LIBRARY OF THE UNIVERSITY OF CALIFORNIA. GIFT OF s^C/t/u ::i s i%a,nw.^Wi*S b >To' fi ;a ( ^'^.;-a^S51i^/sr^-^ *!^ifi3KirvL^CkI ^,% PHYSICAL MEASUREMENTS IN PROPERTIES OF MATTER AND HEAT BY ELMER E. HALL Assistant Professor of Physics, University of California AND T. SIDNEY ELSTON Instructor in Physics, University of California BERKELEY, CALIFORNIA 1908 Entered According to Act of Congress in the Year 1908, by ELMER E. HALL and T. SIDNEY ELSTON, In the Office of the Librarian of Congress at Washington. Printed by The Times Publishing Company Palo Alto, California PREFACE. This manual is printed for the use of students in the Fresh- man year in the various colleges in the University of Califor- nia, and represents the laboratory side of a three-unit course consisting of one lecture, one recitation, and one laboratory period per week throughout the year. The course is preceded by a matriculation course in Elementary Physics and is a part of a two-years course in General Physics, the work in Sound, Light, and Electricity being given during the Sophomore year. The present form of the course is a revision of the work as given heretofore. The predecessors of the present writers, Professor Harold Whiting, A. C. Alexander, G. K. P>urgess, Bruce V. Hill, and A. S. King, have all left their imprint on the course. The writers are especially indebted to Dr. King, whose mimeographed directions they have freely used. Sev- eral experiments have been taken from the Sophomore course of Professor E. R. Drew, written before the present division of subjects was made. Free use has been made of the pub- lished texts and manuals, as is indicated by the references given. ELMSR E. HALL, Berkeley, Cal., July, 1908. T. SIDNEY ELSTON. 205947 LIST OF EXPERIMENTS. 1. Use of the Balance. Density of a Solid. 2. Density by Jolly's Balance. Archimedes' Principle. 3. Boyle's Law. 4. The Volumenometer. 5. Density of Air. 6. Relative Density of Carbon Dioxide. 7. The Force Table. 8. Equilibrium Conditions for Three Forces at a Point. 9. Uniformly Accelerated Motion. 10. The Force Equation. 11. The Simple Pendulum. 12. The Principle of Moments. 13. The Model Balance. 14. Surface Tension by Direct Measurement. 15. Capillarity. Rise of Liquids in Tubes. 1 6. Rise of Liquids Between Plates. 17. Pressure and Radius of a Soap-Bubble. 1 8. Viscosity. Flow of Liquids in Tubes. 19. Relative Densities of Gases. Time of Efflux. 20. Calibration of a Thermometer, Absolute. 21. Calibration of a Thermometer, Relative. 22. Variation of Boiling Point with Pressure. 23. Expansion of a Liquid by Archimedes' Principle. 24. Comparison of Alcohol and Water Thermometers. 25. Expansion of Mercury by Regnault's Method. 26. Expansion of Glass by Weight Thermometer. 27. Expansion of a Liquid by Pycnometer Method. 28. Expansion Curve of Water. 29. Specific Heat of a Liquid by Method of Heating. 30. Specific Heat of a Liquid by Method of Cooling. LIST OF EXPERIMENTS V .31. Mechanical Equivalent of Heat by Method of Percus- sion. 32. Mechanical Equivalent of Heat by Method of Puluj. 33. Cooling through Change of State. 34. Heat of Fusion. 35. Heat of Vaporization at Boiling Point. 36. Heat of Vaporization at Room Temperature. 37. Freezing Point of Solutions. 38. Heat of Solution. 39. Heat of Chemical Combination. 40. Expansion of Air at Constant Pressure by Flask Method. 41. Expansion of Air at Constant Pressure. Constant-Pres- sure Air-Thermometer. 42. Constant- Volume Air-Thermometer. 43. Variation of Pressure, Volume, and Temperature of a Saturated Vapor. 44. Hygrometry. 45. Density of the Air by the Barodeik. 46. Coefficient of Friction. 47. Conservation of Momentum. Coefficient of Restitution. 48. Young's Modulus by Stretching. 49. Hooke's Law for Twisting. Coefficient of Rigidity. 50. Centripetal Force. 51. Friction Brake. Power Supplied by a Motor. 52. Absorption and Radiation. 53. Ratio of the Two Principal Specific Heats of a Gas. REFERENCES. References are given throughout the manual to the follow- ing books. Except in cases of confusion the reference gives the author's name, omitting the title of the book. Ames and Bliss, A Manual of Experimental Physics. Edser, Heat for Advanced Students. Ferry and Jones, A Manual of Practical Physics, Vol. I. Franklin, Crawford, and MacNutt, Practical Physics, Vol. I. Glazebrook and Shaw, Practical Physics (Third Edition). Hastings and Beach, General Physics. Kohlrausch, Physical Measurements (Third English Edi- tion). Miller, Laboratory Physics. Millikan, Molecular Physics and Heat. Preston, Theory of Heat (Second Edition). Watson, Text-Book of Physics (Fourth Edition, 1903). Watson, Text-Book of Practical Physics. PHYSICAL MEASUREMENTS, PROPERTIES OF MATTER AND HEAT. This book is intended to be mainly a manual of directions. It is expected that the student will consult the manuals given in the list of references (a liberal number of copies of which are available) for general notions regarding physical measure- ments, the discussion of results, the effect of errors in obser- vation and methods for their complete or partial elimination. Mimeographed directions regarding the method of writing up and handing in the record of the experiments, a list con- taining the required experiments, and the order in which they are to be performed will be given each student. The satis- factory completion of thirty experiments fulfills the labora- tory part of the requirement for the year's work. i. USE OF THE BALANCE. DENSITY OF A SOLID. References. Glazebrook and Shaw, p. 91; Miller, p. 55; Kohl- rausch, p. 30; Franklin, Crawford, and MacNutt, p. 23. In weighing with a sensitive balance the indications are made by means of a long pointer attached to the beam and arranged to vibrate in front of a fixed scale. If the balance is sensitive the pointer will swing many times back and forth before it finally comes to rest at a definite point which marks the position of equilibrium. Time would be wasted in waiting for it to stop, and even then the indications of the moving pointer are more trustworthy than those of one which has come to rest, because the latter may not be in its true position 2 USE OF THE BALANCE. DENSITY OF A SOLID. f I of equilibrium, or rest-point, owing to friction. To obtain the rest-point, the pointer is allowed to vibrate and the turning- points of a number of consecutive swings are read, the num- ber being so chosen as to give an even number of turning points on one side and an odd number on the other. A little consideration will show that under this condition the point halfway between the mean of all the left-hand and the mean of all the right-hand readings is the true rest-point. This way of getting the rest-point is known as the "method of vibra- tions.'' The sensitive balance must be handled with the greatest care, since any jarring or rapid vibration of the beam is likely to injure the knife-edges upon which the beam rests. On this account the beam should be lowered each time before a mass is placed on the pan or removed from it, and also when the weighing is completed. To illustrate the use of the balance, let it be required to find the density of a solid (a hard rubber cylinder, for ex- ample). (a) With the pans of the balance empty, raise the beam slowly and allow the pointer to swing over four or five scale divisions. Take and record an even number of turning points on one side and an odd number on the other (respectively 4 and 3, say), and from them determine the rest-point. Make two determinations in this way, and take the mean as the zero rest-point. The door to the glass case should always be closed when determining the rest-point. (b) Place the hard rubber cylinder on the left-hand pan of the balance, and add masses to the other pan until the pointer does not swing off the scale when the beam is raised. In making trials for the correct mass on the right-hand pan, raise the beam only high enough to see which side has the greater mass, in order to avoid violent rocking of the beam. Use the fractional masses to bring the pointer approximately to the zero rest-point, and then determine the rest-point by i] USE; OF THE BALANCE:. DE)NSITY OF A SOLID. 3 the method of vibrations. To determine how much must be added to, or subtracted from, the masses on the right-hand pan to take account of the fact that the rest-point with the loaded balance does not coincide with the zero rest-point, add a 5 or 10 mg. mass to either pan and determine the sensitive- ness of the balance, that is, determine the mass which, added to either pan, will move the pointer through one division. From this, the difference in rest-points, and the masses in the right-hand pan, find the exact mass which will balance the hard rubber cylinder in air. (In fine weighing it is often con- venient to use a centigram "rider." By properly placing the rider on the graduated scale attached to the beam, the equiv- alent of any desired mass from i to 10 mg. may be added to either pan. Final adjustments can thus be made without open- ing the balance case. The rider should never be moved without first lowering the balance beam,.) (c) Unless the body whose mass is sought has the same specific weight as the masses used to balance it, the body will be buoyed up by the air either more or less than the masses are buoyed up, and this will introduce an error which is by no means negligible in careful measurements. To correct for air-buoyancy: Measure the dimensions of the cylinder with vernier calipers, and compute its volume. Calculate the vol- ume of the standard masses from their marked values and the density of brass (8.4 gms. per cc.). From these volumes and the density of air at the temperature and barometric pressure at the time of the experiment (see Tables), determine the correction for air-buoyancy. Calculate the density of the cyl- inder. (d) If the arms of the balance are unequal in length, "double weighing" is necessary. Place the cylinder in the right-hand pan and find the mass as before. The true mass is then given by m=\/ni 1 m 2 , where m t and m 2 are the values obtained by the two weighings. The proof of this by an appli- cation of the principle of moments is left to the student. 4 DENSITY OF A SOLID BY JOLLY'S P.. \I..\\CK. [2 2. DENSITY OF A SOLID BY JOLLY'S BALANCE. References. Ames and Bliss, p. 197; Ferry and Jones, p. 97. Jolly's balance consists of a long" spiral spring suspended in front of a graduated mirror. As adapted to this experiment the upper end of the spring may be raised or lowered : the lower end carries two light pans. The lower of the two pan- is always immersed in a beaker of water standing on a small platform which moves up and down along the frame of the apparatus. The use of the balance to measure weight depend-* on the fact that the spring obeys Hooke's law closely for small elongations, i. e., the elongation is proportional to the change in the stretching force. Thus, with the lower pan immersed in water, we note that the weight of a certain mass, sav 10 grams, in the upper pan produces an observed elongation of the spring. If then we substitute any solid for the known mass in the upper pan, giving an elongation not very different from the first, the mass of this solid may be readily computed. If the solid is then placed in the lower pan and immersed in water, the new elongation will give the weight of the solid in water ; or we may add masses to the upper pan until the spring is stretched to the same length with the solid immersed as when it was in the upper pan. Then the weight of the masses we have added gives the apparent loss of weight by the solid in water, or the weight of the water displaced. The mass added is the mass of the water displaced, so from this and the density of the water the volume of the solid can be calculated. (a) By the method outlined above, find the weight jn air and in water of the solids furnished. In taking readings a bead or point of wire at the lower end of the spring is brought into coincidence with its image in the mirror scale. Take care that no air-bubbles cling to the lower pan or to the solid when immersed. Take the temperature of the water at the beginning 3] BOYLE'S LAW. 5 ami at the end of the experiment, use the mean, and from the Tables find the denisty of water at this temperature. From the data thus obtained calculate the density of each solid in grams per cc. (b) Find the density of a solid which floats in water. For this purpose a sinker must be used, but this may be left in the lower pan throughout the experiment. (c) Find the density of a salt-solution by using a solid hung by a thread instead of using the lower pan. Explain the method used. (*>'* 5] DSNSITY"OF AIR. Eliminating d 2 from (i) and (2), *~[ f& mP 1 (3) < V(P, - P s ) In applying the last equation it is essential that the tempera- ture should be the same during the two weighings. If the temperature is not the same, the observed pressure in the sec- ond case must be corrected so as to give the pressure that would have existed had the temperature been the same as dur- ing the first weighing. If T t and T 2 are the temperatures (on the absolute scale) of the air during the first and second weigh- ings respectively, and P 2 ' is the observed pressure in the sec- ond case, the equation P 2 /P 2 ' = T 1 /T 2 , by Charles' law, gives the corrected value of P 2 , the expansion of the glass flask being neglected. The volume, V, is obtained by weighing the bulb full of water at a known temperature. The density, d , at oC. can be obtained from dj by applying Charles' law, where t is the temperature, centigrade, of the air used. (a) Carefully dry the flask by exhausting it several times and admitting air through a calcium chloride drying tube. Ask an assistant for instructions in regard to manipulating the pump. If moisture is visible inside the flask, it may be necessary to put in a little alcohol, rinse the flask, vaporize the alcohol over a Bunsen burner, and pump out. With the dried flask in connection with the manometer and the drying tube, admit air at atmospheric pressure, reading the pressure. Close the stop-cock and carefully weigh the flask. Note the temperature. (b) Pump the air out until as low a pressure as possible is obtained and weigh again at this reduced pressure. Again note the temperature. (c) Fill the flask completely with water up to the stop- IO RELATIVE DENSITY OF CARBON DIOXIDE. [6 cock, taking care to have no water above the stop-cock. Ask an assistant to show you how to fill it. The temperature of the water should be recorded and its density found from a book of Tables. Dry the outside of the flask and then weigh. Calculate the volume of the flask. (d) Using the results obtained in (a), (b), and (c), find the density of the air, in grams per cc., at the given tempera- ture. Calculate the density, d , at oC. Take the value given in the Smithsonian Tables and calculate your percentage error. Point out the chief sources of error in the experiment. 6. RELATIVE DENSITY OF CARBON DIOXIDE. Reference. Millikan, p. 114. The relative density of carbon dioxide compared with air as a standard is to be measured. The method employed is that used in Exp. 5. Using the same symbols as there used, and making the weighings and noting the pressures as there indi- cated, we have for the air, mP, d . = vTPT^p,)' If the measurements are then repeated for the carbon dioxide, m'P/ ** = WT-^ the symbols having the same meaning as in the case of air. From (i) and (2), if D is the relative density of the carbon dioxide, ~d, -mP, (P.'-P, 1 )' from which we see that a determination of the volume of the flask is unnecessary. (a) Read the directions given under Exp. 5. Ask an assist- ant for instructions in the use of the pump. Carefully dry the 7] THE: FORCE: TABLE n flask, and fill it with dry air admitted through the calcium chloride tube. Using a sensitive balance, weigh the flask full of air at atmospheric pressure, noting pressure and tempera- ture. In weighing follow the method given in Exp. i. (b) Pump the air out to a low pressure and weigh the flask again at the reduced pressure. If the temperature is not the same, within o.5, the observed pressure should be corrected as in Exp. 5. (c) Fill the flask with dry carbon dioxide at atmospheric pressure. This can best be done by pumping out the flask and admitting the gas from the generator several times in succes- sion. Take care not to allow any air to pass through the acid into the generator, and keep the stop-cock closed when not using the generator. When the flask is filled with carbon diox- ide at a known pressure and temperature, weigh it as before. (d) Pump the carbon dioxide out to a low pressure, as in the case of the air, and weigh again. ( the acceleration produced, we have, as the result of defini- tion and experiment, ( i ) p = kmp. This equation is called the Force Equation, or the Equation of Motion ; and the purpose of the present study is to verify it experimentally, k, in the equation, is a constant numerical factor, whose value depends upon the system of units used. This equation states ( I ) that, if two forces act on bodies of the same mass, the acceleration produced will be directly pro- portional to the forces; and (2) that, if two forces produce the same acceleration in two bodies of different mass, the masses will be directly proportional to the forces. Let M, M be two equal masses suspended from a cord passing over a pulley whose friction and rotational inertia we will assume to be negligible. The total mass suspended is 2M, the resultant l8 THK I'OKCK EQUATION. [ 1O force acting on it is zero. Let a mass m l be added to one side. The resultant force, P,, now is kn^g, and it will cause the three masses to move in its direction with an acceleration, Pi ; hence, by equation (i), (2) P 1 =k(2M + m 1 )p 1 . If a different force be applied by replacing m l with a mass m 2 , the resultant force, P 2 , will be km 2 g, and it will produce an acceleration, p 2 ; hence, by equation (i), (3) P 2 Hence knvg_ P, _ k( 2 M -j m,)p nciicc -: -pr- -, ', or, km 2 g P. k( 2 M 4- m,)p 2 ,^ -f m, ( 2 M -f- m,) P ; An experimental verification of equation (4) will constitute a verification of equation ( i ) , though it will not, of course', determine the value of the constant, k. The apparatus used in Exp. 9 is employed, with the addi- tion of a pulley-attachment at the top over which a cord passes, from one end of which the fork is suspended and from the other end a number of masses just sufficient to balance the fork and the friction of the pulley. Note the precautions given in Exp. 9. Special care should be taken to insure as little fric- tion as possible. (a) Adjust the apparatus so that a good trace may be obtained and so that a slight tap will cause the fork to descend without acceleration. The forces, including friction, are then just balanced. Whiten the plate with the preparation fur- nished. (b) Remove a mass, m,, from the balancing weights. Note the total mass, M,, of the moving system. Obtain two good traces. (c) Repeat with a different mass, m 2 , removed, the total mass of the system now being M 2 . CT L Il] THE SIMPLE PENDULUM. 19 (d) Repeat again with a third mass removed. (e) Measure the traces as explained in Exp. 9, using ten vibrations of the fork as the interval of time. Calculate the accelerations p lt p 2 , p ?> , corresponding to (&), (c), (d) above. Then, by equation (4), we should have m^/m 2 = M^/M-gp., and m,/m 3 = M^/M^. Test this, calculating the percent- age error in each case. Name the principal sources of error in the experiment, and account for any discrepancies in your results. IT. THE SIMPLE PENDULUM. References. Ferry and Jones, p. 66; Millikan, p. 95. For vibrations of small amplitude the period of a simple pendulum is given by the equation, T= 2 where T is the time of one complete vibration, 1 is the length of the pendulum, and g is the acceleration due to weight. If T and 1 are known for any place, g can be determined for that place. In the present experiment T is to be measured by compar- ing by the "method of coincidences" the period of the simple pendulum with that of a clock pendulum of known period. An electric circuit is completed through an electric bell, the clock pendulum, the simple pendulum, and the mercury contacts at the bottom of each pendulum. Assume that the period of the clock pendulum is two seconds, that is, that the time of a single swing or half-vibration is one second. If the period of the simple pendulum were the same and the two pendulums be started together, they would vibrate in coincidence and the bell would ring with every passage. If, however, the time of a single swing of the simple pendulum were less than one second, say by i/n th of a second, it would gain on the clock 2O TUK SlMl'LK 1'KMH'l.r.M. [ll pendulum and thus fall out of coincidence with it, so that the bell would cease to ring until n seconds later, when the two pendulums would be in coincidence again. Let us sup- pose that the time between these successive coincidences is 100 seconds, then we know that in this time the clock pen- dulum has made one hundred half-vibrations and the simple pendulum one more, or 101 half-vibrations. In other words, the simple pendulum has made 101 half-vibrations in 100 sec- onds, hence the value of its half-period is 100/101 seconds. If, on the other hand, the simple pendulum had been observed to fall behind the clock pendulum, and the time between suc- cessive coincidences remained the same, we would know that its half-period is 100/99 seconds. (a) The simple pendulum used consists of a brass sphere suspended from a knife-edge by a wire so that the length is adjustable. The mercury contact below should be so adjusted that the platinum point on the ball passes freely through it. Adjust the pendulum so that its length is either greater or less, by 2 or 3 cm., than that of a pendulum beating seconds. Two different lengths (in successive determinations) should be used such that one is greater and the other less than that of a pendulum beating seconds. In getting the length it is well to measure with a meter rod and square to the top of the ball, and then to determine the diameter of the ball with the cali- pers. After adjusting, start the ball swinging in an arc of about 10 cm., and record, from the clock in minutes and sec- onds, the times of six successive coincidences between the simple pendulum and the clock pendulum. If the bell rings for more than one swing during each coincidence, take the mean of the times of the first and last rings as the time of the coincidence. Find the difference in time between the first and fourth coincidences, the sec- ond and fifth, the third and sixth, and take the mean. From this calculate the period. Be sure to note whether the pendulum was gaining or losing on the clock. 12] THI$ PRINCIPLE OF MOMENTS. 21 Calculate the value of "g" for Berkeley for the two cases and take the mean. (&) What effect would be produced upon the vibration of a pendulum by carrying it, (i) to a mountain top, (2) from the equator to the pole of the earth? In what way does the pendulum used in this experiment fall short of the require- ments for a simple pendulum? What is the object of taking a small amplitude of vibration ? 12. THE PRINCIPLE OF MOMENTS. References. Millikan, p. 29; Ames and Bliss, p. 118. The purpose of this experiment is to determine the condi- tion which must be satisfied if a body, acted upon by three or more forces in the same plane, is to remain in equilibrium with reference to rotation. In order that zrbody at rest shall remain at rest, or a body in motion remain in motion with con- stant linear and angular velocity, the vector sum of all of the forces acting upon it must be zero and the algebraic sum of the moments of these forces about any axis must be zero. In the case where all the forces are in the same plane, the second of these conditions, sometimes called the Principle of Moments, requires that the sum of the moments of all the forces about any point in the plane selected as a center of moments shall be zero. To prove this it is only necessary to show that the sum is zero for one selected point, provided that this point is so chosen as not to lie in the line of any of the forces. (The proof of this for the case where there are three forces is left to the student.) If a point in the line of any force were chosen, the moment of that force with reference to that point would be zero no matter what the value of the force ; hence, the result would not be a test of the principle. The apparatus used to test the principle consists of a circu- lar table with a movable disk resting on bicycle balls. The 22 THE PRINCIPLE OF MOMENTS. [l2 disk may be pivoted in the center if desired. To pegs, placed at will in the disk, cords are attached which pass over pulleys clamped at different points around the circular table. From the ends of the cords are suspended known masses whose weight produces the forces required. (a) Pivot the disk in the center and place a sheet of manila paper upon it. Attach cords to the disk at three different points chosen at random ; and, placing the pulleys at any con*- venient points, add masses until the three forces are of conven- ient values. See that the disk is free to move on the bicycle balls ; then mark points or lines on the paper to indicate the directions of the forces. Note the magnitude of the forces, counting in the weight of the pan in each force. (b) Remove the paper, trace the lines of direction of the forces, and make the measurements necessary to determine their moments about the pivot as an axis. Find the sum of the moments, taking those as positive which tend to produce a counter-clockwise rotation about the given axis and those as negative which tend to produce a clockwise rotation. (c) Choose any arbitrary point on the paper used in (a) and (b), and find the sum of the moments about this point as a center of moments. Why is not the sum zero? (d) Remove the pivot, and repeat (a) and (b) once, selecting in turn as centers three points as widely separated as possible. Find the sum of the moments as before. Also find the vector sum of the forces by the method of the closed polygon. (e) Repeat (d), using four forces instead of three. (/) From the data of (a) and (b) determine the vector sum of the three forces used in that case. If this sum is not zero, it means that the pivot itself exerted a force on the disk in the same plane with the three forces. What do you con- clude is the magnitude and direction of this force? Draw- its line of direction on the paper, and then repeat (c), includ- 13] THE MODEL, RALANCE. 23 ing now in your sum the moment of the force clue to the pivot. Is the sum now approximately zero? (g) In the various cases of equilibrium considered above, what do you find the vector sum of the forces to be? What have you found to hold true for the moments of these forces? Calculate the percentage error for one case. 13. THE MODEL BALANCE. Reference. Glazebrook and Shaw, p. 83. A model balance is a simplified beam balance used to test the relation between the sensitiveness of the balance and its dimensions and load. By "sensitiveness" is meant the facil- ity with which the pointer of tthe balance can be deflected when there is a small difference between the masses suspended from the two sides of the beam. The sensitiveness of the bal- ance depends upon the length and mass of the beam, the load in the pans, the distance between tRC^enter of weight of the beam and the central supporting knife-edge, and upon whether the beam is straight or curved up or down. To obtain an expression showing the character of this dependence we need to apply the principle of moments. Let m = the mass of the beam, 1 = the length of the beam-arm (the two being as- sumed equal), M = the mass hung on each side, including the mass of the scale-pan, h = the distance from the central knife-edge to the center of weight of the beam, x = a small excess mass placed in one pan, a = the deflection produced by the addition of x, ft = the angle, for the given load, between a horizon- tal line and the line drawn from the central knife-edge to the knife-edge at either end, when the beam is so placed that the two angles which can be thus formed are equal, ft will be 24 THE MODEL BALANCE. [13 positive if the beam is concave upwards, negative if the beam is concave downwards. Applying 1 the principle of moments for the case of equilibrium, the central knife-edge being the center of moments, we have (l) (M-fx)gl COS ( a) MglcOS ( + a) mgh sin a = o. Expanding, collecting terms, and transposing, [mh (2M -{- x) sin ft] sin a = Ix cos (3 cos a, or (->\ tan a 1 cos /S ~x~ " mh (2M + x) 1 sin 0' If the beam is straight, ft = o and tan a _ ]_ x " mh' The sensitiveness is measured by the ratio, tan o/x. Hence, in the case of a straight beam, it is independent of the load and increases with any arrangement which makes the fraction, 1/mh, larger. In the case of a curved beam the sensitiveness is dependent upon the load and also upon the extent and direction of the curvature. The model balance provided allows ample modification of the several quantities in equation (2). The following possi- bilities are at once apparent : ( i ) the center of weight of the beam may be raised or lowered according as the central knife- edge is placed in the lower or upper hole in the beam; (2) the length of the beam may be varied by hanging the masses at different distances from the center; (3) the mass of the beam may be increased by inserting a brass cylinder in the hole which marks the center of weight of the beam ; (4) the points of application of the masses may be placed level with, above, or below the central knife-edge, thus making the beam straight, or curved up or down; (5) the ratio of the lengths of the beam-arms may be varied. 13] THE MODEL BALANCE. 25 (a) Starting with the knife-edge in the upper hole of the beam, adjust so that the pointer hangs at the middle of the scale. Hang successively several one-gram masses from the peg at one end of the upper row, noting if the deflection is -jM / approximately proportional to the number of masses used, dr y Why should the deflection not be strictly proportional to the number of masses ? Remove the masses and hang an unknown mass in their place. From the deflection produced, calculate the mass of the unknown. V^>?* (b) Hang the one-gram masses from the inner peg of the upper row. Compare the results with those of (a), and state how the sensitiveness depends upon the length of the beam- arm, other conditions remaining the same. (c) Place the brass cylinder in position at the center of weight of the beam, and thus increase its mass. Determine how this affects the sensitiveness. (d) Remove the brass cylinder and change the knife-edge to the lower hole of the beam. Test the sensitiveness and compare with (a). How does the sensitiveness depend upon the position of the center of weight of the beam with refer^ ence to the central knife-edge? (e) Change the knife-edge back to the upper hole of the beam. Hang a 5o-gram mass on each end peg of the upper row. Test the sensitiveness and compare with (a). Does the sensitiveness for this form of beam change with the load. If so, account for it. (/) Transfer the 5<>gram masses to the end pegs of the lower row, and test the sensitiveness. Compare with (e). How is the sensitiveness affected when the beam is curved down? (g) Repeat (/) with the 5o-gram masses removed. Com- pare with (a). When the load is small what effect has the curvature on the sensitiveness? A sensitive chemical balance is usually made with the beam curved slightly upwards when there is no load in the pans A 26 SrKI'.U'K TK.XSIOX r,V DIRECT MKASrkKMF.XT. [14 medium load straightens the beam and an excess load causes a downward curvature. From the results above show how the sensitiveness changes with the load on account of the curvature of the beam. Illustrate by a diagram, applying the principle of moments to explain the action. 14. SURFACE TENSION BY DIRECT MEASURE- MENT. References. Watson, p. 189; Hastings and Beach, p. 138; Milli- kan, p. 195. The method employed will be that of making the force due to surface tension in a given case the equilibrant of a known force. A wire rectangle is hung from the spring of a Jolly's balance and allowed to dip in a soap solution which forms a film across the rectangle. When equilibrium is established the force due to surface tension in the two surfaces of the film must just balance the tension in the spring. By knowing the force which will stretch the spring the same amount we have a measure of the product of surface tension (which 1 is the force per centimeter width exerted by the surface film) and twice the width of the rectangle where it is cut by the surface of the solution. If T is the value of the surface tension, 1 the width of the rectangle along the surface of the liquid, and F the force exerted by the spring, write the equation giving the value of T. The Jolly's balance used is a very sensitive one, and must be handled with great care. Ask for directions if its opera- tion is not already understood. The catch above the pan of the balance allows only a small motion of the lower end of the spring, the extension of the latter being produced by raising the upper end of the spring by means of a telescoping tube moved by rack and pinion, the scale and vernier on this tube measuring the extension. To make a measurement the clamp holding the catch is raised or lowered (the platform holding the beaker being adjusted at the same time) until the spring 14] SURFACE TENSION BY DIRECT MEASUREMENT. 27 almost supports the weight, after which an exact setting is made by means of the telescoping tube. If any change is then made in the pull on the end of the spring, this change is measured by the amount the spring must be shortened or lengthened by the telescoping tube to restore the equilibrium. Wire rectangles of different sizes and a wide beaker are pro- vided. The greatest care must be taken that the beaker and rectangles are clean. They should be washed in caustic potash and rinsed thoroughly in hot water before being used and before changing to another liquid. Do not touch the inside of the beaker, the liquid, or the part of the rectangle on which the film is formed. (a) Suspend a rectangle, 2 cm. wide, from the spring, and let it be partially immersed in a beaker of soap solution. Read the extension of the spring when there is no film in the rect- angle, and again with a film across it. Take three sets of read- ings. Note whether the pull of the film depends on the depth to which it is immersed. If it does depend on the depth, be careful to have the same depths in all measurements. Repeat these measurements using rectangles 4 cm. and 6 cm. wide. (b) Calibrate the balance by observing the extension pro- duced by known masses. (c) Use the rectangle 4 cm. wide, cleaning it and the beaker thoroughly, and repeat (a) with water fresh from the tap. As no film will form with pure water, take the reading of the balance when the under side of the upper wire of the rectangle is just above the surface of the water and not in contact with it ; and again, after immersing the upper wire of the rectangle so as to wet it, take a reading when it breaks away from the surface. Take three sets of readings. (rf) Repeat (c), using water to 50 C. or higher. (e) Repeat (c), using alcohol. (/) From the data taken in (a), state how the total tension in the film varies with its width. Calculate the surface ter- 28 CAPILLARITY. RISE OF LIQUIDS IN TUBES. [15 sion, T, in dynes per cm., for the liquids used in (a), (c), (d) t and (e), comparing the values obtained and pointing out how the surface tension is affected by the temperature. 15. CAPILLARITY. RISE OF LIQUIDS IN TUBES. References. Watson, p. 194; Watson's Practical Physics, p. 139; Millikan, p. 194; Miller, p. 113. In the present experiment the values of the surface tension of water and of alcohol are to be measured by observing the rise of these liquids in capillary tubes. When the inner sur- face of a tube is wet by a liquid, the surface tension of the latter may be considered as acting vertically upward at all points around the circumference of the tube. The total upward force is then 27rrT, where r is the radius of the tube and T the surface tension. If the tube is of small bore, the liquid will rise inside the tube, equilibrium being established "when the weight of the liquid within the tube above the level of the liquid outside equals the total upward force due to surface tension. If d is the density of the liquid, h its height in the capillary tube above the surface-level, write the expression which gives the weight (in dynes) of the column of liquid. Equate this expression to 2?rrT. From the equation thus formed T, the surface tension in dynes per centimeter, can be found. (a) Capillary tubes of different sizes are provided. These may be thermometer tubes or larger glass tubing drawn out to a fine bore. In either case every precaution must be taken to have the tubes perfectly clean and free from all traces of grease. They may be cleaned with caustic potash solution, then rinsed with tap water and dried by drawing a stream of air through them with the jet-pump. With a rubber band fasten the tubes side by side to a glass scale, and place the scale and tubes vertically in a small dish of distilled water. Lower the tubes first to the bottom of the dish so as to wet l6] RISE OF LIQUIDS BETWEEN PLATES. 29 the inside for some distance above the point to which the water will rise. Then clamp them with the ends below the surface, and note on the scale the point to which the water rises in each tube. To obtain the reading for the water sur- face in the dish a wire hook is provided, which should be brought up so that the point is just even with the surface. Then read the height of this point on the glass scale. (b) Measure the inside diameter of the tube with a microm- eter microscope. If drawn-out tubing is used, scratch the tube with a file at the point to which the water rises, break it and measure the diameter of the end. Calculate the surface tension of water and compare this value with the value found in Experiment 14. (c) In the same way find the surface tension of alcohol. (d) From the data taken in (a) calculate the difference in pressure on the two sides of the surface film. Does this dif- ference in pressure, taken in connection with class-room work or reading, suggest a relation other than that used in (a) for finding the surface tension. If so, calculate the surface ten- sion of water by this method. Would the water rise as high in the tubes had the experi- ment been performed in a 'Vacuum"? Explain. 16. RISE OF LIQUIDS BETWEEN PLATES. Reference. Hastings and Beach, p. 146. . ;.; i .; . .; ';.' In the present experiment the surface tension of water and of alcohol is to be measured by means of the rise of the liquid in a wedge-shaped space between two plates of glass. Two plates of glass are separated by two thin pieces of brass placed between the edges at one side, and a single thicker piece on the. opposite side. The plates are clamped together and placed upright in a shallow dish of liquid. If the liquid wets the plates, it will rise in the wedge-shaped space. The general effect is similar to that obtained by a row of small 30 RISE OF LIQUIDS BETWEEN PLATES. [l6 tubes of gradually decreasing bore. We may consider any very small rectangular prism of the liquid of length x, thick- ness d (the distance between the plates at the point chosen), and height h. Assuming the surface tension to act vertically upward where the liquid wets the plates, the whole upward pull along the two edges of the prism will be 2Tx, where T is the surface tension. This force must equal the weight of the prism of the liquid, which is hxdDg, where D is the density of the liquid. From this relation T can be found. (a) Clean the plates very carefully with caustic potash solution, and rinse with water. Clamp them together as indi- cated above, and upon one side of one of the plates place a thin sheet of white paper. Stand the plates upright in a shallow vessel of distilled water, and looking through the paper and the plates toward the light, trace the surface of the liquid between the plates, the surface of the liquid in the dish, and the outlines of the three pieces of metal. Removing the sheet of paper, draw a line on the paper through the position of the outer edge of the two thin pieces of metal. This line should be perpendicular to the line representing the surface of the liquid in the dish. Also draw a line, parallel to the first, through the inner edge of the thicker piece of metal. Select any point, P, on the curve representing the surface of the liquid between the plates. From this point draw a line per- pendicular to the line representing the surface of the water in the dish, and call its length h. Draw another line through P perpendicular to the first line, and let the length along this line from P to the line which coincides with the outside edge of the two thin metal pieces be 1. Let the whole distance be- tween the lines through the edges of the metal pieces be L. Measure the thickness of these metal pieces with a micrometer caliper, calling the thickness of the inner ones d lt and that of the thicker one d 2 . Then at the point P, I 1 /] PRESSURE AND RADIUS OF A SOAP-BUBBLE. 3! Derive this equation. From the values of d and h thus found, calculate the surface tension of water. Repeat the measure- ments and calculation for one or two other points on the curve. (b) Repeat (a), using alcohol instead of water, and find the surface tension of alcohol. 17. PRESSURE AND RADIUS OF A SOAP-BUBBLE. References. Watson, p. 193; Watson's Practical Physics, p. 143. The purpose of this experiment is to determine how the excess pressure inside of a soap-bubble depends upon its ra- dius. The surface tension in the outside and inside surfaces of a soap-bubble tends to contract it, and does contract it until this force is counterbalanced by the excess pressure of the compressed air within the bubble. To determine the relation between this excess pressure and the radius of the bubble, let us consider that a weightless, air-tight, rigid surface is made to pass horizontally through the bubble so as to divide it into two hemispheres without changing the pressure, the film of each hemisphere attaching itself to this surface all the way around. Let the lower of these hemispheres now be removed, leaving the other undisturbed. Consider the forces acting on the bottom of the undisturbed hemisphere. The lower side of this plane is acted upon by an upward force of P-n-r 2 dynes, where P is the atmospheric pres- sure in dynes per sq. cm., and r is the radius of the hemisphere. The upper side of the same plane is acted upon by two forces, one downward and due to the inside air-pressure, the other upward and due to the tension in the soap-bubble film ; the first is equal to (P -f p)^ 1 " 2 dynes, where p is the excess pressure inside the bubble above the value of the air-pressure outside ; the second is equal to 2T(27rr) dynes, where T is the value of the surface tension per centimeter length of the circumference. 32 PRESSURE AND RADIUS OF A SOAP-BUBBLE. [if Since there is equilibrium the algebraic sum of these forces is zero; hence, PTrr 2 + 47rrT ( P + p)7rr 2 = o. From which This shows that the excess pressure within the bubble is in- versely proportional to its radius. The apparatus consists of two essential parts (see Watson's Practical Physics, Fig. 58) : the first is an enclosed box with dish of soap solution, and the tube for blowing the bubble, with a mirror and movable thread (on graduated scale) for meas- uring the diameter of the bubble; the second is the delicate pressure-gauge. This last consists essentially of a sort of U-tube containing a light liquid and having arms at almost 1 80 to each other. One arm is always open to the air, while the other may be placed in communication with the air or with the inside of the soap-bubble. A very small difference in pres- sure may cause a considerable motion of the liquid. A scale on the board on which it is mounted enables one to read the position of the two ends. With both ends of the tube open to the air the position of these ends is read, then one end is placed in communication with the bubble. A micrometer screw at the end may now be moved so as to tilt the board on which the tube is mounted and bring the ends of the liquid column back to the same points on the scale. If H is the vertical dis- tance through which the micrometer screw has been moved to do this, L the distance from the screw to the hinge of the board, and a the angle through which it has been inclined, then H/L = tan a. If h is the height of one end of the liquid col- umn above the other end, and 1 is the slant distance between them, h/1 = sin a. Since a is small, we take the sine and tan- gent as equal, hence h = Hl/L ; and since the pressure equals Dhg, where D is the density of the liquid, we have i8] VISCOSITY. FLOW OF LIQUIDS IN TUBES. 33 DgHl (2) p = --. Ask for instructions in manipulating both pieces of apparatus. (a) Blow several bubbles of different sizes, say five or more, and measure the pressure and diameter of each. First determine the zero-readings at the ends of the liquid column in the pres- sure-gauge, then blow the bubble, put it in communication with the gauge, and make the setting as indicated above. As nearly at the same time as possible, measure the diameter of the bub- ble by making settings with the thread on each side of the bub- ble, putting it in line with its image in the mirror behind. (b) Determine whether the pressure varies inversely as the radius of the bubble. Calculate for each case the surface ten- sion of the soap solution. 18. VISCOSITY. FLOW OF LIQUIDS IN TUBES. References. Watson, p. 196; Watson's Practical Physics, p. 145; Ferry and Jones, p.143. The dependence, of the rate of flow in tubes, on the diameter and length of the tube, and on the temperature of the liquid and the kind of liquid used, is to be observed. When a liquid flows through a tube, if the liquid wets the walls of the tube, the layer of liquid in immediate contact with the wall generally remains at rest. The speed with which the liquid moves increases from the surface of the tube to the axis of the tube. Hence, if. we imagine the liquid to consist of a number of hollow cylinders coaxial with the tube, the fluid within each of these cylindrical shells will be moving more slowly than in the shell immediately inside, and faster than in the shell immediately outside. This relative motion of adjacent layers of the liquid is connected with the internal friction or viscosity of the liquid. Viscosity varies greatly with the kind of liquid used, this dependence upon the character of the liquid being indicated by the "coefficient of viscosity." If a 34 VISCOSITY. FLOW OF LIQUIDS IN TUBES. fl8 liquid is very viscous, like syrup, its coefficient of viscositv is high ; if like alcohol, its coefficient of viscosity is low. For a given liquid at a given temperature, the coefficient of viscosity is a constant. In the case of a liquid flowing through a long, narrow tube, the volume V, issuing per second from the end, depends upon the difference in pressure p, between the two ends of the tube, the radius r of the tube, its length 1, and the coefficient of vis- cosity c of the liquid. These quantities are connected by the relation V == 8lc* To compare the coefficients of viscosity of two different liquids, it is evident, if the above relation be accepted, that, for equal times of flowing, the coefficients will be in inverse proportion to the volumes, or c t : c 2 = V 2 : Vj. Three small-bore tubes are provided, two being of the same length but of different bores, and the third being longer but of the same bore as one of the two shorter ones. The reservoir used consists of a large bottle through whose cork are fitted two glass tubes, long enough to reach about two-thirds of the way to the bottom. The outside end of one of these tubes is connected by rubber tubing with the tube through which the flow is to be measured ; the other tube is left open to the air. Both tubes must extend some distance below the level of the liquid in the bottle, and the cork must be air-tight. By means of this arrangement a constant head of pressure may be obtained. (a) Clean the tubes thoroughly with chromic acid and rinse by drawing clean water through them with a jet-pump. Attach one of the tubes to the siphon-tube from the reservoir, letting the lower end dip into water in a beaker. Weigh the beaker and contained water on the trip-scales. Before replacing the beaker in position, nearly fill the reservoir with water at the room temperature, start the siphon, and let the water run into 19] RELATIVE DENSITIES OF GASES. TIME OF EFFLUX. 35 a waste vessel until the air begins to bubble from the lower end of the open tube up through the water in the reservoir. Then replace the beaker, record the height of the water-level in it, and allow the water to flow for two minutes. Weigh the beaker again to determine the volume which has run through. The head of pressure will be given by the difference in height of the lower end of the open tube in the reservoir and the mean of the initial and final levels in the beaker. Point out clearly why the head is measured from the end of the open tube and not from the water-level in the reservoir. Make two indepen- dent trials. (b) Repeat with each of the other tubes. Measure the diameters of the tubes with the micrometer microscope, or by weighing mercury which occupies a known length of the tube. What do your results show concerning the dependence of the rate of flow on the radius and length of the tube? Calculate the coefficient of viscosity of the water for the three cases, and take the average value. (c) With one of the tubes, use water at 5o-6oC. in the reservoir, and compare with previous results to observe the effect of temperature on viscosity. (d) Repeat (c) with a ten per cent solution of sugar, and, if there is time, with a ten per cent salt-solution. Discuss the results, comparing them with those of (a) and (b), noting the effect upon viscosity of different sorts of dissolved sub- stances. 19. RELATIVE DENSITIES OF GASES. TIME OF EFFLUX. References. Kohlrausch, p. 64; Ferry and Jones, p. 107. The ratio of the densities of two gases, under the same conditions as to pressure, is equal, very approximately, to the inverse ratio of the squares of the speeds with which the 36 RELATIVE; DENSITIES OF GASES. TIME OF EFFLUX. [19 gases escape through a fine opening in a diaphragm. Since the time of escape of a given volume will be inversely as the speed of efflux, it follows that the ratio of the densities of two gases is equal to the direct ratio of the squares of the time of efHux of equal volumes under the same conditions. This rela- tion was experimentally discovered by Bunsen. For a proof of it, from the energy relations, see the second reference given above. (a) The gas-holder consists of a glass cylinder, at the top of which is a three-way stop-cock and a diaphragm with a fine opening. The cylinder is placed in a reservoir of mer- cury. The three-way cock allows communication to be made with the outside for filling or with the diaphragm. Within the cylinder is a float which indicates when the desired volume of gas has escaped. First fill the cylinder with dry air. To do this, turn the stop-cock so as to put the cylinder in communication with the air, and lower the cylinder as far as it will go. This drives out most of the contained gas. Connect the cylinder with a calcium chloride drying-tube, and raise the cylinder. This operation will fill the cylinder, and by repeatedly emptying and filling the cylinder it will become practically freed of the moist air or other gas previously contained in it. Close the stop-cock, and lowering the cylinder, clamp it in position. Turning the stop-cock so that the gas in the cylinder is in com- munication with the diaphragm, note the time when the upper point of the float is on a level with the surface of the mercury or with a mark on the cylinder. Again note the time when the second mark on the float is on the same level. Repeat, mak- ing two or three determinations of the time of efflux for the given volume of air, and take the mean. (b) Repeat (a), filling the cylinder with illuminating gas, following the directions there given for filling the cylinder, the cylinder being connected directly to the source of the gas used. Note the time of efflux between the same two positions 2OJ CALIBRATION OF A THERMOMETER, ABSOLUTE. 37 for the float as used in (a). This insures the same conditions as to pressure in the two cases. (c) Repeat (b), using dry carbon dioxide. (d) Calculate the relative densities, referred to air, of the gases used in (b) and (c). Taking the density of dry air under standard conditions to be 0.001293 gms. per cc., find the density, under standard conditions, of the gases used. What "Laws" have been used, or assumptions made, in answering the requirement of the preceding sentence? 20. CALIBRATION OF A THERMOMETER, ABSOLUTE. References. Kohlrausch, p. 81 and p. 86; Watson's Practical Physics, p. 162; Miller, p. 160; Edser, p. 23. The object is to determine the fixed points and the correc- tions to the scale-readings of a mercurial thermometer. (a) Determination of the Lower Fixed Point. Put the thermometer through the cork in a test-tube, having filled the latter about half full of distilled water. Place the tube in a freezing mixture of shaved ice and salt, and stir the water around the thermometer until it begins to freeze. Read the thermometer. By warming the tube in the hand and repeating the freezing process, obtain several readings of the true zero- point. (b) Determination of the Upper Fixed Point. Place the thermometer in the cork in the tube at the top of the boiler, though with the bulb well above the surface of the water. Boil the water so that the steam issues freely, but not with any perceptible pressure, from the upper vent. Read the thermometer when it becomes steady. Allow the boiler to cool slightly and repeat, making in all three readings. If the instrument be provided with a water-manometer, take the manometer reading simultaneously with the temperature read- ing. Read the barometer and determine the pressure of the 3^ CAU1JKATION OF A THERMOMETER, ABSOLUTE. [2O steam, and find the true boiling 1 point for this pressure from the Tables. (c) Let the thermometer cool slowly to about the tempera- ture of the room, and repeat (a). If the freezing point ob- served now is different from that observed in (a), use the mean of the two values in the calibration that follows. Assum- ing the correct freezing point to be o, write the corrections of the thermometer at the zero-point and at the boiling point Record these two corrections by points on coordinate paper, having as abscissae degrees centigrade from o to 110, and as ordinates the corrections of the thermometer at the corre- sponding temperatures in tenths of a degree but on a larger scale. Corrections should be plus ( + ) if they are to be added to the observed to give the true temperatures, minus ( ) if they are to be subtracted. Connect these two points by a straight line. The ordinate of this straight line at any point gives the correction of the thermometer at that temperature on the assumption that the bore of the thermometer is uni- form throughout the whole range. In general this assump- tion is not justified, and there must be added to this correc- tion at each point another correction due to the inequalities of the diametef of the bore. In order to determine this latter correction it will be necessary to calibrate the tube. (d) Calibration of the Tube. Break off a portion of the thread of mercury about ten degrees in length. ( Both Kohl- rausch and Miller give directions for breaking the thread at any desired point, but if you cannot readily succeed, ask for assistance.) Place the lower end of the thread, approximately ten degrees long, at the zero-point of the scale and read the position of the upper end to tenths of a degree. Then place the lower end at 10 and read the upper end. Repeat with the lower end at the successive points 20, 30, 40, etc., up to 90 ; then come down again with upper end at 100, 90, 80, etc., reading the lower end each time. 2O] CALIBRATION OF A T H^RMOMKTER, ABSOLUTE. 39 . (c) Record the observations and calculations in tabular form in six columns as follows : 1 i ) The reading of the lower end, o, 10, 20, 30, etc. (2) The reading of the upper end. (3) The length of the thread in each position. (4) The mean length, 1, for each position. By the mean length, 1, for each position is meant the mean of the readings over a certain interval going up (say from 30 to 40) and over the same interval (40 to 30) coming down. Find the mean value of all these lengths through- out the whole range and record this as the "mean length," L. (5) The correction, (L 1), for the length of each inter- val, that is, the difference between the mean length for all intervals and the observed length of each interval. (6) The correction for the upper end of each interval. This is the correction for the lower end of the interval plus the correction for the length of the interval, since a correction at any point evidently affects all points above this. The cor- rection thus found for any point represents the magnitude of the inequalities of the bore up to that point. It must be added to the observed reading for that point to give the correct reading, it being assumed that the fixed points are properly placed. The corrections should be recorded with proper signs. (See Watson's Practical Physics, p. 168.) (/) To construct a final table of corrections it is neces- sary to add, algebraically, the corrections found in (c) and in (c, 6). This can best be done by plotting. On the plot made in (c) record at the points 10, 20, etc., the correc- tions at these points obtained from (e), measuring the cor- rections up or down from the slanting line already drawn, according as the sign is plus or minus. Draw a smooth curve through the points. The ordinate of this curve at any point, measured from the horizontal base line, is the total correction to the reading of the thermometer at the corres- ponding temperature. 40 CALIBRATION OF A THERMOMETER, RELATIVE. [21 21. CALIBRATION OF A THERMOMETER, RELATIVE. Most varieties of glass expand at different rates at differ- ent temperatures, hence, even with a thermometer whose bore has been carefully calibrated by some such method as given in Exp. 20, the reading can be relied upon only within certain limits. After having obtained a thermometer whose calibra- tion curve is accurately known, so that it may be taken as a "standard," the most convenient method of calibrating other thermometers is by direct comparison with the standard. If the calibration curve of the standard thermometer can be relied upon, all irregularities of the thermometer calibrated can be corrected. The thermometer to be calibrated in this experiment is a 50 thermometer reading to o.i. Tie the thermometer to be calibrated to the "standard" with soft cotton twine, wind- ing it between the stems so as to separate them slightly. Put the bulbs nearly opposite each other ; and see that correspond- ing divisions are as nearly opposite as is consistent with this condition. Suspend the two securely, with the bulbs in the middle of a kettle of water, and steady the stems by catching them loosely, without pressure, in a clamp. The thermom- eters are to be read by a short-focus telescope, which slides easily on the vertical rod of its stand. This should be set with its object-glass at a distance of about 50 cm. from the thermometers, which should be perpendicular to its axis. When taking a reading always set the telescope so that the top of the mercury column appears in the middle of the field of view, not near its upper or lower edge, in order to avoid parallax. (a) Take a careful series of readings to o.oi. at intervals of 2 or 3 from about 5 to 45. Keep the water well stirred, and keep the temperature fairly constant for a few minutes before each reading. A good plan is to take a preliminary 22] VARIATION OF BOILING POINT WITH PRESSURE. 41 reading of each thermometer in order to see about where the reading is going to come. The two exact readings can then be made so quickly as to be practically simultaneous. Read again in a few seconds, taking the thermometers in reverse order. Repeat if necessary until the differences obtained for two such readings agree fairly well. (b) Let the observers change places, and take a similar descending series, cooling the water by dipping out hot and adding cold water. (c) Ask to see the calibration curve of the standard used, and from it construct a table of corrections for the thermom- eter you are calibrating. Plot a calibration curve, recording the number of the thermometer. In your future work with a ther- mometer of this type, use the one you have calibrated. 22. VARIATION OF BOILING POINT WITH PRESSURE. References. Edser, p. 191; Millikan, p. 157. If, after reading the references, the arrangement of the apparatus is not understood, ask an assistant for directions. Before turning off the air-pump, be sure to let air into the ap- paratus by opening the pinch-cock nearest the pump, other- wise water will flow back into the tubing. In boiling the water do not play the flame on the flask directly below the glass beads, but rather to the side, but never above the water- line. (a) The water should first be started through the steam condenser. This is a glass or metal tube used to jacket the tube leading from the boiling-flask, thus condensing the steam as it comes from the flask. The thermometer should be passed through the stopper of the flask and so regulated that its bulb will be in the rising steam, but not in the water. The con- nection with the large glass bottle serves to equalize sudden changes in pressure. 42 EXPANSION OK A i.inrin r,v ARCH IMKDKS' PKI \an.K. [23 First boil the water at atmospheric pressure, reading the manometer and noting the temperature. Then take a series of readings at intervals of about 5 cm. pressure, until the "bumping" becomes so violent as to render further reading impracticable. Before each reading, after pumping to the pres- sure desired, close the stop-cock over the jet-pump, wait a short time for the pressure to reach equilibrium, and then make the reading of boiler temperature and corresponding pressure. Put the pump again in connection, obtain a new pressure, and repeat the readings. (b) Take a series of readings with increasing pressures up to atmospheric pressure, choosing values different from the previous ones. (c) Plot the observations on coordinate paper, using pres- sures as ordinates and temperatures as abscissae. From the curve find the boiling point of water at a pressure of 1/2 atmosphere. Discuss the phenomena of this experiment in con- nection with the difficulties experienced in cooking food at high altitudes. Could determinations of the boiling point of water be used to measure altitude, and how ? . 23. COEFFICIENT OF EXPANSION OF A LIQUID BY ARCHIMEDES' PRINCIPLE. The coefficient of expansion of a heavy oil is to be obtained by observing the change in the buoyant force acting on a metal cylinder when immersed in the oil at different temperatures. A brass cylinder is suspended from one arm of the balance and carefully weighed, first in air, then in water at a known temperature. The oil is then placed in a calorimeter con- sisting of one beaker inside another, and the cylinder is weighed when immersed in the oil, the temperature of the oil being noted, which should be the same as that of the water, or nearly so. Since the oil thickens if cooled, it is convenient to make the first weighings at the room temperature. 24] COMPARISON OF ALCOHOL- AND WATER-THERMOMETERS. 43 After weighing in the cool oil, the inner beaker is removed and placed in a water-bath heated to 60 or 70 C. Replacing the beaker with the heated oil in the calorimeter beaker, the cylinder is again weighed in the oil, the temperature of the oil during the weighing being carefully noted. Let M = the mass balancing the cylinder when in air, m t = the mass balancing the cylinder when in water, nio = the mass balancing the cylinder when in cool oil, m., = the mass balancing the cylinder when in hot oil, t 1 = the temperature of the cool oil and the water, and t 2 = the temperature of the hot oil. Then V lf the volume of the cylinder at the lower temperature, is (M mJ/D, where D is the density of the water at this temperature (see Tables). If V 2 is the volume of the cylinder at the higher temperature, t,, and a the coefficient of cubical expansion (3 times the coefficient of linear expansion found in the Tables), we have V 2 = V, f I + a(t, t t )]. The den- sity, D,, of the cool oil is (M m 2 )/V 1 . The density, D 2 , of the hot oil is (M m,,)/V 2 . If ft be the coefficient of cubical expansion of the oil for the temperature range used, we have D 1 = DJ I -f- p(t. 2 t,)], from which (3 can be found. Make two determinations of the density of the oil at the two temperatures. From the mean of the values for each density calculate ft. Point out the sources of error in the experiment. If the cylinder had an internal cavity, show what its effect upon the value of /? would be. 24. COMPARISON OF ALCOHOL- AND WATER- THERMOMETERS. In this experiment the relative expansions of water and alcohol are to be studied, and the behavior of these liquids when used in thermometers to be observed. 44 COMPARISON OF ALCOHOL- AND \\ATER-TH ERMOMETERS. [24 (a) Two thermometer bulbs are to be filled, one with water, the other with ethyl alcohol by the aid of the reservoir-tube. The reservoir is fitted on the end of the thermometer-stem, filled with water (or alcohol), and warmed. The liquid is then introduced into the thermometer-bulb by alternately heating the bulb to drive out the air and allowing it to cool to admit the liquid. Ask for assistance if the' method is not understood. The water should be heated to drive out the oxygen held in so- lution, before filling the reservoir with it. Take care not to ignite the alcohol. The liquid in each thermometer should stand i or 2 cm. above the lower end of the stern when the bulb is in melting ice. (b) Glue or otherwise fasten a strip of stiff paper along the back of each stem, to be used as a scale. Then place the ther- mometers in clamps with their bulbs in shaved ice or in a mix- ture of water and ice. When the reading becomes steady, indi- cate the position of the meniscus of each by a sharp line on the card. Mark the line zero. This is the first fixed point of the thermometer. (c) To determine the second fixed point, place the bulbs in a beaker of wood alcohol which is itself placed on a sup- port in a bath of water. Heat the water-bath slowly until the wood alcohol begins to boil. Be very careful not to bring the alcohol itself to the flame, and avoid inhaling the fumes of wood alcohol. When steady, again indicate the position of the meniscus on each stem by a sharp line. Mark this point 66, which is the boiling point of wood alcohol on the centi- grade scale. (d) Lay the stems of the thermometers on a flat surface, measure the distance on each between the two fixed points, and divide this distance into 66 equal parts, calling each part a degree. Put the marks for each degree on the scale and number every tenth one. (e) Place the two arbitrarily calibrated thermometers in a water bath of o, as recorded by each thermometer. Gradu- 25] COEFFICIENT OF EXPANSION OF MERCURY. 45 ally raise the temperature of the water-bath and note the read- ings of the two thermometers, at first at short intervals, then at longer intervals, until- the upper fixed point is reached. (/) Plot a curve on coordinate paper, having as abscissae the temperatures by the alcohol thermometer, and as ordinates the corresponding temperatures by the water thermometer. Draw a smooth curve through the points. This curve gives the relation between the tem- peratures as recorded by the two thermometers. What inferences can you draw from the curve? If alcohol be taken as the standard substance, what can you say of the uni- formity of the expansion of the water? If the water be assumed as the standard, what of the expansion of the alco- hol ? Which would be the best substance to use in a practical thermometer, and why? 25. COEFFICIENT OF EXPANSION OF MERCURY BY REGNAULT'S METHOD. References. Watson, p. 221; Edser, p. 71. This method was originally devised by Dulong and Petit, but improved and made practical by Regnault. It is an abso- lute method in which the effect of the expansion of the con- taining vessel is eliminated. The method is applicable to any liquid. Two glass tubes, AA' and BB', are surrounded by metal cylinders, L and M respectively, in which baths of dif- ferent temperatures may be placed. These glass tubes are connected by a horizontal tube, ACB, (from which there ex- tends an upright open tube, C), and also by an inverted U-tube at the bottom between A' and B'. Mercury is poured into the glass tubes until it stands at some point in C just above the horizontal level, AB. This insures the height remaining the same, or very nearly the same, at A and B, and gives a means of measuring the height without observing the meniscus at A or B. In the bend of the tube GK there is compressed air 4 6 COEFFICIENT OF EXPANSION OF MERCURY. [25 so that the pressure is always the same at the meniscus G and the meniscus K. The levels G, K, and C may be measured. Cold water is passed through M, or a mixture of ice and water placed in it, and steam is passed through L, thermome- ters at A and B indicating the temperatures. Suppose that the temperature of M is oC. and that of L, t^ . The temper- ature of the mercury in both branches of the inverted U-tube may be assumed to be the same as that of M. Let D be the density of the mercury in M (at oC.), and D l be the density of the mercury in L (at t^C.), H the vertical height of C above the level of A'B', and h the vertical distance OK. Then, since the pressure at G is the same as that at K, we have in the tube A'E a pressure one way due to the hot mercury col- umn AA', and balancing it a pressure due to the cold column equivalent to (BB' DK + GE), that is, to a column of height (BB' GK) or (H h). Therefore, if P be the atmospheric pressure, we have, D,SH + P = D g(H h) + P. from which (i) D /D 1 = H/(H h). Now, if ft be the coefficient of cubical expansion of mercury, v,, the volume of a given mass m at oC., v, the volume of the 25] COEFFICIENT OF EXPANSION OF MERCURY. 47 same mass at t,C., we have v l = v,,(i -\-ftti). Since v = m/D and v, =111/0,, Z^*-:pk{*-Hh ftt,) or i + 0t t = D /D 1 =H/(H h). Therefore ( 2 ) /? =h/ ( H h ) t t . If the mercury in M and in the U-tube is at room tempera- ture (.say to ) instead of at o, ft calculated from (2) will be the average coefficient of cubical expansion between t, and t l , and not the coefficient between o and t^ (or 100) as given in the Tables. In order to find this zero coefficient a slightly different equation should be used. We not only have D = D,(i + jSti) and D = D 2 (i + 0t 2 ), but also D 2 /D l = H/(H h), where D 2 is the density at t 2 . From these three relations (3 is obtained. (a) Pass cold running water through the jacket M and steam through L, keeping thermometers on the two sides. Wrap a cloth about the horizontal tube A'E, and keep this wet with cold water to make the conduction of heat to the column EG as small as possible. Note the difference in height of the menisci at G and K when the temperatures become steady. The distance, h, must be very accurately determined at each setting, whereas a reading of the height, H, to I mm. is sufficiently accurate. Do not forget to read both heights and the temperatures when there is a change. (b) The pressure of the gas in the inverted U-tube, GK, should now be changed. Do not attempt to do this yourself until an assistant has shown you the method. The height of the mercury in C may require adjusting by adding or taking out a little mercury. Only perfectly clean mercury should ever be used. Read h and H again, making four or five different set- tings. (d) Calculate for each setting the value of the mean coeffi- cient of expansion between the temperature of cold water and the steam, and take the mean. Derive the equation for ft, the average coefficient of cubical 48 EXPANSION OF GLASS BY WEIGHT THERMOMETER. |>6 expansion between oC. and iooC., and calculate its value from your results. What percentage difference is there between this value of ft and the one calculated by equation (2) ? Calculate the percentage error in your value of ft, taking the value from the Smithsonian Tables as the correct one. Considering AA'EG as one U-tube with balancing columns of liquid, write the equation for equal pressures at the level A'E ; do the same for the branch BB'DK, and from these see if equation (i) can be derived. 26. EXPANSION OF GLASS BY WEIGHT THERMOMETER. References. Watson, p, 219; Watson's Practical Physics, p. 195; Edser, p. 66. The purpose of this experiment is to determine the coeffi- cient of cubical expansion of glass by means of the weight thermometer. The weight thermometer consists of a glass tube closed at one end and ending in a curved capillary at the other end. It is filled with mercury at oC., and the mass of the mercury measured. When later placed in a bath of higher tempera- ture, some mercury overflows, since mercury expands more rapidly when heated than does glass. The mass of this over- flow is measured. Let M = the mass of the mercury filling the weight ther- mometer at oC., V = the volume of M, and hence of the weight thermom- eter at oC, ^ and y = the coefficients of expansion respectively of mer- cury and glass, and m= the mass of the mercury which overflows when the temperature is raised from o to t. Then V (i + ftt) = the volume of the mass M at t. 26] EXPANSION OE GLASS BY WEIGHT THERMOMETER. 49 and V ( i -f- yt) = the volume of the weight thermometer at t ; hence V (i + 0t) V ( I + yt) == V (/3 y)t = the vol- ume of the mass m at t. If D and D t be the densities of mercury at o and tC., respectively, then (1) D = M/V , (2) D t = m/V (0 y)t, (3) D = D t (i+0t). By eliminating- D and D t from equations (i), (2), and (3), we get Besides containing the known masses, M and m, this equation contains the three quantities, /?, y, and t. Any two of these three quantities being known, the third will be given by the equation. In the present experiment ft and t are known, and y is to be calculated. (a) Weigh the empty weight thermometer to 10 mg. Then fill it with mercury. In doing so it should be held by a clamp, or suspended in a gauze jacket, and heated by a flame held in the hand. The end of the capillary dips under the surface of mercury in a porcelain dish. The mercury in this dish should first be heated, and then the weight thermometer heated until the air bubbles out through the mercury. On allowing the bulb to cool, some mercury will run into it. The process is then repeated. When considerable mercury is in the bulb, heat it until it boils vigorously, but be careful not to heat too hot that portion of the glass where there is no mercury. Keep the mercury in the dish hot, otherwise the glass is apt to crack just as the cooler mercury rushes in. The tube must be completely filled to the end of the capillary, the last bubble of air being expelled. (b) Keeping the end of the capillary in the dish of mer- cury, allow the weight thermometer to cool in the air suffi- ciently so that you can bear your hand on it. Then surround 5O EXPANSION OF A LIQUID HY PYCNOMETER MKTHOD. [2/ it with shaved ice and leave it long- enough to contract as much as it will. Assume that its tem- perature is now o C. Carefully remove the dish and brush the mercury off the end of the capillary. Place a watch-glass under the end to catch the mercury as it begins to expand and flow out. Now remove the ice-bath and warm the bulb with the hand until its temperature is raised to the temperature of the room. (c) Place the weight thermometer in the boiler provided. Heat it to the boiling point of water by passing steam over it until no more mercury comes out. Read the barometer and calculate the temperature of the steam. Very carefully weigh the mer- cury in the watch-glass to i mg. Weigh the weight ther- mometer and contained mercury to 10 mg. (d) Using your values of M and m, and taking the coeffi- cient of expansion of mercury as found in Exp. 25, calculate the coefficient of cubical expansion of glass. What additional measurements would you need to make in order to measure the room temperature with your weight thermometer ? 27. EXPANSION OF A LIQUID BY PYCNOMETER METHOD. Reference. Edser, p. 81 and p. 86. The method consists in determining the mass of the liquid (alcohol) filling a pycnometer at each of several different tem- peratures, and from the data calculating the coefficient of ex- pansion of the liquid. Four determinations should be made, at intervals of about 8. beginning with the room tempera- ture. (a) Fill the pycnometer with alcohol and set it on a plat- form in a kettle of water, so that the water comes well up to the neck of the pycnometer. Hang a 50 thermometer in the bath alongside the pycnometer, and keep the bath well stirred 27] EXPANSION OF A LIQUID BY PYCNOMETER METHOD. 51 for about five minutes. The temperature of the bath, which must be a little above that of the room, should remain con- stant within o.i during this time, and at the end of it the alcohol will have the same temperature within o.i. Take the pycnometer out of the bath, wipe the outside dry, and weigh to i mg. (b) Repeat with the bath at about each of the higher tem- peratures selected, holding the temperature steady for ten minutes by holding the lamp under the kettle for a few sec- onds occasionally. Careful trial has shown that after this treatment the temperature of the alcohol at the center of the pycnometer is about o.i lower than that of the bath, and therefore the average temperature of the alcohol is the same as that of the bath to within o.i. (c) Empty the alcohol into the bottle from which it was taken, dry the pycnometer with a jet-pump, and weigh. (d) Determine the mass of the alcohol filling the pycnom- eter at each temperature. Plot the results, with tempera- tures, starting from o, as abscissae, and changes in mass as ordinates. Assuming that the expansion is uniform, draw the straight line which best represents the plotted points, and from it find the mass filling the pycnometer at o. (e) From the masses filling the pycnometer at o and at 40, calculate the relative volume of a given mass of alcohol at 40, refered to its volume at o. From this calculate the coefficient of cubical expansion of the alcohol for the given range of temperature. (/) What has been found is not the absolute coefficient of expansion, since the pycnometer also expands. Find, from your own work in Exp. 26, or from the Tables, the coefficient of cubical expansion of glass, and by applying it to the above result find the absolute coefficient for the alcohol. 5 2 EXPANSION CURVE OF WATER. [28 28. EXPANSION CURVE OF WATER. The variation of the volume of a given mass of water, as the temperature is raised by steps from the freezing point, is to be studied, taking- the expansion of mercury as the tem- perature standard. It should be remembered that our choice of a thermometer and scale of temperatures is entirely arbi- trary. The statement that a certain substance expands "uni- formly" can mean only that it expands uniformly with the change of some property of a particular substance chosen as a standard. Taking the expansion of mercury as a standard, we wish here to determine how water changes in volume with change of temperature. (a) The bulb of the water thermometer can be filled by the aid of a reservoir. The reservoir is filled with warm water and the end of the thermometer tube introduced, the bulb being below the reservoir. The bulb is then alternately heated to drive out the air and allowed to cool to admit water. Ask an assistant for directions if there is difficulty. Fill the ther- mometer until the water stands in the stem, at oC., about 2 cm. above the bulb. Fasten the tube to the face of a metric scale and place the bulb in the water-bath. The bulb is first to be surrounded with shaved ice. When conditions become constant, take a reading of the height of the water meniscus and also of the mercury thermometer placed in the bath near the bulb. Melt the ice and gradually raise the temperature of the bath very carefully, at first reading the mercury and water thermometers at every degree between o and 8C., then at approximately TO, 15, 20, and every ten degrees thereafter as far as the water thermometer will permit. (b) Determine the volume of the water in the bulb at o C. by weighing the bulb with the water in it and then weigh- ing it empty and dry. (c) Determine the diameter of the bore either by direct measurement with the micrometer microscope or, better, by 28] EXPANSION CURVE OF WATER. 53 placing in the tube a thread of mercury, measuring- its length and then weighing the mercury. ((/) From the determinations of the volume of the bull) and the diameter of the bore of the tube, calculate the volume, in cu. cm., of the water at each of the temperatures observed, making no allowance for the expansion of the glass. On coordinate paper plot the results and draw a curve, hav- ing for abscissae the temperatures as recorded by the mercury thermometer, and for ordinates the corresponding volumes of water. In doing this, choose as large a scale for volumes as possible, so that the total change of volume will about cover the. width of the paper. Since your observed changes of vol- ume are only apparent changes, the volume-change of glass must be added in order to obtain the true expansion of the water. To do this, calculate what the volume-increase of the water thermometer is between o and iooC., due to the expansion of the glass. At the point on the temperature axis corresponding to 100 erect an ordinate equal to this expan- sion. Draw a slanting line through your origin of coordi- nates and the upper end of this ordinate. The lengths of the ordinates between this line and the temperature axis represent the expansion of the glass for the corresponding tempera- tures. Then, from various points along the apparent expan- sion curve of water, measure, vertically upward, distances equal to the volume-increase of the glass corresponding to this temperature. Draw a smooth curve through all of these points. This curve referred to the horizontal axis will give the true expansion of the water. (e) State any conclusions that can be drawn from an ex- amination of the curve in regard to the behavior of water as its temperature is raised from o to the highest point reached. By the use of the curve determine the average cubical coefficient of expansion of water (i) between o and ioo ; (2) between o and 20, (3) between o and 8. Also deter- mine the cubical coefficient of expansion of water at 15. 54 SPECIFIC HEAT OF A LIQUID BY METHOD OF HEATING. 29 29. SPECIFIC HEAT OF A LIQUID BY METHOD OF HEATING. Reference. Miller, p. 188. In this experiment a heating coil, composed of high resist- ance metal through which an electric current is passed, is immersed for a given time, first in one liquid and then in another. If the same current passes through the coil in the two cases, equal quantities of heat should be generated in equal times. Noting the mass of the liquid in each case and the rise in temperature, the two quantities of heat may be equated and the specific heat of one liquid calculated, if that of the other is known. Water, taken as a standard, will be one liquid used. A second liquid, properly labeled, will be found upon the laboratory table. The method is applicable to any liquid which is not a conductor of electricity and which does not act chemically upon the material of the coil or calor- imeter. (a) Place the bottle, containing the liquid to be used in (b), in a vessel of ice-water to cool. Weigh a quantity of ice-water in the calorimeter cup. Set up the calorimeter and immerse the heating coil, having the temperature of the water about 12 below the room temperature. Allow a few moments for the contents of the cup to come to a uniform temperature, then note the temperature, and turn on the current in the coil. Record the time when the current is started, and also the time for each degree rise in temperature of the water until it reaches a temperature as far above that of the room as it started below. Keep the water well stirred, and do not place the thermometer very close to the heating coil. (&) Repeat (a), using the liquid furnished instead of water in the calorimeter cup. Unless known from previous work it will be necessary to find the water-equivalent of the calorimeter cup, stirrer, and thermometer. 30] SPECIFIC HEAT OF A LIQUID BY METHOD 0$ COOLING. 55 (c) Plot on the same sheet of paper the results of (a) and of (b), using temperatures as ordinates and times as abscissae. What would be the form of the curve for each case, if the con- dition were fulfilled that equal quantities of heat are gained by the liquids in equal times? Compare the part of the curve near the room temperature with the end portions. Erect two perpendiculars to the time axis, making these lines cut the curves as far apart as possible. From these intersections obtain the range of temperatures passed through by the water and the other liquid in equal times. Take the quantities gained by the two liquids in this tim'e as equal, and form an equation from which the specific heat of the liquid may be calculated. From the result just obtained calculate what mass of the liquid will be "equivalent" to the water used in (a). (d) If you have time, take the amount of liquid found in (c) to be equivalent to the water used in (a), and repeat (b). From the data obtained calculate the value of the specific heat. Why should this value be more reliable than the one found in (0? 30. SPECIFIC HEAT OF A LIQUID BY METHOD OF COOLING. References. Millikan, p. 206; Miller, p. 186; Watson's Practical Physics, p. 224. This method is a comparison of the quantities of heat lost by two liquids, one of which is water, when equal volumes are allowed to cool under exactly the same conditions through a certain range of temperature. The conditions of radiation being the same for both, if a liquid of mass m l and specific heat Sj cools through a certain temperature-range in T x sec- onds, and a second liquid of mass m 2 and specific t heat s c requires T, seconds for the same temperature-change, then the quantities of heat lost will be proportional to the times, i. e., Q,/Q 2 = Tj/Ta. If w denote the water-equivalent of 56 SPECIFIC 1I1CAT 01? A LIQUID BY METHOD OF COOLING. [30 the containing vessel, thermometer, and stirrer, the above relation becomes m^ -f- w _ T, m.,s 2 + w ~" T./ from which the unknown specific heat can be determined. (a) A large jar is used, having a wooden cover from which is suspended a smaller vessel. The space between these is made a water-jacket by putting enough water in the larger so that when the cover is put in place the space between the two vessels will be filled with water. The liquid used, tur- pentine, is now heated in a water-bath to about 85 and poured into a small copper cup, closed by a cork through which the thermometer and stirrer pass. The cup is then passed through the wooden cover and hangs suspended from the cork. Make the necessary weighings on the trip-scales. Allow the turpentine to cool to about 50, recording the time for every two degrees fall in temperature at first, and later for each one degree fall. (b) Put fresh water in the jacket, and repeat (a) with water in the cup instead of turpentine, using as nearly as possible the same volume. (c) Plot the cooling curves of turpentine and of water on coordinate paper, using temperatures as ordinates and cor- responding times as abscissae. On these curves take a cer- tain range of temperature by drawing two lines parallel to the axis of abscissae, each line cutting both curves. Make this temperature-interval as long as possible. The intersections of these lines with the curves will give the times required. Cal- culate the specific heat of turpentine. If the water had not been changed between the two sets of observations, in what way would the value for the specific heat of t turpentine have been affected ? What source of error still remains even if the water in the jacket is changed before the second measurement? Suggest a way to avoid this uncertainty. 31 ] MECHANICAL EQUIVALENT OE HEAT. 57 31.- MECHANICAL EQUIVALENT OF HEAT BY METHOD OF PERCUSSION. References. Preston, p. 39; Edser, p. 269; Watson, p. 310. In this experiment the mechanical equivalent of heat is to be determined from the work done by a falling- body, the fall- ing taking place inside a bamboo tube. (a) Take two bottles, each containing a kilogram of shot, and cool the shot about three degrees below room temperature by placing the bottles in a vessel of ice-water. Pour the shot from one bottle into the tube, and measure inside the tube to determine the mean distance the shot will fall when the tube is turned end for end. To correct partially for radiation, let the first trial be a preliminary one to determine approximately what the rise of temperature will be. Cool the shot down below the room temperature far enough, so that when it becomes heated it will rise the same number of degrees above room temperature. To determine the rise in temperature, close the tube securely, take the temperature of the shot by inserting the thermometer through the small hole in the side near one end, lay the tube on the table, and, resting one end on the table, raise the other end quickly so that the shot does not fall until the tube is vertical. Avoid any up or down motion of the tube as a whole. Do not hold the tube by the ends. Let the shot fall through the tube 100 times in this way, then take the temperature again through the small hole. Pour the shot back into the bottle and cool it once more. Take the shot from the second bottle, and, if it has become too cool, shake it in the bottle. Repeat with this shot while the tube is still warm. Make four or five measurements in this way. After the tube is warm it may be necessary to take some other initial temperature to properly correct for radiation as suggested above. 58 MECHANICAL KOU1VAI.KNT OF H KAT. [32 (b) From the work in ergs and the calories of heat gained by the shot, calculate the mechanical equivalent of heat for each measurement. Give reasons why you consider some values better than others. 32. MECHANICAL EQUIVALENT OF HEAT BY METHOD OF PULUJ. Reference. Miller, p. 194. This method of determining the mechanical equivalent of heat involves the measurement of the work done in raising a given mass of mercury through a measured range of tempera- ture. The apparatus consists of two hollow cones, one within the other, mounted on a rotating apparatus. Mercury is placed in the inner one and a thermometer suspended in the mercury. The outer cone is rotated and tends to carry the inner cone with it. This is prevented by a lever attached to the inner cone. From one end of the lever a cord passes over a pulley to a scale-pan on which a mass may be placed. The deflection of the lever may be read by means of a scale under the shorter end. If the outer cone is rotated with con stant angular velocity, the pointer of the lever will be held at a constant deflection. The force-moment opposing the rotation of the inner cone may then be measured. Let M be the mass of the pan and its contents, and f the friction of the pulley. Then the total force acting on the end of the lever is F = Mg + f Let L be the length of the force-arm ( not of the wooden lever). The force-moment is then FL= (Mg-f- f)L. If the c. g. s. units are used, the work in ergs in each revolution is W, = 27rL(Mg -f f), and in n revolutions is W n = 27rLn(Mg+f). In making n revolutions, let us suppose that the temperature has risen from t, to t 2 . Let the sum of the masses of the two cones be m lt and the specific heat of the steel Sj. Let m 2 be the mass of the mercury in the inner cone, s, its specific heat. OF 321 MlvCHANICAI, EQUIVALENT OF HEAT. 59 and w the water equivalent of the thermometer. Then the quantity of heat in calories generated is Q== (nvs, + m 2 s 2 + w)[(t 2 t,) +R1, where R is a temperature correction to allow for the heat lost by the cones through radiation and conduction to surrounding objects. From these values the mechanical equivalent of heat may be obtained from the relation, J = W/Q. (a) The masses of the steel cones are given. Partially fill the inner cone with mercury, but not fuller than one centi- meter below the top, and weigh. Put the inner cone, ther- mometer, lever, and scale-pan in position. Rotate the outer cone and adjust the mass on the scale-pan so as to give a steady deflection of 20 degrees or more. When the lever is deflected, the pulley should be moved so that the cord is par- allel to the axis of the machine. (b) Read the temperature, T 1? of the mercury, remove the thermometer and lever (but not the inner cone), and per- form 200 rotations with the same speed as before. Record the final temperature, T 2 . This is done to determine the rate at which the temperature is changing, due to radiation and conduction, before the test is made. (c) Replace the lever and thermometer, read the tempera- ture. t,, and rotate the cone steadily as before until the tem- perature has risen about 5, and again record the tempera- ture, t 2 . Record the number of rotations, n. (d) Remove the lever and thermometer and again per- form 200 rotations, recording the temperatures, T 3 and T 4 , as in (b). This enables one to determine the rate at which the temperature is changing, due to radiation and conduction, after the test is made. (e) Detach the cord from the end of the lever and attach masses just sufficient to balance the mass of the scale-pan when the cord is hung over the pulley. Now put additional 6O COOIJXC, THKOIV.H C'HAXGK Or STATIC. masses on one side till the system just begins to move. Call this extra mass p ; then f = pg. (/) From (b] the fall in temperature before the test, due to losses during one rotation, is R, = ( T, T.,)/2OO. and from (d) the loss per rotation after the test is R 2 = (T 3 T 4 )/200. Hence, for the n rotations of (r), the loss due to radiation and conduction is R = - (T, T, + T, T 4 ). 2 200 V Explain fully how R is given by the measurements of (b) and (d) and the last calculation. Calculate J from the equa- tion previously given. 33. COOLING THROUGH CHANGE OF STATE. I. Melting Point and Cooling Curve of Paraffine. (a) Take several pieces of capillary tubing, each 2 or 3 cm. long, dip the ends in melted paraffine, and let them fill by capillarity. Fasten them around the bulb of a thermometer by means of a rubber band. Immerse the thermometer bulb in a test-tube of water and heat this in a water-bath. Note the temperature at which the paraffine melts. Remove the test tube containing the thermometer and note the temperature at which the paraffine solidifies. Take the mean of these two as the melting point. (b) Put a thermometer in a test-tube together with enough paraffine to cover the bulb when melted. Heat the test-tube in a water-bath until the thermometer registers about 70. Remove the test-tube from the bath, clamp it in a stand, and allow the paraffine to cool slowly in air to about 38 C. Record the time for each degree or half -degree fall, or at shorter intervals when the cooling is evidently not uniform. Plot on coordinate paper, using temperatures as ordinates and times 34 1 MAT OF FUSION. 61 as abscissae. Explain the form of the different parts of the curve. II. Cooling Curve of Acetamide. Place a thermometer in a test-tube and surround the ther- mometer with crystals of acetamide, filling the test-tube about one-third full. Place the test-tube in a water-bath and heat to the temperature of boiling water. Remove the test-tube from the bath, clamp it in a stand, and either note the time for each degree or half-degree of fall, or record the tempera- ture every half-minute. Continue the readings until the tem- perature falls to about 40. Plot on coordinate paper, using temperatures as ordinates and times as abscissae. Explain the form of the different parts of the curve. 34. HEAT OF FUSION. References. Watson, p. 246; Ferry and Jones, p. 231. The substance whose heat of fusion it is desired to measure is an alloy known as Wood's fusible metal. Its composition is lead 15.9 parts, cadmium 7 parts, bismuth 52.4 parts, and tin 14.2 parts. The alloy is a solid at ordinary temperatures, but readily melts in hot water. The method employed will be the method of mixtures. A known mass of the metal is placed in a nickel crucible of known mass and specific heat, and is heated to iooC. The whole is then plunged into a known quantity of cold water in a calorimeter cup and the change in temperatures noted. From these data, if the spe- cific heat of the metal in the solid and in the liquid state be known, the heat of fusion may be found. Let M be the mass of the alloy; m the mass of the nickel crucible; W the mass of the water in the calorimeter cup plus the water-equivalent of the cup, thermometer, and stirrer; s,_ the specific heat of the liquid alloy ; s 2 that of the solid alloy ; s 3 that of the nickel crucible; T the melting point of the alloy; t the initial tern- 62 HEAT OF FUSION. [34 perature of the alloy and crucible ; t the initial temperature of the calorimeter, t the final temperature; and L the heat of fusion of the alloy. Write the proper equation representing the transfers of heat in the above process, using- the symbols indicated, and solve the equation for L. (a) First determine the mass, in grams, of those things whose mass it is necessary to know. Place the alloy in the crucible and determine, within 3, the melting point of the alloy. To do this, stand the crucible (in a clamp provided for it) in a vessel of water. Heat the water, taking care after the water has reached the temperature of 60 that the heating be done slowly, so that the metal will be at the same tempera- ture as the water. The thermometer must be placed in the water and not in the metal. The liquid metal "wets" glass, hence metal would be withdrawn with the thermometer and relatively large changes in mass introduced. If the thermom- eter were placed in the metal and left there, there would be great danger of its breaking on the solidification of the metal. After the melting point has been found, bring the water to the boiling point, taking care that no water gets inside the crucible. Remove the crucible, quickly wipe the outside, and carefully drop it in the calorimeter, right side up, with its contained metal, letting the water, if it will, run into the cru- cible and thus more quickly cool the metal. Note the time taken for the temperature to become uniform, then note the rate of cooling and correct for radiation. When cold, the metal can be dumped out of the crucible, leaving the crucible clean. The specific heats of the solid and liquid alloy will be given, or if time permits they may be found by the method of mix- tures. The specific heat of nickel may be found in a book of Tables. Make two or three determinations, as outlined above, of the heat of fusion of Wood's alloy. (b) Why is only a rough determination of the melting 35 1 HEAT OF VAPORIZATION AT BOILING POINT. 63 point necessary? Discuss the relative accuracy with which the different masses used must be determined for a given accuracy. Point out the sources of error in the experiment. 35. HEAT OF VAPORIZATION AT BOILING POINT. References. Edser, p. 150; Watson's Practical Physics, p. 237. In this experiment Kahlenberg's modification of Berthe- lot's apparatus 1 is used. (a) Determine the boiling point of the liquid used, by care- fully heating a small quantity in a test-tube or beaker by means of a water-bath. (b) Weigh the calorimeter, first dry and empty, then about two-thirds full of water. Carefully dry and weigh the worm, together with the two corks which fit its ends. Set up the calorimeter with stirrer, worm, and thermometer. The boiler consists of a test-tube to which is fitted a rubber stopper. A glass tube extends through the stopper to the bottom of the test-tube; two wires also pass through the stopper, and are connected to a coil of wire which loosely surrounds a part of the glass tube. When in use the test-tube is inverted, enough liquid being placed in it to completely cover the coil of wire after the tube is inverted. An electric current is then sent through the coil, furnishing the heat to boil the liquid. The vapor from the boiling liquid passes downward through the glass tube and enters the worm, when the boiler is placed in position over the calorimeter. Care should be taken to use enough liquid so that the heat- ing coil is covered throughout the experiment. Never allow the heating current to be closed through the coil while the coil is not completely covered with liquid. Do not place the boiler over the calorimeter until the liquid boils and the vapor is issuing freely from the tube. See that the cork is removed journal of Physical Chemistry, 1901, Vol. 5, p. 215. . -, 64 HEAT OF VAPORIZATION AT ROOM TEMPERATURE. [36 from the free end of the worm, as the boiling must be done at atmospheric pressure, otherwise the temperature of the vapor will not be that found in (a). When all is ready, note the temperature of the calorimeter, and place the boiler in its proper place so that the vapor enters the worm. Gently stir the water in the calorimeter, and read the thermometer at one-minute intervals until the temperature has risen about 5. Turn off the current, remove the boiler, cork the ends of the worm, and continue to read the ther- mometer at one-minute intervals for five minutes. Remove the worm from the calorimeter, carefully dry the outside, and weigh. Pour the contents of worm and boiler into the proper bottle, and empty the calorimeter. See that the electric cir- cuit is disconnected. (c) From the series of temperatures taken determine the rise of temperature of the calorimeter, correction being made for radiation. Determine the water-equivalent of the calor- imeter and contents, including the stirrer, thermometer (if the equivalent of the thermometer is not known, it can be found by the method given in Exp. 36), empty worm, and water. The necessary specific heats may be obtained from the Tables. Calculate the amount of heat gained by the calorimeter. Knowing the mass of the vapor condensed, the change in temperature of the liquid, and the specific heat of the liquid (see Tables for the specific heat), calculate the heat transferred to the calorimeter, and determine the heat of vaporization of the liquid at its boiling point. 36. HEAT OF VAPORIZATION AT ROOM TEMPERATURE. The heat of vaporization of a liquid varies with the temper- ature at which vaporization takes place. In nature vaporiza- tion takes place, for the most part, at atmospheric tempera- ture rather than at boiling temperature. The object of this 36] HEAT OF VAPORIZATION AT ROOM TEMPERATURE. 65 experiment is to find the amount of heat necessary to vapor- ize one gram of a liquid at the room temperature. To do this, dry air is made to bubble through the liquid, thus increasing the free surface and producing rapid evaporation. The loss of weight of the liquid gives the amount evaporated, while from the fall of temperature of the liquid and calorim- eter, together with their masses and specific heats, the heat- loss can be determined and the heat of vaporization calcu- lated. (a) Carefully weigh the calorimeter cup when dry and empty, and again when containing about 100 grams of the liquid whose heat of vaporization is desired. Place the cover on the calorimeter, w r ith the thermometer bulb in the liquid and arranged so that dry air can be forced through the liquid by means of a small foot-bellows. Have the initial tempera- ture of the liquid 2 or 3 above the room temperature. Pass the dry air through the liquid, allowing ample room for the vapor-charged :air to escape, until the temperature is as much below room temperature as the initial temperature was above room temperature. Weigh the calorimeter and remaining liquid. A 50 thermometer graduated in o.i should be used. Wet the thermometer, with the liquid used, about as high as the depth to which it will be placed in the liquid in the calor- imeter, so that as much liquid will be introduced at first as will be withdrawn later when the thermometer is removed from the calorimeter. (b) Repeat the work of (a), this time drawing dry air through the calorimeter by means of a jet-pump. When finished, empty and dry the calorimeter. If a liquid other than water was used, the liquid should be poured back into its proper bottle. (c) From the amount of liquid evaporated, the fall in temperature, and the water-equivalent of the thermometer, calorimeter, and liquid used, determine the heat of vaporiza- tion in (a) and (b), taking the mean as the final value. The 66 FREEZING POINT OF SOLUTIONS. [37 amount of liquid cooled may be taken as the mean of the initial and final amounts. (d) Point out the chief sources of error. What reason can you suggest for the increase of the heat of vaporization of a liquid as the temperature of the liquid is decreased ? Evaporation takes place from dry ice at temperatures below the freezing point. This change from solid to vapor is called sublimation. By what amount would you expect the heat of sublimation to exceed the heat of vaporization for any given substance? 37. FREEZING POINT OF SOLUTIONS. References. Watson, p. 268; Watson's Practical Physics, p. 258; Edser, p. 167. The object of this experiment is to observe the lowering of the freezing point of water caused by dissolving salt and sugar in it to form solutions of different concentrations, and to determine the molecular weights by means of this low- ering. (a) Using a 50 thermometer, determine the freezing point of pure water with the same apparatus as that employed in the calibration of the 100 thermometer. Then determine the freezing point of a 4 per cent solution of common salt in water. By percentage solution is here meant the number of grams of dissolved substance per 100 grams of the solution. Repeat for an 8 per cent and for a 12 per cent solution. (b) Repeat (a) with aqueous solutions of sugar of 12, 20, and 40 per cent concentration. (c) Tabulate the results of (a) and of (b), and for each case calculate the lowering, per gram of dissolved substance, of the freezing point of a given mass of water. What rela- tion seems to hold between the change of freezing point of a given mass of water and the mass of dissolved substance? 38] HKAT OF SOLUTION. 67 Note the difference of freezing point for 12 per cent solu- tions of salt and sugar. Calculate the molecular weights of salt and of sugar from the relation, M = Ks/St, where s is the number of grams of dissolved substance, S is the number of grams of the sol- vent, t is the depression of the freezing point, and K is 1850 for aqueous solutions. 38. HEAT OF SOLUTION. The quantity of heat absorbed in the solution of one gram of a substance in a large amount of the solvent is called its heat of solution. If heat is given out in the solution, the quantity is considered negative. If the temperature of the salt after solution be different from that at which it was poured into the water, it will be necessary to consider its specific heat also. According to the following method the heat of solution and the specific heat are both determined, although the former is the main object of the experiment. (a) Calculate the thermal capacity of the calorimeter cup and stirrer from their masses and specific heats, the latter of which may be found in a book of Tables. To this must be added the thermal capacity of the immersed portion of the 50 thermometer used. This may be experimentally deter- mined with sufficient accuracy as follows : Set up the cal- orimeter with water at the room temperature in both cup and jacket, having first weighed the cup and contained water. Record the temperature when it has become steady, then take out the thermometer and immerse it, to the same depth as usual in the cup, in water at about 45. After a few min- utes read the temperature to o.i, then as quickly as possible take out the thermometer and put it into the usual position in the calorimeter, shaking off superfluous water on the way. 68 1 1 HAT OF SOLUTION. [38 Stir until the temperature becomes steady and record the reading. Then calculate the thermal capacity of the ther- mometer. (6) On one of the Becker balances weigh out on pieces of dry paper two portions of salt, each of about 15 grams, to o.oi gram. Make sure that the salt is quite dry and finely pulverized, and be careful not to leave any in the balance-pan. This amount of salt, if sodium hyposulphite be used, when dissolved in 200 grams of water will lower its temperature a little over 3. It is best to have the cup about 3 warmer than the jacket, because the larger part of the salt dissolves in a few seconds, so that the loss of heat by radiation during this time is small ; and the temperature being then reduced to about that of the jacket, there is no loss by radiation dur- ing the longer time required for the complete solution of the salt. (c) Set up the calorimeter, with the jacket filled with water at the room temperature, and the cup containing 200 grams of water about 3 warmer. Keep the stirrer moving slowly and read the temperature of the cup at intervals of one minute for about five minutes. Pour in the salt one minute after the last observation, stir rather vigorously to hasten solution, and record the final temperature. From the series of observations, calculate the temperature of the cup at the time when the salt was poured in. The tem- perature of the salt at that time may be assumed to be that of the room. (d) Make a similar trial with a second portion of salt, having the cup at about 40 C. Make sure that there is the proper difference between cup and jacket at the time the salt is poured in. (e) Call the specific heat of the salt x, and the heat of solution y. Write for each case an equation involving the following quantities : i. Heat lost by water in cup. 39] HEAT OF CHEMICAL COMBINATION. 69 2. Heat lost by thermometer, cup, and stirrer. 3. Heat gained by salt in changing temperature. 4. Heat absorbed during solution of salt. Solve the two equations for x and y. Why does this method give widely varying results for the specific heat, while the results for heat of solution are fairly consistent? Caution : Do not leave the solution standing in the cup. Wash it out as soon as possible. 39. HEAT OF CHEMICAL COMBINATION. Determination of the heat generated by the combination of sodium hydroxide with hydrochloric acid to form sodium chloride. A 0.5 normal solution of each of the above compounds is furnished. By a normal solution is meant one which, in 1000 grams of the solution, contains a mass of the element (which is to enter into the new combination) equal in grams to its atomic weight. Thus the normal solution of sodium hydrox- ide is a solution which contains, in looo grams of the solution, 40 grams (23-}- T 6 + i) of sodium hydroxide, or 23 grams of sodium. The 0.5 normal solution contains one- half as much sodium in the same quantity of solution. It is evident that if equal masses of these solutions be mixed, the reaction will be just completed, and the result will be a neutral solution of sodium chloride. The solutions are to be mixed in the calorimeter cup at as nearly as possible the same temperature, and the resulting rise of temperature noted. The alkali should be placed in the cup, and the acid added to it. The acid, being immediately neutralized, will then have no action on the metal of the cup. (a) Weigh out 100 grams of the sodium hydroxide solution in the cup, and the same amount of the hydrochloric acid solu- tion in the beaker. The latter should be weighed out roughly at first, poured back into the bottle, then the wet beaker counter- 70 EXPANSION OF A GAS BY FLASK METHOD. [40 poised and the amount weighed accurately. This will then be, fairly accurately, the mass which is afterward poured into the calorimeter. A small error is introduced by taking the second thermometer out of the beaker after reading its temperature, but this may be neglected. If care has been taken not to handle the cup and beaker any more than is necessary, the two temperatures should be very nearly the same when ready for use. Since the amounts used are equal, it may be safely assumed that the resulting solution of sodium chloride has risen to the final temperature from the mean of the two initial temperatures. A direct determination of the specific heat of the sodium chloride solution is impracticable. The value, 0.987, which has been calculated by interpolation from tabulated results, may be used for this case. Make two trials, and calculate for each the quantity of heat which would have been evolved if 1000 grams of nor- mal solution had been used in each case. (b) Repeat the work, if there is time, with similar solu- tions of potassium hydroxide and sulphuric acid, and com- pare the results. 40. COEFFICIENT OF EXPANSION OF A GAS AT CONSTANT PRESSURE BY FLASK METHOD. (a) Thoroughly dry the flask or bulb by rinsing out ten times or more with dry air, then hang it in the boiler with the bulb down and with the stop-cock open. If there is any chance for the steam to enter, attach a rubber tube to the open end and place the other end of this tube where the steam cannot enter. Boil the water, causing the steam to pass around the whole flask until the air inside is at the temperature of the steam. Then close the stop-cock, remove from the boiler, and allow to cool. Next place it under the surface of the ice-water until it has assumed that temperature. Then open 41 ] CONSTANT-PRESSURE AIR-THERMOMETER. 71 the stop-cock under water, allowing the water to enter but not the air to escape. Raise the bulb so that the level of the water inside is the same as that without, thus assuring the same pressure. Close the stop-cock, remove and dry, and then carefully weigh. In order to obtain the volume, the flask must now be weighed full of water, and then again empty and dry. It is best to fill with ice-water and to make the weighings when it is cold, so as to get the volume at oC. In drying the flask, great care should be taken not to break the stop- cock by the heat. These weighings will enable you to deter- mine the volume of the air in the flask when under atmos- pheric pressure and at oC. From the results of the above measurements and the coefficient of expansion of glass find the coefficient of expan- sion of air. (b) Repeat with some available gas other than air, and compare the result with that of air. 41. COEFFICIENT OF EXPANSION OF AIR. CON- STANT-PRESSURE AIR-THERMOMETER. References. Edser, p. 108; Watson's Practical Physics, p. 209. The object of the experiment is to study the variation of the volume of a gas when heated under constant pressure (Gay Lussac's Law), and to determine the average coeffi- cient of cubical expansion of air between o and iooC. In the form of constant-pressure air-thermometer used, the air (carefully dried) is contained in a glass tube graduated in cu. cm. and closed at one end. The graduated tube is con- nected to an open glass tube by rubber tubing, forming a "U" containing mercury. The pressure on the enclosed air can be regulated by raising or lowering the open glass tube. Surrounding the graduated tube containing the air is a ves- sel, covered by an asbestos jacket, in which a bath of water 7 2 CONSTANT-PRESSURE AIR-THERMOMETER. [4! may be placed or through which steam may be passed. The graduated tube is vertically adjustable through a sleeve in the bottom of the vessel, so that the meniscus of the mercury may be seen outside and the volume read. (a) Fill the vessel with a mixture of ice and water, and, when the enclosed air has had time to come to the tempera- ture of the bath, read the volume, after adjusting the pres- sure so that it is equal to atmospheric pressure. Fill the vessel with water at ioC., adjust the pressure, and again read the volume. In this way raise the temperature by steps, reading the volume of the air at 10, 20, 30, 45, 60, 80, taking care each time to wait long enough (three min- utes or more) for the enclosed air to come to the same tem- perature as the bath, and each time adjusting the pressure so that it is equal to atmospheric pressure. The mercury meniscus on the closed-tube side should always be as close to the bottom of the jacket as will permit of reading the level. After closing the top, pass steam througn the vessel, in at the bottom and out at the top, and take another reading of the volume, pressure conditions being the same as before. It will probably be necessary to wait longer in this case than in the other cases for the air to reach the temperature of the steam. (b) Make another and similar series of observations at a pressure 10 cm. above atmospheric pressure. (r) Plot the observations of (a) and of (b) on the same sheet of coordinate paper, using temperatures (centigrade) as abscissae and volumes as ordinates. From the volume at oC. and the volume at 100 C., as taken from the curve, calculate, for each curve, the average apparent coefficient of expansion of the air between those temperatures. Take the mean of the two results, correct for the expansion of the glass, and obtain /?. the absolute coefficient of expansion of air. 42] CONSTANT-VOLUME AIR-THERMOMTER. 73 (d) Show how the volume varies with the absolute tem- perature, the pressure being constant. Taking some particular volume on the two curves, com- pare the corresponding temperatures and pressures. Do the same for another volume. Knowing the pressure corre- sponding to each of the two curves, how does the pressure vary with the temperature when the volume is constant? Taking the same temperatures on the two curves, find how the volume varies with the pressure. (e) Write an equation connecting the temperature, pres- sure, and volume of a gas which might be inferred from the three parts of (d). 42 CONSTANT- VOLUME AIR-THERMOMETER. References. Watson's Practical Physics, p. 203; Millikan, p. 125 and p. 131; Miller, p. 166. The object of this experiment is to study the law of vari- ation of the pressure of a given quantity of enclosed air at constant volume as the temperature is changed, and also to determine the pressure coefficient of the gas. The air is enclosed in a glass bulb placed inside a vessel so that the bulb may be surrounded by a water-bath, by shaved ice, or by steam. A thermometer is placed in the bath to determine its temperature. The pressure on the enclosed gas is regu- lated by means of a mercury column in a rubber tube, con- necting on the one side with the glass tube which forms an extension of the bulb, and on the other with an open ver- tical glass tube whose position is vertically adjustable. The pressure on the enclosed air may be determined from the dif- ference in level of the mercury on the two sides and the barometric reading. The volume of the air in the bulb is made the same before each reading by bringing the mercury men- iscus in contact with a glass point inside the glass tube attached to the bulb. 74 CONSTANT-VOLUME AIK-THICRMOMETKR. | 4-' Caution : The mercury on the bulb side should always be lowered some distance before changing to a lower tem- perature. Be especially careful to do this before removing the steam when you have taken a reading at the boiling point. (Otherwise, on cooling, the mercury will run into the bulb.) Do not hurry in taking the readings after chang- ing the temperature, but wait until the meniscus set at the glass point does not move. (a) Without any bath in the reservoir, while all is at the room temperature, bring the mercury to the glass point and determine the pressure. Record the room temperature, and the barometric reading. (b) After having lowered the mercury on the bulb side, surround the bulb with shaved ice, and then determine the pressure with the meniscus at the glass point. The tempera- ture may be taken as oC. Melt the ice, and then make a series of determinations of the pressure with the water in the vessels having in turn temperatures of 10, 20, 30, 45, 60, and 80 (approxi- mately). Pass steam through the vessel, making another determin- ation. This temperature may be found by determining the boiling point from the known atmospheric pressure. Arrange all observations in tabular form. (c) Plot on coordinate paper the results of (b), using temperatures as abscissae and the corresponding pressures as ordinates. Draw a smooth curve through the points of the plot. Calculate the mean increase of pressure per degree increase in temperature from o to 100, using values taken from the plot. This is the temperature coefficient (ft) of pressure of a gas. Write it as a decimal and find its reciprocal. The negative of this represents what point on the absolute scale of temperatures ? 43] VARIATION OF A SATURATED VAPOR. 75 (d) Write an equation connecting P , the pressure at o ; P, the pressure at t ; t ; and j3. Using this equation and the pressure of (a), calculate the temperature of the room, thus using the apparatus as" a ther- mometer. Compare the result with the room temperature as read from a mercury thermometer. (e) Show from your results how the pressure of the gas varies with the absolute temperature, the volume remaining constant. 43- VARIATION OF PRESSURE, VOLUME, AND TEMPERATURE OF A SATURATED VAPOR. References. Edser, p. 220; Watson, p. 251. The apparatus is the same as that used in the experiment on Boyle's Law. Instead of a gas, there is in one closed tube some liquid ether and in the other some water. (In some of the instruments the tubes contain carbon bisulphide and water in place of ether and water. Note which you are using.) The pressure on the vapor above each liquid is produced by a mer- cury column in a rubber U-tube, one side of which is connected to the closed tubes, the other being connected to an open glass tube. By measuring the height of the mercury menisci on the two sides, and noting the barometric pressure, the pressure on the vapor may be obtained. In doing this there should be added to the reading of the meniscus in the closed tube the mercury-equivalent of the height of the liquid column in the tube. The volume may be read directly from the graduated tube, and may be changed by raising or lowering the mercury. The temperature may. be regulated by means of a water-bath surrounding the closed tube. (a) At the temperature of the room, without any water around the closed tubes, make five different readings, chang- ing the volumes and reading the corresponding pressures. 76 VARIATION 01? A SATTRATED VAPOR. [43 Record the volume and pressure each time for both closed tubes. After changing the volume each time, wait until con- ditions become steady before taking the readings. (b) Surround the tube with a mixture of ice and water. Arranging the tubes so that you can read the mercury men- iscus, repeat (a), making, however, only two readings of vol- umes and corresponding pressures. Melt the ice gradually and increase the temperature to 5, and take two more read- ings, having one volume the same as with the temperature at o. Keeping one volume the same at each temperature, re- peat the readings, making two settings at each temperature, with the bath at TO, 20, 30, 35, 40, and 45. Do not go higher with the instruments containing ether. If carbon bisulphide be used, omit the readings at 35 but make a read- ing at 50 and at 55, but at no higher temperature. (r) Put a little ether (or carbon bisulphide, if this be the liquid in the tube) in a test-tube, and boil it over a water-bath with a thermometer placed in the ascending vapor. Xote the boiling point. Obtain the boiling point of water from the Tables. (d) What relation is found to exist in each case between the volume and pressure of a saturated vapor at constant tem- perature? Does Boyle's Law hold? From the observations for each vapor, plot on coordinate paper a curve, with temperatures as abscissae and correspond- ing pressures as ordinates, for some volume which is the same in the different sets of readings. How do you find that the ''vapor pressure" varies with the temperature, the volume being constant ? From the curves of ether and water, or carbon bisulphide and water, find the temperatures corresponding to atmos- pheric pressure. Compare them with the boiling points of ether and water found in (c). To what is the vapor pressure of any liquid equal at its boiling point? 44] HYGROMETRY. 77 44. HYGROMETRY. References. Miller, p. 197; Millikan, p. 164; Watson's Practical Physics, p. 247; Edser, p. 240. In this experiment the dew-point and the relative and abso- lute humidity of the air are to be determined. The absolute humidity, d, is the density of the water-vapor present in the air, usually expressed in grams per cubic meter. The rela- tive humidity is the ratio of the amount of water actually pres- ent in the air to the amount required to saturate it at the same temperature, the latter quantity being the maximum amount of water-vapor that can be held in suspension at that tempera- ture. The dew-point is the temperature at which the amount of water actually present in the air would saturate it, that is, the temperature to which the air must be lowered before the condensation of water will begin. The pressure of water- vapor is the pressure which it would exert by itself if there were no air present in the space considered. By Dalton's Law this is the pressure it actually does exert when mixed with air. In a given volume the mass of vapor is proportional to the pressure, so that the relative humidity is equal to the ratio of the pressure, p, of the water-vapor in the air to the pressure, P, of saturated water-vapor at that temperature, that is, rela- tive humidity is equal to p/P. (I) Regnault's Hygrometer. (a) Partially fill one of the hygrometer tubes with ether and insert a thermometer with its bulb in the liquid. Force a current of air through the ether with a bicycle pump. The rapid evaporation of the liquid causes the temperature to fall. When the tube and the air immediately about it are cooled to the dew-point, moisture appears on the tube, this being detected more easily by comparison with the other tube. Note the tem- perature at which dew begins to form. Allow the tube to become warm and record the temperature at which the dew /8 DENSITY OF THE AIR BY THE BARODEIK. [45 disappears. Take the mean of these two as the dew-point. Make three such determinations of the dew-point. (b) From Whiting's Tables find the pressure of saturated water- vapor at the dew-point and also at the temperature of the room, and calculate the relative humidity. The absolute humidity may be found by multiplying the relative humidity by the number of grams of saturated water-vapor in a cubic meter of air at the room temperature, found in Whiting's Tables. II. Wet- and Dry-Bulb Hygrometer, or Augusta's Psy- chrometer. In the wet- and dry-bulb hygrometer, one bulb is covered with wicking which dips into water, so that the bulb is cooled by evaporation. After the two thermometers come to constant temperatures, record the temperature of the dry bulb, t, and of the wet bulb, t 18 Read the barometer. The following empirical formula may then be used: p = pi 0.0008 b ( t t x ) , where p is the pressure of water- vapor present in the atmos- phere, pi the pressure of saturated vapor at the temperature of the wet bulb (obtained from Whiting's Tables), and b is the barometric pressure, all three being expressed in millimeters of mercury. Find the pressure of saturated water-vapor at the room temperature from the Tables, and calculate the rela- tive humidity. Find then the absolute humidity as in (b). From Table 15, p. 868, Whiting's Tables, find the dew-point from the readings of the wet- and dry-bulb hygrometer. Compare the values obtained in I and II for the humidity and the dew-point. 45. DENSITY OF THE AIR BY THE BARODEIK. I. To find the difference between the barodeik reading and the true density of the air. The barodeik is an ordinary balance, having a hermetically 45] DENSITY OF THE MR BY THE BARODElK. . 79 sealed flask suspended from one scale-pan, and from the other (as a counterpoise) a glass plate so chosen as to have a surface about equal to the exterior surface of the flask. The reading of the balance-pointer on a properly graduated scale gives the density of the surrounding air. (a) Set and read the barometer with great care. Read the wet- and dry-bulb hygrometer. From Table 15, Whiting's Tables, calculate the dew-point and also the pressure of water- vapor in the air. Remember that "dew-point" means the temperature at which the water-vapor now in the air would be saturated ; or the temperature at which the existing pressure of water-vapor in the air would be the maximum pressure. (&) From (a) calculate the density of the air. The mass of one cu. cm. of dry air, at oC. and 76 cm. pressure, is 0.001293 grams. The mass of the same volume of water-vapor, under the same conditions, is 5/8 as much. Then, if H be the barometric height, f the pressure of water-vapor, and t the temperature, the mass of dry air in one cu. cm. of moist air is, by the general gas law, PV = RmT, i H f where a is the coefficient of expansion of a gas. The mass of water-vapor in the same volume is M, = (s/ 8 ) 0.001293 ^~ ~- The sum of these two is the required density. (Deschanel, p. 400.) (c) Read the barodeik. Do not touch the instrument, but, by moving the hand near the flask, set up a small vibration ; then close the case, and determine the resting-point of the pointer, which is the density of the air as indicated by the instrument. (d) Record the difference between the reading thus ob- 8O DENSITY OF THE AIR BY THE BARODEIK. [45 tained and the true density found in (b), with the proper sign, so that when added algebraically to the observed read- ing it will give the true density of the air. This is the abso- lute correction for the scale-division to which it applies. II. Relative Calibration of the Barodeik Scale. (a) Read the instrument as in I (c). Repeat with the rider at division 2 on its scale to the right of the center, which is equivalent to adding 2 mg. to the right-hand pan of the balance; then use the rider in the corresponding position on the left-hand side. (b) Repeat the readings with the rider at division 5 ; first on the right-hand, then on the left-hand side. (c) Using the exterior volumes of flask and plate as given on the instrument, calculate the changes in the density of the air which would produce the same effects on the instru- ment as the putting of the separate masses on the right pan, and on the left pan. From these results construct a table of corrections, with the proper signs, for the different resting- points observed. Note that this is a relative calibration ; that is, it gives the corrections to be applied to certain readings, as compared with one reading (namely, that when no weights were used), which is assumed correct. (rf) In part I the absolute correction for a certain reading was found. That reading was the same as, or not far from, the one assumed correct above, so the same absolute correction may be applied to the latter. By means of this, convert the table of relative corrections, (c), into a table of absolute corrections. This completes the absolute calibration of the instrument. (e) Plot on coordinate paper the readings of the barodeik scale as abscissae and the relative corrections of (c) as ordi- nates, but on a much larger scale. Show how the curve can be made to indicate absolute corrections instead of relative, by moving the horizontal axis of reference up or down by 46] COEFFICIENT OF FRICTION. 8 1 a proper amount. This converts it into an absolute calibra- tion curve for the instrument, enabling one to find the density of the air at any time by merely reading- the resting-point of the pointer. 46. COEFFICIENT OF FRICTION. References. Watson, p. in; Ferry and Jones, p. 75. When one body is caused to slide over the surface of another the force which is brought into play to oppose the motion is called "friction."' This force is parallel to the sur- face and opposite in direction to the motion. When the sliding body is on a level plane, the normal force is equal to the weight of the body ; when on an inclined plane it is equal to the component of the body's weight normal to the plane. In either case the force of friction is equal and oppo- site to the force necessary just to produce motion (starting friction), or to keep the body moving at constant speed (moving friction). If P is the normal force between the two surfaces and F is the force of friction, the ratio F/P is called the coefficient of starting or moving friction, as the case may be, and is usually denoted by the Greek letter p. Ry measuring these forces and calculating their ratio the coefficient may be determined. A second method of deter- mining the coefficient of friction is to vary the inclination of the plane until the body by its weight just begins to move (starting friction) or moves down the plane with constant speed (moving friction). If the angle of inclination at which this occurs is a, prove that the coefficient of friction is equal to tan a. (a) The coefficient of friction is to be found between blocks provided and the surface of a plane whose inclination can be varied. Take one of the blocks and weigh it. Deter- mine the force of starting friction and also of moving friction 82 COEFFICIENT OF FRICTION. [46 on a level surface by applying forces to it by means of the shot-bucket and string and pulley. Calculate the coefficient of friction for the two cases. (b) Determine the coefficient of friction for the same block and surface from the tangent of the angle obtained by varying the inclination of the plane until ( i ) motion com- mences, and (2) continues at constant speed. (c) Set the plane at the angle giving constant speed down the plane, and find the force that will cause the block to move up the plane at constant speed. Calculate the coefficient of friction. (d) Set the plane at an angle of 30 and find the force necessary to move the block up the plane at constant speed, and then, if possible, the force necessary to make it move down the plane at constant speed. Then, by calculating the force perpendicular to the plane, find the coefficient of fric- tion. If this process is not entirely clear, repeat with the plane at an angle of 60. (c) Repeat (a), for starting friction, having the block "loaded" by placing a known mass on top of it. Compare the coefficient of friction found with that found in (a). (/) Take a block having three or more surfaces of differ- ent areas but of the same smoothness, and determine (by any method) the force of friction as the block slides or is moved successively on the three surfaces. (g) Take a block with surfaces of different degrees of smoothness, and determine the coefficient of starting friction for two or more sides. (h) Compare the results obtained from (a), (b), (c), (rf), and (c), stating your conclusions. What do you con- clude from (/) ? From (g) ? Upon what does the friction between two surfaces depend ? 47] CONSERVATION OF MOMENTUM. 83 47. CONSERVATION OF MOMENTUM. COEFFI- FICIENT OF RESTITUTION. References. Ames and Bliss, p. 90; Millikan, p. 58. In any system of bodies not acted upon by any outside force, and in which the several bodies may be moving with different velocities and in different directions with frequent collisions, the vector sum of the momenta remains constant. This is known as the Law of Conservation of Momen- tum. In our present study the number of bodies will be limited to two and the velocities restricted to the same straight line, the collisions taking place centrally. Let us suppose that we have two bodies, A and B, suspended by strings so that they hang in contact when at rest. Let A be dfiawn aside and then released. At the lowest point of its swing it strikes the ball B. Let m t be the mass of A and i^ its veloc- ity just as it strikes B. Its momentum then at this instant is m l u l . The ball B will instantly start off with a velocity v 2 , say, and a momentum m 2 v 2 , if m 2 is its mass. The ball A may continue on with a diminished velocity, v x ; or remain at rest, if it loses all of its momentum ; or it may rebound, in which case v, is negative. After impact the two balls will move away from each other with a relative velocity which is greater the greater their elasticity. The elasticity is taken into account in a factor called the "coefficient of restitution." The coefficient of restitution is numerically equal to the ratio of the relative velocities with which the bodies move apart after impact to that with which they approached each other before impact, i. e., it is given by the equation, CD e = = "i u, where the velocities before impact, u^ and u 2 , and the veloc- ities after impact, v x and v 2 , are all in the same straight line. One or more of the velocities mav be negative, or the 84 CONSERVATION OF MOMENTUM. [4- particular value of a velocity may be zero, as in the case just outlined lu = o. The value of e always lies between zero and unity. For "perfectly elastic" bodies e= i. but for all actual bodies e < i. For inelastic bodies e = o. Tn any case, whether the bodies are elastic or inelastic, the conserva- tion of momentum holds, i. e., (2) mjUj + rn 2 u 2 = m,Vj + m 2 v 2 . This may be verified by determining the masses and the veloc- ities. In order to find the velocity of a suspended ball as it col- lides, or just after collision, we make use of the fact that the velocity is the same as the ball would have acquired if it had fallen the same vertical distance that it has descended in its swing before collision, or that it has risen in its swing after collision, as the case may be. Let the height be h; then, as the case may be, u or v equals V 2 gh- If the angle of the half swing is a and the length of the pendulum is 1, we have, (3) u, or v, = V 2 gl( I cos ) (a) The numbers in the scale of the frame from which the balls are suspended represent degrees of arc. First use the two large ivory balls and see that they are adjusted so as to hang fully in contact and so that their centers are in line. Record the zero-reading for each ball. Draw one aside through about TO or 15 and fix it in position with a thread. Record the reading. Release it by burning the thread. Note carefully the extremity of the swing of each ball after impact. This can be done by placing a slider in the position for each ball. Several trials will be necessary to accurately determine these points. From these, and the zero-readings, the arcs of the swings are found, and then by measuring the length of the cord (to the center of the ball) the velocities Uj, v v and v 2 can be determined. Repeat for two other starting points from which the ball is released. Determine the masses of the 48] YOUNG'S MODULUS BY STRETCHING. 85 balls, and then calculate for the three cases the momentum of the system before and after impact. (b) Use one large ivory ball and one small one, and repeat for one or two starting points, releasing the large ball. (c) Repeat (b), reversing the process by releasing the small ball. (d) Use two lead balls and repeat for one or two starting points, or place a layer of paraffine on each large ivory ball on adjacent sides and use them as inelastic balls. (e) Calculate the coefficient of restitution for all the cases above. Does it appear to depend on the size of the balls or only on the material? For the same cases calculate the percentage difference in the momentum before and after impact. Does the "Conserva- tion of Momentum" appear to hold equally well in all cases ? Calculate the percentage loss of kinetic energy for each case. What becomes of the energy apparently lost? Is the loss greater in the more or in the less elastic bodies? 48. YOUNG'S MODULUS BY STRETCHING. References. Watson's Practical Physics, p. 99; Millikan, p. 65; Ferry and Jones, p. 120. Hooke's Law states that in elastic bodies, within their elas- tic limits, the strain or deformation produced is propor- tional to the stress or distorting force. In particular it states that if different forces be applied to a wire, e. g., by sus- pending it and hanging masses from it, the amount of stretching will be (within certain limits) proportional to the applied force. For a wire of any given material the ratio of the stress per unit area of cross-section to the increase in length per unit length is a constant, and is known as Young's mod- ulus. (The word stress strictly applies to the force inside the wire which opposes the distorting force.) 86 YOUNG'S MODULUS BY STRETCHING. [48 A wire is held vertically between two clamps. To the lower clamp, C, is attached the end of a rod whose upper end is loosely held in a support. From the lower end of the wire or the clamp C, masses may be suspended and the wire stretched. Above the upper end of the rod is a screw with a divided head so that small fractions of a turn may be read. When the screw comes in contact w r ith the rod an electric bell rings. If the wire is then stretched, the screw must be advanced again before ringing will occur. In this way the change in length is readily determined, if the pitch of the screw is known. (a) Hang first a 1500 gm. mass from the wire and leave it there for all of the zero- readings. This will insure the wire being straight at the beginning. Make a setting with the screw and read it to o.oi mm. or less. Then increase the load by adding 500 gms. at a time (each time reading the screw), until the wire carries a load of about 4000 gms. (b) Take the masses off, 500 gms. at a time, reading the screw each time. (c) Repeat (a) and (b) at least once. Record the obser- vations in tabulated form. Calculate and record the elonga- tion produced by each 500 gms. (d) Measure the length of the wire between the clamps, and also the diameter of the wire. (e) Repeat (a), (&), (c), (d) with a wire of different diameter. (f) From the mean elongation for a stretching force of 500 grams-weight, calculate separately Young's modulus for the two wires. Does Hooke's Law hold for these wires according to the results in your tabular form? If there is any variation from the law, assign a reason if you can. 49] HOOKK'S LAW FOR TWISTING. 87 49. HOOKE'S LAW FOR TWISTING. COEFFICIENT OF RIGIDITY. References. Millikan, p. 71; Ames and Bliss, p. 168. If a cylindrical wire be fixed at one end, and the free end be twisted about the axis of the wire, no change of volume will occur, but the strain in the wire is found to be one of shape only. For a wire of given material, length, and diam- eter, the force-moment producing the twisting is found to be (within certain limits) proportional to the angle of twist. This statement may be deduced mathematically from Hooke's Law which states that in elastic bodies (within their elastic limits) the strain or deformation produced is proportional to the stress or distorting force. The mathe- matical reasoning establishing the relation between the angle of twist, the force-moment producing the twisting, and the material, length, and radius of the wire is not simple, involv- ing integral calculus. The object of this experiment is to establish the relation experimentally. If M is the moment of the twisting force, a the angle of twist in radians, 1 the length of the rod, and r its radius, we ma write a == , ?rnr 4 In the above equation, n is constant for a given material and is called the "coefficient of rigidity,'' or sometimes the "mod- ulus of torsion." (a) The torsion lathe, as described in the first reference given, will be used. The support with graduated wheel, and also the fixed support, should be clamped firmly to the table. Clamp the smallest rod provided, with one end in the wheel and the other in the fixed support, being careful to have the rod straight and the scale on the wheel adjacent to the ver- nier. Add masses to the pan, preferably 100 grams at a 88 CENTRIPETAL FORCK. [50 time, and read the angle of twist for each twisting moment used. Reverse the process, reading the angle each time a mass is taken off. Record, in tabular form, the masses used, the corresponding angular deflections, and the increase in deflection for each loo-gram mass added. Measure the diam- eter of the wheel, the diameter of the rod, and the length of the rod between the clamps. (b) Repeat the observations of (o) with the same rod clamped so as to use one-half the length there used. (r) Repeat with a rod of the same material and length as that used in (a), but having a different radius. Repeat with a rod of different material, but having the same length and radius, approximately, as that used in (a). (d) From your results show how the angle of twist varies with the twisting moment, with the length of the rod, and with the radius of the rod. Calculate the coefficient of rigidity for each case, (a), (&), (c), using the average angular deflection corresponding to the force-moment produced by the weight of 100 grams. If the radius of the wire be measured to the nearest o.oi mm., with what accuracy should the length be measured? 50. CENTRIPETAL FORCE. Reference. Millikan, p. 100. The object of this experiment is to determine the force necessary to keep a body of given mass in a circle of given radius, while it moves with constant speed. Experience shows that a body in motion will continue to move with the same speed in the same straight line, unless acted upon by some outside force. An outside force, if acting in the direction of the motion, will cause a change in speed ; if acting at right angles to the direction of motion, it will cause no change in speed, but will cause a change in the direction of the motion. A body in motion always moves in a straight line, unless there 50] CENTRIPETAL FORCE. 89 is a. force applied causing it to leave the straight line. If the force, perpendicular to the line of the motion, be momentar- ily supplied, the direction of the motion is changed, but the body continues to move in a straight line at an angle with its former direction. If the force be continually supplied, the body moves in a curved path. If the body be kept in a circu- lar path, a force of definite magnitude must be continuously applied to the body, the direction of the force being always per- pendicular to the instantaneous direction of the motion. Since the instantaneous direction of motion is along the tan- gent, the force perpendicular to the direction of motion must be along the radius of the circle. If the force ceases to be supplied, the body ceases to leave the straight line and hence continues to move in the tangent to the circle at the position occupied by the body at the instant the force ceased to act. This is illustrated by whirling a stone at the. end of a string the string supplies the force necessary to keep the stone in a circular path. If the string breaks, the necessary force is no longer supplied, and the stone is no longer pulled out of the straight- line path. It moves away, therefore, at a tangent to its former circular path. This cental force is called the Centripetal Force, or the Normal Force. It is called the normal force because it is always normal to the curved path. It is always directed toward the concave side of the curve. If the path is a circle, it is directed inward along the radius. For a circular motion the magnitude of the normal force is directly proportional to the mass of the body and the square of its speed, and inversely proportional to the radius of the circle. This arises from the fact that the force is pro- portional to the product of the mass and the acceleration of the body, and the acceleration is inward along the radius and of magnitude v 2 /r, where v is the speed of the body and r is the radius of the circle. (a) To a rotator is attached the "centripetal force" appar- atus. Two masses, m l and m 2 , are arranged to slide along 9O FRICTION BRAKE. POWER SUl'lM.lKD I'.V A MOTOR. [51 the horizontal guides. They are attached, by means of cords passing over pulleys, to a large mass, M, which can slide up and down along the vertical rod. As the speed of rota- tion is increased, more and more force must be supplied to irij and m 2 in order to hold them to a circular path. Finally, when the speed passes a certain value, the force necessary to keep the masses moving in their circular paths is greater than the weight of M can supply, and the mass M is lifted. The speed may be so regulated that M remains about half- way up the rod. Its weight, Mg dynes, represents the nor- mal or centripetal force supplied to the masses m x and m. 2 . Write the equation representing this relation. The masses M, m lt and m 2 must be determined, and the distances of m l and m 2 from the axis of rotation. The speeds of m, and m 2 may be calculated, provided the number of rotations in a given time be counted. Make several determinations of the speed, varying its value by altering the masses, m x and m,, or by changing their distances, i^ and r 2 , from the axis. (b) In each case test the equality of the weight, Mg, and the calculated centripetal force required. Determine, in each case, the percentage difference. Point out the principal sources of error in the experiment. In the case of circular motion what term is commonly applied to the reaction against the centripetal force? 51. FRICTION BRAKE. POWER SUPPLIED BY A MOTOR. Reference. Watson, p. 116. The object of this experiment is to measure by means of a friction brake the power delivered by an electric motor and to study the effect of altering the friction of the different parts. An electric motor, a bank of incandescent lamps arranged in parallel, and a key are connected in series with 51 ] FRICTION BRAKE. POWER SUPPLIED BY A MOTOR. 91 the no-volt power-circuit. The circuit is made by pressing the key. The resistance can be decreased by introducing more lamps into the circuit. A Prony brake is used. The Prony brake consists of a lever, one end of which is bound around a revolving shaft in such a way that the friction produced will tend to rotate the lever in the direction in which the shaft revolves. This tendency to rotate is balanced by a spring balance acting at right angles to the lever or by the weight of masses hung from the lever. If P is the force in dynes acting on the lever to prevent rotation, and L the distance from the line of P to the center of the shaft, the power absorbed by the brake, or the work per second, will be 27rLnP, where n is the number of revolutions of the shaft per second. (a) Suspend a spring balance from the iron stand, and then attach it below to the lever of fhe brake so that, when the motor is running, the balance will oppose any tendency of the brake to rotate. Note the reading of the balance when the motor is not running. Then start the motor by gradually decreasing the resistance given by the incandescent lamps, and, with the motor running at less than full speed, tighten the belt connecting the motor to the shaft of the brake. Allow the motor to run at full speed with the belt taut, and record the number of revolutions of the shaft in three min- utes, as given by the speed counter. Note the reading of the spring balance while the shaft is rotating. Take two more readings with the balance at different points along the lever. (b) Tighten the screws which bind the wooden blocks of the brake against the shaft, and take measurements with three different lever arms. Note if the lamps grow brighter when the friction is increased. If so, what can be said about the dependence of the power consumed by a motor on the load? The effect may also be observed by tightening the belt con- necting motor and brake-shaft. (c) Calculate the power delivered by the motor for each 92 - ABSORPTION AND RADIATION. [52 of the six measurements. In what units is the power ex- pressed, if the force of the balances is in dynes and the lever arm in centimeters? Reduce the results to horse-power. If you know the method by which electrical power is computed, show how the efficiency of the motor may be calculated. (d) Disconnect the friction brake, attach the spring bal- ances to a cord, and hold or suspend them above the motor so that they will pull in parallel lines, thereby pressing the cord against half of the periphery of the motor-wheel. Allow- ing the motor to run at moderate speed, record the difference in the readings of the two spring balances as the cord presses against the wheel. (e) Repeat (d) for the other pulley-wheel on the motor- shaft, exerting as nearly as possible the same tension as be- fore. Measure the diameter of each of the wheels and see what relation exists between the friction and the radius of the wheels, the angle of contact being the same in the two cases. 52. ABSORPTION AND RADIATION. References. Watson, p. 304; Edser, p. 436. It is well known that a dull black surface absorbs light more readily than a white or light-colored one. This is shown by the difficulty in illuminating a photographic dark room or a room with dark-colored hangings. The purpose of this experiment is to see whether the relations which hold for light apply also to the vibrations of longer period which are manifest to our senses only through the sensation of heat. That is, it is proposed to study the rate of absorption of heat by black and by polished surfaces, and also the rate at which heat is radiated by these surfaces to a colder body. (a) A box lined with tin has an opening in the side in which three thermometers may be set and read from the 53] RATIO OF THE: SPECIFIC HEATS OP A GAS. 93 outside. The bulb of one of the thermometers is bare, another is silvered, and the third is coated with lampblack. All three thermometers should register the temperature of the room. Record the room temperature. Heat water to boiling in a kettle and pour into the vessel in the box, arranging this so that the steam will not reach the thermometer bulbs and con- dense on them. Record the readings of all three thermometers each minute until a steady temperature is reached. Then at an even minute remove the hot water and continue the read- ings till the thermometers again register the temperature of the room. (b] Make a good freezing mixture in a large beaker, and place this in the box close to the thermometer bulbs, the thermometers being equally distant from the freezing mix- ture. Read the temperatures each minute until they cease to fall. Remove the freezing mixture and read the ther- mometers as they return to room temperature. (c) Plot the results of (a) and (b) on coordinate paper, using times as abscissae and temperatures as ordinates, mak- ing the scale as large as possible. Discuss the form of the curves and the relation between the several curves. What re- lation exists between absorption and radiation at the highest and at the lowest temperatures reached? Connect the re- sults with the fact that stoves are made black and the fender and knobs of the stove are nickeled. 53. RATIO OF THE TWO PRINCIPAL SPECIFIC HEATS OF A GAS. References. Watson, p. 328; Watson's Practical Physics, p. 267. The object of this experiment is to obtain, by the appli- cation of the laws of thermodynamics, the value of y, the ratio of the specific heat of a gas at constant pressure to its specific heat at constant volume. Air is compressed, adia- 94 KATIO OF THK SI'KCIKIC HKATS <>!' A C.AS. [53 hatically, in a large balloon-flask to a pressure greater than atmospheric pressure, and is allowed to cool to room temperature, t r The pressure, p,, is then measured on the manometer. Let the volume of unit mass of gas in the flask- be v,. The stop-cock is now opened and the air allowed to expand, the volume, v,, increasing to v 2 , the pressure falling to atmospheric pressure, p , and the temperature to t 2 . The stop-cock is again closed and the temperature allowed to rise again to t lf the volume remaining the same, v 2 . and the pres- sure rising to p 2 . The gas has now been in three conditions, as follows : Condition. Pressure. Vol. of i gm. Temperature. I. p, Y! t : II. Po V 2 t, III. p 2 V, t, The change from I to II was adiabatic (i. e., no heat passed in or out ; see Watson, p. 326) , hence (1) 1W = p v./, or (V 1 /v 2 )>'=p /p 1 . The change from I to III is isothermal, hence, by Boyle's Law, (2) PiV 1 = p 2 v 2 , or ( Vl /v 2 K= (Po/Pi)''. Hence, (p 2 /p,) y = (Po/Pi) '< or, solving for y, log Po- log P. log p, log p, Perform the experiment as indicated above, making several determinations of the pressures, reading the manometer and the barometer. Calculate y. Take the value given in Watson (p. 325) as correct, and calculate the percentage error of vour result. oj^gSt^ C - UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 6 Dec'49AP 51