Cop GeiL minum 0.000023 0.000019 0.000017 0.000018 0.000009 0.000015 0.000012 0.000029 0.000013 0.000009 0.000019 0.000011 0.000023 0.000029 0.001100 0.001630 0.000182 0.000112 0.22 0.093 0.093 0.095 0.19 0.031 0.11 0.031 0.11 0.032 0.05H 0.12 0.054 0.094 0.58 0.54 0.0332 0.50 0.48 1.00 0.42 0..36 0.25 0.93 0.08 0.0015 0.70 O.lM 0.08 0.14 0.17 1.09 0.09 0.15 0.30 0.62' ' 0.0057 657 900 1084 1000 1100 1064 1300 327 1470 1775 961 1350 232 419 -110 —118 -38.8 0.0 ss man Silver. ...... Glass Gol Iror iisa Nicl Plat Silv Stee Tin Zinc i 30 i eel 4.6 27 21 1 13 28 2^8' 80 2i6' 90 62.0 930 r Alcohol 78 3 Ether 34.9 (D- Mercury jce 357 Water 0.000180 0.000940 0.0014 537 70 100 ^Turpentine 159 0) Note: — The coefficients of expansion are for cubical expansion. 162 SPECIFIC RESISTANCE IN OHMS PER CUBIC CENTI- METER OF VARIOUS SUBSTANCES, TOGETHER WITH THEIR TEMPERATURE COEFFICIENTS. SUBSTANCE Specific Besistance Tunperatnn CoaBciiat Aluminxiin Antimony Bismuth Brass Carbon, gas Copper, annealed Copper, liarfl drawn _ German Silver Gold Iron Iron, telegraph wire . Lead . Manganin Mercury . Nickel Platinum Silver, annealed Silver, hard drawn __. Zinc 0.00000289 0.00000450 0.00001200 0.00000800 0.00500000 0.00000157 O.dOOOOlfil 0.00002080 0.00000208 0.00000964 0.00001500 0.00001960 0.00004750 0.00009430 0.00000950 0.00001898 0.000001490 0.000001620 0.00000570 0.00310 0.00388 0.00354 0.00400 -0.00050 0.00390 (l.dO.'iiN) 0.00034 0.00365 0.00550 0.00387 + 0.00001 0.00088 0.00600 0.00247 0.00377 0.00377 0.00360 Ebonite 2.8xl0'« Gutta-percha 4.5x10" Glass 9.1x1011 Mica 8.4x101^ Paraffin 3.4xl0i« ■Selenium 6.0x10* Shellac 9.0x10" 163 SPECIFIC INDUCTIVE CAPACITIES OF VARIOUS SUBSTANCES. SOLIDS. LIQUIDS. Ebonite 2.80 Aoptone - 21.8 (iliiss, hard crown 6.96 Alcohol, ethyl 2.5.0 Glass, light flint (i.72 Benzol 2.27 (rlass, dense flint 7.38 Carbon disulfid 2..18 Glass, extra dense flint 9.90 Ether, ethyl 4.35 Gutta-percha 2.50 Glycerin -1.27 Mica 7.00 Petroleum Paraffin ^ 2.20 Turpentine Rubber, soft 2.40 Water Shellac _ 3.40 Sulfur 3.84 GASES. Air 1.000 Alcohol, methyl 33.0 Carbon dio-xidc 1.0008 Chloroform 5.2 Hydrogen 0.9998 V:n-niHii 0.9985 2.06 2.2.3 75.5 SOME OF THE WAVE-LENGTHS OF THE VISIBLE SPECTRUM WITH THE ELEMENTS WHICH PRODUCE THEM. Wave Length Element in Cm. K __1 0.0000405 Itl, 0.0000421 Ca 0.0000423 cs _...._. „ _ - o.oo()04."'>(; Sr 0.0000461 Tl _. 0.0000535 Ca 0.0000553 Xa 0.0000589 Ca 0.0000619 C;, ^ 0.0000027 LI 0.0000671 K 0.0000768 164 , INDEX OF REFRACTION OF SOME COMMON SUBSTANCES. For the Sodium Wave-length. Substance. ludex. Substance. Index. Air 1.0003 Glass, flints 1.651 Alcohol 1.362 Glass, dense flint 1.963 Calcite, ordinary ray 1.659 Quartz, ordinary ray 1.544 Calcite, extraordinary ray_ 1.486 (Quartz, extraordinary ray- l.'>~>:^ Carbon disulfld 1.629 Rock Salt 1.544 Glass, crown 1.524 Water 1.333 SOME USEFUL PHYSICAL CONSTANTS. 1°=0.0174.53 Radians. 1 Radian^=57°.296. ,r=3.1416 ,r==9.8696 Log,r=0.4971 4^=12.566 1 Inch=2.5400 cm. 1 Foot=30.48 cm. 1 Meter=39.37 in. 1 MiGron=0.001 mm. 1 Ounce (Av.)=28.35 grams. 1 Pound (.Av.)=453.6 grams. 1 Kilogram=2.204 pounds. I Gram weight in Salt Lake City=979.8 dynes. 1 Atmospheric pressure^ 1033.6 grams. Volts X Amperes=Watts. Volts x Amperes x Seconds^ Joules. 1 Joule=10'' ergs. 1 Joule=0.239 calorie. 1 Horse-power =746 ^^''atts. Mechanical Equivalent of Heat=4.184xl0'' ergs. Velocity of Light in Vacuum=2.9989xl0^'' cm per second= 186,000 miles per second. Velocity of Sound in Air at 0''C=331.8 meters per second. INDEX 167 INDEX. Absolute expansion of mercury, coefficient of . .■ 101 Acceleration of gravity, determined by Kater's pendulum 43 Acceleration of gravity, determined by physical pendulum 41 Acceleration of gravity, determined by reversible pendulum 39 Air thermometer 102 Air thermometer, the Jolly 103 Angle of a prism, measurement of 146 Angular acceleration 26 Angular acceleration apparatus 26 Angular harmonic motion apparatus 30 Angular harmonic motion, verification of laws of 30 Atwood's machine 38 Atwood's machine, Gaertner pattern 35 Back lash in a screw 13 Boyle's law 68 Boyle's law verified 68 Calibration of a glass prism -, 150 Calibration of a mercury thermometer 83 Calibration of a screw 12 Calibration of the bore of a glass tube 14 Calibration of the bore of a thermometer 85 Calorimeter, water equivalent of 86 Cathetometer 20 Center of mass, formula of verified 51 Centrifugal force, verification of law of 50 Charles' law 103 Coefficient of absolute expansion of mercury determined by Reg- nault's method 101 Coefficient of apparent expansion of mercury determined by the weight thermometer 99 Coefficient of expansion of air at constant pressure 106 Coefficient of expansion of air at constant volume by the Jolly air thermometer 103 Coefficient of expansion of mercury 99 Coefficient of friction 45 Coefficient of linear expansion 97 Coefficient of linear expansion determined 96 Coefficient of restitution 62 Coefficient of rigidity by method of oscillations 67 Coefficient of rigidity by twisting 65 Coincidence of swing of two pendulums 41 Comparator 16 Concave mirror, focal length determined 142 Concave mirror, verification of the layvs of 142 Current measured by electrolysis 132 Density determined by the Nicholson hydrometer 59 Density determined by the specific gravity bottle ,. 57 Density of a liquid by balanced liquid columns 60 12 168 Density of a liquid by the pyknometer , 58 Density of a solid 55 Diagonal scale, use of 11 Diameter of a glass tube measured ". . . 19 Dip of the earth's magnetic field determined 121 Dispersion of a prism as a function of the wave-length ISO Earth's magnetic field, horizontal intensity of at different places, compared 122 ■Earth's magnetic field, horizontal intensity of, determined by electrolysis ; : . 134 Earth's magnetic field, horizontal intensity of, determined by mag- netometer 123 Eiectrocalorimeter in determining mechanical equivalent of heat. . 134 Electrostatic induction 115 Equilibrium of a body under the action of parallel forces, condi- tion of 51 Equilibrium of forces acting at a point, verification of laws of . . . 47 Equipotential lines defined 137 Equipotential lines determined 137 Falling bodies, determination of the laws of 23 Focal length of a lens 143 Focal length of a concave mirror 142 Force table, the 47 Frequency of vibration of a tuning fork, absolute determination of .79 Friction, coefficient of, determined , 45 Fusion of ice, heat of, defined 86 Fusion of ice, heat of, determined 86 Fusion point of a solid defined 95 Fusion point of a solid determined 94 Gay Lussac's law 103 Glass prism, calibration of ISO Grating, plane 151 Grating, plane, used in measuring wave-length of light 151 Gravity, determination of by Atwood's machine 37 Heat of fusion of ice 86 Heat of .fusion of ice, determined 86 Heat of vaporization apparatus 89 Heat of vaporization of water defined 88 Heat of vaporization of water determined 88 Heat of vaporization of water determined, electrical method 89 Hooke's law 63 Hooke's law, verification of, by bending 64 Hooke's law, verification of, by stretching 63 Impact, elastic SO Impact, inelastic 50 Index of refraction determined by u microscope 141 Index of refraction of a prism .' 148 Jolly air thermometer 104 Jolly air thermometer in determining the coefficient of expansion of air at constant volume 103 Joly steam calorimeter 92 Jbly steam calorimeter in determining specific heat . 91 Joule's law 135 m Kater's pendulum in determination of gfivity. '; . . . 43 Kundt's method of determining velocity of sound in metals ' 75 Linear acceleration 23 Linear acceleration apparatus 24 Linear expansion, coefficient of 97 Linear expansion, coefficient of, determined 96 Linear harmonic motion, verification of the laws of 29 Linear momentum, verification of the laws of 40 Lines of current flow defined 137 Lines of current flow determined 137 Magnetic dip, determination of 121 Magnetic fields, plotting of 120 Magnetometer in determining the ratio of M to H 123 Magnifying power of a microscope 152 Mance's method of measuring the resistance of a battery 131 Maximum vapor pressure of a liquid 107 Mechanical energy, loss of on impact 62 Mechanical equivalent of heat, determined by electrocalorimeter. . 134 Mechanical equivalent of heat, determined by Puluj's method.... 109 Mechanical equivalent of heat, value and definition of 109 Micrometer caliper, use of 11 Micrometer microscope, use of 19 Minimum deviation, angle of, of a prism measured 148 Molecular depression - 96 Molecular depression, determination of 95 Moment of inertia by angular acceleration '. 34 Moment of inertia of a regular solid about a given axis 32 Moments, verification of the laws of 51 Motion of a body under a constant force, verification of the laws of 35 Optical lever 63, 97 Parallelogram of forces, verification of the law of, for the addi- tion of two forces 48 Physical pendulum 41 Physical pendulum in determining the acceleration of gravity. ... 41 Physical pendulum, variation in periodic time as position of knife- edges is changed 44 Pitch of a screw 12 Pressure coefficient of air at constant volume, determined by the air thermometer 102 Probable error 23 Puluj's method of determining mechanical equivalent of heat.... 109 Pyknometer in determining density of a liquid 58 Radiation from a warm body 93 Radius of curvature of a lens or mirror, measurement of 21 Ratio of M to H by magnetometer method 123 Regnault's expansion of mercury apparatus 101 Regnault's method of determining the coefficient of absolute ex- pansion of mercury 101 Resistance measured by the Wheatstone bridge 127 Resistance of a battery measured by Mance's method 131 Resistance of a galvanometer measured by method of shunts.... 130 Resistance of a galvanometer measured by Thomson's method 129 Resonance 77 Restitution, coefficient of 62 170 Reversible pendulum in determination of gravity 39 Rigidity, coefficient of, determined by method of oscillations 67 Rigidity, coefficient of, determined by twisting 65 Sensibility of a balance 52 Sensibility of a balance, determination of 54 Shunts, law of verified 126 Sonometer in verifying the law of vibrating strings 74 Specific gravity 56 Specific heat- of a solid determined 90 Specific heat of a solid deteijmined by the Joly steam calorimeter 91 Spectrometer, adjustment of 146 Spherometer 21 Stage micrometer 19 Stationary vibrations 73 Tangent galvanometer, constant of, determined 132 Thermometer, calibration of 83 Thermometer, calibration of the bore of 85 Thomson's method of measuring the resistance of a galvanometer 129 Tocpler-Holtz machine 117 Traveling microscope 12, 14 Uniformly accelerated motion of rotation, test of the equations of 26 Uniformly accelerated motion of translation, test of the equa- tions of .23, 35, 37 Vaporization of water, heat of, defined ?8 Vaporization of water, heat of, determined 88 Vaporization of water, heat of, determined by the electrical method 89 Vapor pressure of a liquid, the maximum, determined 107 Velocity of sound in air determined 11 Velocity of sound in metals, determined by Kundt's method 75 Vernier caliper, use of 11 Vibrating air columns 76 Vibrating strings, law of verified by sonometer 74 Volume of a cylinder 11 Water equivalent of a calorimeter defined 86 Water equivalent of a calorimeter determined 87 Wave-length of light measured by means of plane grating 151 Weight thermometer in determining cubical coefficient of appar- ent expansion of mercury 99 Wheatstone bridge in measuring resistance 128 Young's modulus defined 63 Young's modulus determined by bending 64 Young's modulus determined by stretching 63