MVB(sinr OF DAVIS PHYSICAL MEASUREMENTS IN THE PROPERTIES OF MATTER AND .IN HEAT BY RALPH S. MINOR Associate Professor of Physics, University of California AND T. SIDNEY ELSTON Instructor in Physics, University of California BERKELEY, CALIFORNIA 1910 OT Copyrighted in the Year 1910 by RALPH S. MINOR and T. SIDNEY ELSTON In the Office of the Librarian of Congress, Washington PREFACE This manual is printed primarily for the use of the fresh- man students in the various engineering colleges of the Uni- versity of California, 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 the first part of a two-years course in General Physics, the second part which deals with Sound, Light, and Electricity being given during the sophomore year. The predecessors of the present writers, Professor Harold Whiting, Professor Elmer E. Hall, A. C. Alexander, G. K. Burgess, Bruce V. Hill, and A. S. King, have all played a part in the evolution of this manual. Its present form, how- ever, is largely a revision of the manual printed in 1908 by Prof. Hall and Dr. Elston. In this revision the present writers have been guided by a desire to so simplify the work as to make possible the performance of all of the experiments, with a few exceptions, during a two-hour laboratory period. Free use has been made of published texts and manuals with- out specific credit being always indicated. Several experiments have been taken from the sophomore course of Professor E. R. Drew, written before the present division of subjects was made. RALPH S. MINOR, T. SIDNEY ELSTON. Berkeley, Cal., August, 1910. LIST OF EXPERIMENTS 1. Sensitive Beam Balance. Density of a Solid. 2. Jolly's Balance. Specific Gravity. 3. Model Beam Balance. 4. Boyle's Law. 5. The Volumenometer. 6. The Force Table. 7. Three Forces in Equilibrium. 8. Density of Air. 9. Relative Density of Carbon Dioxide. 10. Uniformly Accelerated Motion. 11. Centripetal Force. 12. The Principle of Moments. 13. The Simple Pendulum. 14. The Force Equation. 15. Surface Tension by Jolly's Balance. 1 6. Capillarity. Rise of Liquids in Tubes. 17. Rise of Liquids between Plates. 1 8. Viscosity. Flow of Liquids in Tubes. 19. Efflux of Gases. Relative Densities. 20. Absolute Calibration of a Thermometer. 21. Relative Calibration of a Thermometer. 22. Variation of Boiling Point with Pressure. 23. Coefficient of Expansion of a Liquid by Archimedes' Principle. 24. Comparison of Alcohol and Water Thermometers. 25. Coefficient of Expansion of a Liquid by Regnault's Method. 26. Coefficient of Expansion of Glass by Weight-Ther- mometer. 27. Coefficient of Expansion of a Liquid by Pycnometer. 28. Expansion Curve of Water. LIST OF EXPERIMENTS v 29. Specific Heat of a Liquid by Method of Heating. 30. Specific Heat of a Liquid by Method of Cooling. 31. Mechanical Equivalent of Heat by Callendar's Method. 32. Mechanical Equivalent of Heat by Puluj's Method. 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. t 38. Heat of Solution. 39. Heat of Neutralization. 40. Expansion of Air at Constant Pressure by Flask Method. 41. Expansion of Air at Constant Pressure. Constant- Pressure Air- Thermometer. 42. Constant- Volume Air-Thermometer. 43. Vapor-Pressure and Volume. 44. Vapor-Pressure and Temperature. 45. Hygrometry. 46. Density of the Air by the Barodeik. 47. Coefficient of Friction. 48. Conservation of Momentum. Coefficient of Restitution. 49. Young's Modulus by Stretching. 50. Hooke's Law for Twisting. Coefficient of Rigidity. 51. Friction Brake. Power Supplied by a Motor. 52. Absorption and Radiation. 53. Ratio of the Two Specific Heats of a Gas. REFERENCES The following list of books includes all those to which reference is made in this manual. Some of them are text- books, some laboratory manuals, and other books of tables. A few of them have been placed on the window-shelf for gen- eral reference ; the others can be drawn at the desk. Text Books- Duff (Blakiston's) : Text-Book of Physics (Second Edi- tion. Edser : Heat for Advanced Students. Hastings and Beach : General Physics. Preston: Theory of Heat (Second Edition). Watson: Text-Book of Physics (Fourth Edition, 1903). Laboratory Manuals. Terry and Jones: A Manual of Practical Physics, Vol. i. Millikan: Mechanics, Molecular Physics and Heat. Watson : Text-Book of Practical Physics. Books of Tables Landolt and Bernstein: Physical and Chemical Tables. Smithsonian Institute: Physical Tables. Whiting: Physical Tables. PHYSICAL MEASUREMENTS. PROPERTIES OF MATTER, AND HEAT. This book is intended to be mainly a manual of directions. It has been the aim of the authors to make it complete enough^ however, so that when used in conjunction with >Blakiston's Physics, the class text-book, it will cover the minimum re- quirement for the year's work. It is expected that the student will elect to consult the larger reference-books (a liberal num- ber of copies of which are available at the desk) for general notions regarding physical measurements, the discussion of re- sults, the effect of errors in observation and methods for their complete or partial elimination. Printed directions, regarding the method of writing-up and handing-in the record of the experiments, will be found on the folder used as a cover for the record. A list of the re- quired experiments and the order in which they are to be per- formed will be posted on the laboratory bulletin-board. 1. SENSITIVE BEAM BALANCE. DENSITY OF A SOLID. Weighing by Method of Vibrations. In weighing with a sensitive beam balance use is made of a long pointer attached to the beam and arranged to vibrate in front of a fixed scale. The more sensitive the balance the greater the angle will be through which the pointer will vi- brate for a given excess mass placed in either of the pans. The pointer will swing many times back and forth before it finally comes to rest at a definite point which marks the position of 2 SENSITIVE BEAM BALANCE. DENSITY OF A SOLID. [l 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 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 number 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 vibrations." To weigh a body it is necessary first to know the rest-point with the two pans empty. This is the zero rest-point. The body, being placed now in the left-hand pan, enough standard masses are placed in the other pan to balance it. If the rest- point now be the same as before, the weight of the body in air is represented by the weight of the standard masses used. If the rest-point be not the same, it is best then to determine the sensitiveness of the balance, that is, the number of scale-divi- sions through which the addition of i mg. to the pan will shift the rest-point. From this and the difference between the two rest-points, the weight of the body in air may be obtained by calculation. In most sensitive beam balances a centigram rider is used. By properly placing the rider on the graduated scale attached to the beam, the equivalent of any desired mass from I to 10 mg. may be added to either pan. Final adjustments can thus be made without opening the balance case. The rider should never be moved without first lowering the balance beam. The balance must be handled with the greatest care, since any jarring or rapid vibration of the beam may injure the l] SENSITIVE BEAM BALANCE. DENSITY OF A SOLID. 3 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 sensitive beam balance, let it be required to find the density of a cylindrical solid. (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 and the rider to bring the pointer ap- proximately to the zero rest-point, and then determine the rest-point by 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 or empty- pan rest-point, by means of the rider add 5 mg. or so to either pan and determine the sensitiveness of the balance. From the sensitiveness, the difference in the rest-points, and the masses in the right-hand pan, find the exact mass which will balance the hard rubber cylinder in air. (c) Correction for Air-buoyancy. Unless the body whose mass is sought has the same density as the masses used to balance it, the body will be buoyed up by the air either more 4 JOLLY'S BALANCE. SPECIFIC GRAVITY. [2 or less than the masses are buoyed up, and this will introduce an error which is by no means negligible in careful measure- ments. To correct for air-buoyancy : Measure the dimensions of the cylinder with vernier calipers, and compute its volume. Calculate the volume of the standard brass masses from their marked mass values and the density of brass (8.4 gms. per cc.) Rea4 the thermometer and barometer. From the given vol- umes 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 cylinder. (d) If the arms of the balance be unequal in length, "double weighing" is necessary. In such a case the cylinder is placed in the right-hand pan and the mass found as before. The true mass is then given by m = where m^ and m z are the values obtained by the two weighings. The proof of this by an application of the principle of mo- ments is left to the discretion of the student. In most sensi- tive balances the arms are so nearly equal that double weigh- ing is unnecessary. 2. JOLLY'S BALANCE. SPECIFIC GRAVITY. Jolly's balance consists of a. long spiral spring suspended from an upright steel frame. The lower end of the spring car- ries two light pans, the lower of which is always immersed to the same depth in a beaker of water standing on a small plat- form attached to the frame. In one form of the balance the upper end of the spring is stationary, and the lower end car- rying the pans may be raised or lowered and the elongation read from a graduated mirror fixed to the frame just behind the spring. In the other form of the balance the upper end of the spring is movable and the lower end with the two pans 2] JOLLY'S BALANCE. SPECIFIC GRAVITY. 5 is kept at a fixed mark; the elongation is read from a grad- uated sliding scale attached to the upper end of the spring. The use of this balance to determine specific gravity depends on the fact that the spring obeys Hooke's law closely for small elongations, that is, the elongation is proportional to the change in the stretching force. With the lower pan immersed in water we first note the "empty-pan," or zero, scale-reading. The solid, whose specific gravity is to be found, is then placed in the upper pan, elongating the spring and requiring a read- justment along the scale. The scale-reading is again noted and the elongation determined. This elongation represents the weight of the solid in air. The solid is now transferred from the upper pan to the lower pan and the elongation of the spring, again from the zero position, is determined. This latter elongation represents the weight of the solid in water. From these data the elongation equivalent to the weight of water displaced by the solid may be found and the specific gravity of the solid calculated. (a) Hooke's Law Tested. Place a standard 5 eg. mass in the upper pan and determine the elongation. Repeat with larger masses, up to 5 gm. or more. With one form of the bal- ance the readings are taken by bringing a bead or point of wire at the lower end of the spring into coincidence with its image in the mirror scale ; with the other form the readings are taken from the vernier and sliding scale after bringing the mark at the lower end of the spring on a level with the etched line on the glass tube. From the observed elongations determine if the spring obeys Hooke's law or not. (b) Solids Heavier Than Water. By the method outlined above, find the weights, in air and in water, of the solids fur- nished. Take care that the lower pan is immersed to the same depth in the water and that no air-bubbles cling to the lower pan or to the solid when immersed. From the data obtained calculate th^ specific gravity of each of the solids. 6 MODEL BEAM BALANCE. [3 (c) Solids Lighter Than Water. Find the specific grav- ity of a solid which floats in water. For this purpose a sinker must be used, but it may be left in the lower pan throughout the experiment. The following readings will be found nec- essary: first with the upper pan empty, then with the solid in the upper pan, and again with the solid tied to or wedged un- der the sinker in the lower pan. (d) Liquids. Find the specific gravity of a salt-solution by using a solid hung by a thread instead of using the lower pan. (Salt-water will corrode the nickeled surface of the pan.) It will be necessary to find the elongation of the spring equi- valent to the displacement of the solid in water as well as in the salt-solution. Explain the method used. (e) How would you proceed to use Jolly's balance to weigh objects as you use the beam balance? If a bubble of air had been carried down with the solid when immersed, would the calculated specific gravity have been greater or less than it should be? Why is it necessary to keep the lower pan immersed always to the same depth throughout a given experiment? What additional data would you need, if required to calcu- late the density from the specific gravity of any of the objects used in this experiment? Explain. 3. MODEL BEAM BALANCE. The beam balance consists of a metal beam, supported so as to be able to rotate about a central knife-edge located verti- cally above the center of gravity of the beam. Near the ends of this beam, pans are hung from knife-edges. The result is that, wherever the object and the standard masses may be placed in the two pans, the vertical force which keeps them in equilibrium must pass through the knife-edge above, and so the effect upon the balance is the same as if the whole weight 3] MODEL BEAM BALANCE. of the scale-pan and included load acted at some point in the knife-edge from which the pan is hung. The distance from the central knife-edge to the knife-edge at either end of the beam is called the arm of the balance or the length of the beam. A model beam 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 fa- cility with which the pointer of the 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 the center of gravity 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. Fig. i. Let us suppose that in the figure the points A, C, B rep- resent the positions of the three knife-edges and G the posi- tion of the center of gravity of the beam when the two pans are carrying equal loads ; and that the points A', C, B' repre- 8 MODEL BEAM BALANCE. [3 sent the corresponding positions of the knife-edges and G' the corresponding position of the center of gravity when a small excess mass is added to the right-hand pan. L,et m = the mass of the beam, / = 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 gravity 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 positive if the beam is concave upwards, negative if the beam is concave downwards. Applying the principle of moments for the case of equilibrium, the central knife-edge being the center of moments, we have (I) (M + *) gl COS (ft a) -- Mgl COS (ft + a) mgh sin a = o. Expanding, collecting terms, and transposing, [mh (2M -j- JF) sin ft] sin a = Ix cos ft cos a, or , tan /'cos /9 x = mh (2M + *) /sin ,3' If the beam is straight, ft = o and tan / (3) ~T =mh The sensitiveness is measured by the ratio, tan a /x. Since the expression for this ratio does not contain M, it follows that the sensitiveness in the case of a straight beam is inde- 3] MODEL BEAM BALANCE. 9 pendent of the load ; it increases with any arrangement which makes the fraction l/mh larger. In the case of a curved beam, however, it is evident from (2) that the sensitiveness is de-: pendent 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), with the exception of the mass m of the beam. The following possibilities are at once apparent : ( I ) the length / of the beam may be varied by loosening the set screws which clamp the movable end-por- tions of the beam to the tubular central portion; (2) its cen- ter of gravity may be raised or lowered (thus altering h) by sliding the metal bob up or down along the pointer; (3) the load M may be increased by adding masses to the scale-pans j (4) the points of suspension of the two pans may be placed level with, above, or below the central knife-edge, thus mak- ing the beam straight, or curved up or down. Straight Beam. (a) Adjust the sliding bob so that the center of gravity of the beam lies below the central knife-edge. Change the set screws on the beams, if necessary, so as to make the two beam- arms equal in length. Place the terminal knife-edges at the zero mark, thus insuring a straight beam. Hang the two scale-pans in position. Level up the balance by means of the thumb-screws on the legs. Place successively several 2 eg. masses in the right-hand pan, recording the deflections and noting if they are propor- tional to the number of masses used. Why should the de- flection not be strictly proportional to the number of masses? For convenience, select the deflection produced by the first 2 eg. mass as a measure of the sensitiveness of the balance. (b) Change the length of the beam-arm. Test the sensi- tiveness by means of a 2 eg. mass. Compare with (a), and IO MODEL BEAM BALANCE. f3 state how the sensitiveness depends upon the length of the beam-arm, other conditions remaining unchanged; and note if the result is in agreement with the formula. (c) Adjust the length of the beam-arm back to its value in (a), and then change the position of the center of gravity of the beam by sliding the bob up or down. Test the sensitive- ness and compare with (a), giving your conclusions. (d) Bring the bob back again to its position in (a). In- crease the load by placing a 100 gm. mass in each scale-pan. Test the sensitiveness and compare with (a), giving your con- clusions. Curved Beam. (e) Remove the masses from the scale-pans. Raise the terminal knife-edges to the first mark above zero, thus chang ing the beam from a straight beam into one which is curved up. Test the sensitiveness and compare with (a). Does in- creasing the upward curvature of the beam, other conditions remaining unchanged, increase or decrease the sensitiveness ? Show that this is in agreement with the formula. (f) Place a 100 gm. mass in each scale-pan. Test the sen- sitiveness and compare with (e), giving your conclusions. (g) Remove the masses from the scale-pans. Lower the terminal knife-edges to the first mark below the zero, thus changing the beam into one which is curved down. Test the sensitiveness and compare with (a), giving your conclusions. (h) Place a 100 gm. mass in each scale-pan. Test the sen- sitiveness and compare with (g), giving your conclusions. A sensitive chemical balance is usually made with the beam curved slightly upwards when there is no load in the pans. A medium load straightens the beam and an excess load causes a downward curvature. From the results above, state how the sensitiveness of such a balance will change with the load on account of the curvature of the beam. 4] BOYLE'S LAW. n 4. BOYLE'S LAW. Reference. Duff, p. 158. The purpose of this experiment is to study the relation be- tween the volume and the pressure of a given mass of air kept at constant temperature. According to Boyle's law the vol- ume of a fixed mass of gas, kept at constant temperature, var- ies inversely as the pressure in the gas. This relation may be mathematically expressed in various ways : ( I ) The vol- umes of the gas at two different times are inversely propor- tional to the corresponding pressures; (2) The product of the volume and pressure at one time is equal to the corresponding product at another time, or, in other words, the product of the volume and pressure of the gas is a constant, that is, *V it, where p and v are respectively the pressure and corresponding volume, and K is a constant whose value is fixed so long as the temperature, mass, and nature of the gas remain un- changed. This is approximately true for most of the perma- nent gases, provided the pressures are not very large. The higher the temperature at which the gas is held, the more closely does the law hold true. A given mass of dry air is enclosed in an inverted, grad- uated glass tube which is attached to one end of a rubber tube containing mercury. At the opposite end of the rubber tube is an open glass tube. The glass tubes are clamped to verti- cal guides having a meter scale between them. The two clamps can be so adjusted along the guides as to vary the pressure on the enclosed air from values greater than atmos- pheric pressure to those which are smaller. By reading the positions of the menisci of the mercury columns in the two tubes and adding to, or subtracting from, the atmospheric 12 BOYLE'S LAW. [4 pressure as determined by the barometer, the pressure corres- ponding to each setting can be found. The corresponding volume of the enclosed air can be read from the graduated tube containing it. (a) Clamp the tube, enclosing the air, to the vertical guide near the bottom. Raise the open tube as high along the other vertical guide as possible and clamp it to the guide. It may be necessary to pour more mercury into the open tube so as to raise the mercury level on that side. Record the temper- ature, the volume of the enclosed air, and the positions of the two mercury menisci. Care should be taken in reading the position of the mercury meniscus to avoid parallax. For this purpose, in sighting, stand so that the eye, the mercury sur- face and the image of the mercury surface formed by the mirror are in the same straight line. In reading the volume of the enclosed air, account should be taken of the curvature of the mercury meniscus. Read the barometer and record the atmospheric pressure in cm. of mercury. (>) Lower the open tube 10 cm. or so at a time, repeating the readings in (a) for each position. Continue until the two mercury surfaces are in the same level. Record, in tabular form, the meniscus-readings^ the volume v, the total pressure />, and the product p v. (c) Unclamp the two tubes and raise them to points near the upper ends of the vertical guides. In doing so, care should be taken not to allow mercury to overflow. Lower the open tube 10 cm. or so at a time, repeating the readings and tabu- lating as in (b). (d) On separate sheets of millimeter cross-section paper plot the following curves : (i) With pressures as ordinates and volumes as abscissae, plot the data of (a), (&), and (c). Draw a smooth curve which will best represent the average position of the plotted points. What mathematical curve does it resemble? The re- 5] THE VOLUMENOMETER. semblance would be much more marked, if the same length had been chosen to represent the pressure-unit and the volume- unit. This, however, it will not be found convenient to do (2) With the products p v as ordinates and the volumes as abscissae, plot the results of (a), (&), and (c). Draw a smooth curve which will best represent the plotted points. What mathematical curve does it most closely resemble? If 'Boyle's law held strictly, what form should the curve take? (e) The temperature prevailing during the experiment was room-temperature. If, instead, a higher temperature had pre- vailed, state with your reasons how you should expect the curves to be displaced in the plots. What effect would an in- crease in the mass of the air have, other conditions remaining the same as in the experiment? What are the principal sources of error ? Determine from the two plots the volumes of the enclosed air for pressures of 50 and 100 cm. Compare. 5. THE VOLUMENOMETER. The object of this experiment is to find the density of an ir- regular solid by means of the volumenometer and the balance. In the volumenometer, A is a glass tube which may be closed Fig. 2. i.f 14 THE VOLUMENOMETER. [5 at the top by a ground glass plate. It corresponds to the closed tube in the experiment on Boyle's law. As in that experiment, the pressure and volume of the air in A are varied by raising or lowering a tube containing mercury. The pressure is de- termined by noting the difference in the levels of the two mer- cury menisci, and adding to or subtracting from the atmos- pheric pressure as read from the barometer. The volume is unknown. The volume of a portion of the tube between two marks (M and N), however, is known. Let the volume between M and N be k, and that above M be V. By determining the pressures when the volume of air is V (mercury meniscus at M), and again when the volume is V-\-k (mercury meniscus at N), an equation involving Boyle's law may be written containing these two volumes and the cor- responding pressures. From this equation V may be calcu- lated. The volume of air in A may be found in this way both with and without the solid body enclosed whose volume we desire to know. The volume of this solid thus becomes known. From its volume and mass its density can be found. (a) With the tube A uncovered bring the mercury meniscus to M, recording the pressure, evidently just equal to the atmos- pheric pressure. Carefully place the plate on A, so as to in- sure an air-tight joint. The plate must be clean and have on it only a little grease. Lower the mercury meniscus in the tube A from M to N, note the difference in level of mercury menisci in A and L, and again determine the pressure. Test for leakage by allowing the tube to remain a minute or more in this position, and make sure that the heights of the menisci do not change. Calculate k, and then, by applying Boyle's law, find V. (b) Remove the plate, place inside the volumenometer one of the bodies whose density is to be determined, and repeat (a). From the volume V of the air, found in this case, and 6] THE FORCE TABLE. 15 the former volume V, the volume of the body is found. Weigh the body and determine its density. (c) Repeat for at least two other bodies. (d) If it were possible to perfect the measuring instru- ments used in the course of this experiment so that they would be absolutely accurate, it would still be unreasonable to expect that the results obtained for the density would be those given in the Tables. Why ? Determine the precision of measurement of the balance, the barometer, and the volumenometer, and from these determine the precision of measurement or reliability of the final result. What are the advantages and disadvantages of this method of determining density? 6. THE FORCE TABLE. Reference. Duff, pp. 36-41. The purpose of this experiment is to determine the vector sum or resultant of two forces acting on a body in the same plane and along lines not parallel. Let the lines of direction of the two forces, / x and / 2 , intersect in a point, O, making angles, a and a 2 , with an arbitrarily chosen axis, OX. The vector sum of these forces is by definition a force, f, given bv the diagonal of the parallelogram formed by /\ and f z as sides. Let the directional angle of f be a. By taking the projections of f lt f 2 , and / upon the perpendicular axes, OX and OF, we see by construction that (1) / sin a = A sin a : + f 2 sin a 2 , (2) / cos a = / cos ai + f 2 cos a 2 ; whence, by squaring (i) and (2) and adding, (3) f 2 = A 2 + / 2 2 + 2fJ, c 16 THE FORCE TABLE. [6 and, by dividing (i) by (2), (4) tana=/' sina '+/' sing '. /, COS a, -f / 2 COS o a The last two equations give the magnitude and direction of the diagonal of the parallelogram in terms of the magni- tudes and directions of the sides. Equation (3) enables one to calculate the magnitude of the resultant of the two given forces, and by equation (4) the direction of this resultant can be determined. The apparatus used to test these results consists of an ad- justable iron table with circular top graduated in degrees. Pulleys can be clamped to the circumference at any chosen points. From a pin, placed in a hole in the center of the table - top, three cords pass over the pulleys and carry pans upon which known masses are placed. The masses and pans should be weighed on the platform scales. Tested Algebraically. (a) Arbitrarily take f lf f 2 , a , a 2 as equal respectively to (200 + m) gms. wt., (100 + ' m) gms. wt, 35, 85, where m is the mass of the pan holding the masses. Calculate by equations (3) and (4) the value of / and a. It will be found advisable to make these calculations and those in (&) before entering the laboratory to begin the experiment. Set one of the pulleys at 35 and one at 85. With the pin in place, put the requisite masses in the pans to make /^ and / 2 equal to the values chosen for them. Then, if a third pul- ley be set 1 80 from the direction determined by a as calcu- lated above, and masses corresponding to the calculated value of / be added, the three forces acting on the pin should be in equilibrium, since the third force is equal and opposite to the vector sum of f l and f 2 . Pull out the pin and see wheth- er the calculation is correct. 7] THREE) FORCES IN EQUILIBRIUM. 17 (&) In a similar manner calculate and test two other sets of values chosen by you. Tested Geometrically. (c) Select three new sets of values for f lf f 2 , a if and a 2 and proceed as follows with each set: Place a circular sheet of manila paper on the table and run the pin through it. Set the two pulleys at a t and a 2 , and place the requisite masses on the pans. Mark with a pencil the directions of the two strings, then remove the paper and on the lines lay off dis- tances from their intersection proportional to f 1 and / 2 . Com- plete the parallelogram and determine from the diagonal the value of /. Now replace the paper on the table, set the third pulley opposite to /, and adjust the masses on its pan to equal the value of / as determined by the diagonal. Pull out the pin and see if the construction is correct. (d) Point out the principal sources of error in the two methods used above. If three forces acting upon a body hold it in equilibrium, how must their lines of direction intersect? A ladder leaning against a smooth vertical wall is prevented from sliding by the reaction of the ground. What forces are acting on the ladder? Construct the line of direction of the reaction of the ground on the ladder. 7. THREE FORCES IN EQUILIBRIUM. Reference. Duff, p. 78. The purpose of this experiment is to study the conditions which must be satisfied in order to produce equilibrium among three forces, two of which are mutually perpendicular. The simplest case is where the three forces are applied at the same point in the body; but the more general case where the three forces have different points of application in the body is i8 THREE FORCES IN EQUILIBRIUM. [7 essentially the same, in-so-far as equilibrium is concerned, for the lines of direction of the three forces must pass through one and the same point if the forces are in equilibrium. It is quite evident that the three forces must lie in the same plane, for each of the three must be opposite to and in the same straight line with the resultant of the other two. Moreover, the resultant or vector sum of the three forces must be zero, if the forces are in equilibrium. Let OA, OB, OC, (see Fig. 3) represent three forces f lt f 2 , / 3 , which are in equilibrium, the first two being mutually per- pendicular to each other. The third force / 3 , to produce equi- librium, must be equal and opposite to and in the same straight line with the resultant of the other two. Since the diagonal OC represents the resultant of f^ and f 2 , it follows that the line OC which represents f s must be equal and oppo- site to and in the same straight line with OC. This is the point of view considered and verified by Exp. 6. Since the forces are in equilibrium, their resultant effect in any direction must be zero, that is, the algebraic sum of the projections of the forces in that direction must be zero. In the line DA the effect of f 1 is f 1 cos o, the effect of / 2 is 7] THREE FORCES IN EQUILIBRIUM, / 2 cos 90, and the effect of / 3 is / 3 cos a. But the resultant effect along DA is zero, hence /! COS + / 2 COS 90 + / 3 COS a = O. Since cos o = i and cos 90 = o, the equation becomes (1) /, + / 3 COSa = 0. Similarly, the resultant effect along EB must be zero, and hence the sum of the projections of the forces into that line must be zero, or /! sin o + f 2 sin 90 + f s sin a = o. Since sin o = o and sin 90 = i, this equation becomes (2) / 2 + / 8 sin a = o. (It should be noted that sin a and cos a are negative for an angle in the third quadrant.) Interpreted geometrically, equation ( i ) shows that OA and OD are equal and opposite ; equation (2) shows that OB and OH are equal and opposite. The present experiment is intended to verify the relations (i) and (2). The apparatus consists essentially of a spring balance P (see Fig. 4), a compression spring balance Q, and Fig. 4. 2O THREE FORCES IN EQUILIBRIUM. [7 a weight-hanger R. The two balances are attached to a ver- tical rod along which they can be adjusted. The balance P is kept in a horizontal position so as to be constantly at right angles to the vertical force exerted by the weight-hanger. The three forces acting at O are respectively a horizontal pull to- ward the right by the balance P, an oblique thrust upward to the left by the rod belonging to the balance Q, and a vertical pull downward due to the weight of the hanger R. A differ- ent arrangement of forces is obtained by changing the number of masses carried by R, and by changing the direction of Q. On a sheet of paper fastened behind the balance the lines of di- rection of the three forces can be traced. (a) Arrange the balances and the weight-hanger in the manner indicated in the figure. Hang a mass of 9 kg. on the hanger, and adjust the balance P along the rod until the an- gle MOR, as tested by means of a square, is a right angle. Support the weight of the two balances by means of the hands ; note if the balance-readings are materially changed, and if they are, ask for assistance in correcting for the same. Record the readings of the two balances and the total weight of the hanger and masses. On a sheet of paper, held behind the balances, make a trace of the lines of direction of the three forces and determine the angle NOR. The sine and cosine of this angle may be found from the Tables, or di- rectly from a measurement of the distances OM and MN. (b) Repeat (a) twice with different masses on the hanger. Readjust the balance P each time so that the angle MOR shall equal a right angle. (c) Change the length of the cord or chain which con- nects O with the balance P, and thus alter the angle NOR. Repeat the adjustments and measurements of (a). (d) From the results in (a), (&), and (c) determine if the condition of equilibrium is satisfied, first by substitut- ing the recorded values in the equations (i) and (2), and 8] DENSITY OF AIR. 21 again by constructing the triangle of forces in each case and noting if it is closed or not. (e) On one of the traces draw through O a line which does not coincide with any one of the three forces. Deter- mine the effect which each one of the three has in this line and see if the resultant effect is zero, employing the method already outlined in deriving equations (i) and (2). What are the principal sources of error in this experiment? How can equations (i) and (2) be used to calculate the magnitude and direction of the resultant of forces f and / 2 ? What is the form of the equations which would connect /!, / 2 , and / 3 if the angle NOR were not a right angle (see Exp. 6) ? 8. DENSITY OF AIR. Let a glass bulb of volume V be weighed full of air at at- mospheric pressure P , and let M be the mass necessary to balance it. Then let the air be pumped out until the pressure is P 2 , the mass as determined by weighing now being (M w), where m is the mass of air that has been pumped out between the weighings. Then if d and d 2 be the densities correspond- ing to the pressures P x and P 2 , it follows, from the definition of density and the interpretation of m, that (!) Vdi Vd 2 = m. The reciprocal of the density is the volume of unit mass: hence, if Boyle's law is applied to unit mass of the air, the temperature being assumed constant, we have (see Exp. 4), Eliminating d z from (i) and (2), we get (3) 4 -/>) 22 DENSITY OF AIR. [8 This last equation may be employed as a formula to find the density of the air from a knowledge of the volume of the flask, the pressure before and after exhaustion, and the mass of air pumped out. In the application it is essential that the temperature should be the same during the two weighings. (Why?) This condition is approximately satisfied in practice. If the temperature were not the same, the observed pressure in the second case would need to be corrected (through the application of Charles' law) so as to give the pressure that would have existed had the temperature been the same as dur- ing the first weighing. The volume V is obtained by weighing the bulb when empty and then when full of water at a known temperature. (a) Carefully dry the flask by exhausting it several times and admitting air each time through a calcium-chloride drying- tube. Ask an assistant for instructions in regard to manipu- lating 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 rinse with dry air as be- fore. With the dried flask in connection with the drying tube, admit air at atmospheric pressure. Close the stop-cock and carefully weigh the flask. Note the temperature. Read the barometer for the pressure. (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 and record the pressure. (c) Fill the flask completely with water up to the stop- cock, taking care to have no water above it. 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. Calcu- late 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- 9] RELATIVE DENSITY OF CARBON DIOXIDE. 23 ture and atmospheric pressure. From this result the density or dry air under standard conditions (that is, at oC and 760 mm. pressure), may be found through the application of Boyle's and Charles' laws, or a combination of the two. If P lt d^ and T 1 represent the pressure, density and absolute temperature of a given kind of gas at one time, and P 2 , d 2 and T 2 represent the corresponding values at another time, and so on, then it follows from a combination of the two laws that P P (~\ -*! -* v3; j T -TT a i 2 i a i L i for the given kind of gas to the degree of approximation with which it observes the given laws. Making use of this relation, calculate the density of dry air under standard conditions of temperature and pressure, and compare with the value given in the Tables. Point out the chief sources of error and any other reasons for the discrepancy in the results. 9. RELATIVE DENSITY OF CARBON DIOXIDE. The relative density of carbon dioxide compared with air as a standard is to be measured. The method employed is that used in Exp. 8. Using the same symbols as there used, and making the weighings and noting the pressures as there indi- cated, we have for the air, (!) rf, = If the measurements are then repeated for the carbon dioxide, (2) d\ = 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, we get, by division, 24 UNIFORMLY ACCELERATED MOTION. flO 4' _ *^-_^) . - 4 - /> (/Y - /Y) ' from which we see that a determination of the volume of the flask is unnecessary. (a) Read the directions given under Exp. 8. Ask an assist- ant for instructions in the use of the pump. Carefully dry the 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 the pressure and temper- ature. In weighing, follow the method given in Exp. I. (b) Pump the air out until a low pressure is obtained and weigh the flask again at the reduced pressure. If the temper- ature is not the same, within o.5, the observed pressure should be corrected as in Exp. 8. (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 until a low pressure is obtained, as in the case of the air, and weigh again. (e) By the use of equation (3), calculate from your results the relative density of carbon dioxide with respect to air, un- der the given conditions of temperature and pressure prevail- ing in the room. How would you proceed to apply Boyle's and Charles' laws to the result in order to find what the ratio would be under standard conditions? 10. UNIFORMLY ACCELERATED MOTION. The purpose of this experiment is to determine the acceler- ation of a freely falling body from the trace made by a vi- brating tuning fork in touch with the body while falling. 10] UNIFORMLY ACCELERATED MOTION. 25 Conceive of a body moving in a straight line with uniformly accelerated motion, being at a point A at a certain time, A^ one interval of time later, A 2 at the end of the second in- terval of time, etc. Let s l be the distance covered during the first interval of time, s 2 the distance covered during the sec- ond interval, etc. ; and let t be the number of seconds in the given interval of time. Let v be the average velocity of the body during the first interval, v z that during the second inter- val, etc. Then, by the definition of average velocity, we have (i) ^ = ^' *>, = -' etc. Let flj be the average acceleration between the first and sec- ond intervals of time, a 2 that between the second and third intervals, etc. Then, by the definition of average accelera- tion, we have or substituting from (i), we get x \ *o i **'-l **> (3) tfi = JL 7 r- L > , = - i - ? -^ etc. If the body has uniformly accelerated motion, a lt a 2 , etc. must be equal. (If the motion is uniformly accelerated, the velocity will increase at a constant time-rate. Then v v the average velocity for the first interval of time, will be equal to the instantaneous velocity of the body at the middle instant of that interval; similarly v 2 will be equal to the instantaneous velocity at the middle instant of the second interval of time, etc. Between the middle instants of any two successive time-intervals the time elapsing is evidently equal to t. The average acceleration, then, between the middle instants of the first and second intervals is (v 2 v ^)/t> as given above.) (a) There are two forms of apparatus used in the labora- tory for this experiment. In one form, the falling body con- 26 CENTRIPETAL FORCE. [ll sists of a brass frame which falls about 120 cm. along ver- tical guides which offer very little friction. This frame car- ries with it a tuning fork, one prong of which is provided with a stylus which traces a wavy line upon the whitened glass plate clamped vertically in the support. The release of the fork by the lever at the top causes the prongs to vibrate. In the other form, the fork is stationary and the glass plate, upon which the trace is to be made, falls about 50 cm. along the vertical guides. The vibrations of the fork, in this case, are maintained electrically. In both forms, a plumb line is used to adjust the plate and guides for the fork, so that they will be accurately vertical. The plate is first covered with a thin coat of corn-starch and alcohol, which quickly dries. It 5s then placed in the frame and adjustments made. At least three good traces should be obtained. A fine line is next ruled along one edge of the trace; and, starting at any convenient point, points four or five vibrations apart are marked off and their dis- tances apart, s lf s 2 , etc., measured. Tabulate these values of s and their successive differences. Repeat for points ten vibra- tions apart. These measurements should be made for at least two traces. (fr) From the known value of the frequency of the fork find t. Calculate the acceleration for each set of observations and take the mean. Is it constant? Estimate the precision of measurement of the result. Name the principal sources of error. 11. CENTRIPETAL FORCE. References. Duff, pp. 24, 35; 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 H] CENTRIPETAL FORCE. 2 7 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 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 continuously 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, along a tan- gent to its former circular path. This central 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. The acceleration which it, as an unbalanced force, gives the body is also inward along the radius and is called Normal Ac- celeration. 28 CENTRIPETAL FORCE. [ll For a circular motion the magnitude of the normal acceler- ation is equal to v 2 /r, where v is the speed of the body and r is the radius of the circle. By the Force Equation an unbal- anced force acting upon a body is proportional to the product of the mass m of the body and the acceleration produced. We have, then, v 'i Normal acceleration (#) = - v i Centripetal force = kma n = km If the quantities m, v, and r are expressed in C. G. S. units the factor k will be unity and the force will be given in dynes. (a) To a rotator is attached the "centripetal force" appar- atus. Two masses, m and w 2 , are arranged to slide along 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 m l 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, so the mass M is lifted. The speed may be so regulated that M remains about half- way up the rod, or better, slowly rises and falls past this point. Its weight, Mg dynes, represents the normal or centripetal force supplied to the masses m and m 2 . Write the equation representing this relation. The masses M, m lf and w 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 trials, selecting each time a different set of values of the masses or of their dis- tances from the axis. In each case maintain the speed for five minutes or more. 12] THE PRINCIPLE OF MOMENTS. 2Q (b) For each trial, test the equality of the weight Mg and the calculated centripetal force required, and determine the percentage difference. Point out the principal sources of error in the experiment. In the case of a body in circular motion what term is com- monly applied to the reaction against the centripetal force? Does it act on the body, or not? In the case of a skater describing a circle on ice, what sup- plies the needed centripetal force? Are the radial forces act- ing on the skater's body balanced? Are the vertical ones bal- anced ? 12. THE PRINCIPLE OF MOMENTS. References. Millikan, p. 29; Duff, pp. 78-81. 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 a body at rest shall remain at rest, or a body in motion remain in motion with con- stant linear and angular velocity, the vector sum or resultant of all 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 sec- ond of these conditions, sometimes called the Principle of Mo- ments, requires that the sum of the moments of all the forces about any and every point selected in the plane as a center of moments shall be zero. For instance, let us suppose a *' - I k (2M 2^ 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. 10 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 or the glass plate (dependent upon which form of apparatus is used) is suspended and from the other end a number of masses just sufficient to balance the same and the friction of the pulley. Note the precautions given in Exp. 10. Special care should be taken to insure as little friction as possible. (a) Adjust the apparatus so that a good trace may be ob- tained and so that a slight tap will cause the fork or the glass plate to descend without acceleration. The forces, including friction, are then just balanced. Cover the plate with a thin coat of corn-starch and alcohol. Take care to have the sty- lus exert the same pressure against the plate throughout the experiment. (b) Remove a mass m 1 from the balancing masses. Note the total mass (2M m t ) 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 (2M w 2 ). (d) Repeat again with a third mass removed. (e) Measure the traces as explained in Exp. 10, using five vibrations of the fork as the interval of time. Calculate 36 SURFACE TENSION BY JOLLY^S BALANCE. [l$ the accelerations a^ a 2 , a 3 , corresponding to (b), (c), (d) above. Then make two tests of equation (4) by substituting in the same. Calculate the percentage difference between the two sides of the equation in each case. If the masses removed in (&), (c), (d) had simply been transferred from one side of the pulley to the other, what changes would be required in substituting in equation (4) ? 15. SURFACE TENSION BY JOLLY'S BALANCE. References. Duff, p. 146; Millikan, p. 181. The purpose of this experiment is to obtain a direct measure of the surface tension of a liquid by balancing it against the tension in a stretched spring. A wire rectangle is hung from the spring of a Jolly's balance and allowed to dip in a soap solu- tion which forms a film across the rectangle. When equili- brium 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 total force due to surface tension. If T is the value of the surface tension per centimeter width of the film, / the width of the rectangle along the surface of the liquid, and F the force exerted by the spring, then it follows, because the forces are in equilibrium, that F = 2/T Knowing F and /, the value of T can thus be found. The Jolly's balance used is one of the two forms used in Exp. 2. Ask for directions, if its operation is not already un- derstood. Wire rectangles of different sizes and a wide beaker are provided. 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 be- ing used and before changing to another liquid. Do not touch l6] CAPILLARITY. RISK OT? LIQUIDS IN TUBES. 37 with the fingers 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. The rectangle should be immersed to the same depth in the two cases, so as to elim- inate the effect of the buoyancy of the liquid. Take three sets of readings. Note whether the pull of the film depends upon the area of it formed in the rectangle. Repeat these measurements, using rectangles 4 cm. and 6 cm. wide. (b) Calibrate the balance by observing the extension pro- duced by known standard masses. (c) Use the rectangle, 4 cm. wide, cleaning it and the beaker thoroughly, and repeat (a) with water fresh from the tap. As a film of no appreciable height will form with pure water, take the reading of the balance without the film 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. (d) Repeat (c), using water at 5OC. 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 ten- sion, T, in dynes per cm., for the liquids used in (a), (c), (d), and (e), comparing the values obtained and pointing out how the surface tension is affected by the temperature. 16. CAPILLARITY. RISE OF LIQUIDS IN TUBES. Reference. Duff, p. 149. In the present experiment the values of the surface tension of water and of alcohol are to be measured by observing the 38 CAPILLARITY. RISE OF LIQUIDS IN TUBES. [l6 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 upward at all points around the circumference of the tube. The total vertical component of this force is 2-n r T cos a, where r is the radius of the tube, T the surface tension in dynes per cm., and a is the angle of contact between the liquid and the tube. If the tube is of small bore, the liquid will rise inside the tube, equilibrium be- ing established when the weight of the liquid within the tube above the level of the liquid outside equals the vertical force upward due to surface tension. If d is the density of the li- quid, h its height in the capillary tube above the surface level, and g the acceleration due to gravity, it follows that 7rr 2 hdg = 27rrT COS a. From this equation the value of T, the surface tension in dynes per cm., 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 should be cleaned with caustic potash solution, rinsed with tap water and then with the liquid to be experi- mented with (in this case, water). '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 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-surface 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. 17] RISE OF LIQUIDS BETWEEN PLATES. 39 (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. If the tube is uniform in bore, its diameter can be found either with the micrometer microscope, or by means of a thread of mercury drawn into the tube. In case the latter method is used, the length and mass of the thread and the density of mercury are all the data needed for calculating the diameter. Calculate the surface tension of water, and compare this value with that found in Kxp. 15. For pure water and ordinary glass the angle of con- tact is approximately zero. (c) In the same way find the surface tension of alcohol. For the angle of contact in this case see the Tables. Would the water or alcohol rise as high in the tubes if the experiment were performed in a vacuum? Explain. If a thread of water were placed in a horizontal, conical- shaped tube, in which direction along the tube would it move? Explain. If mercury instead of water were used, what would happen, and why? 17. RISE OF LIQUIDS BETWEEN PLATES. Reference. Hastings and Beach, p. 146. 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. The two plates of glass, which are in touch with each other on one side, are separated on the other side by a thin piece of brass placed between the opposite edges of the plates. The plates are clamped together and placed upright in a shallow dish of li- quid. If the liquid wets the plates, it will rise in the wedge- shaped space, forming a smooth curve which extends from the surface of the liquid in the dish, on the side where the 4O RISE OF LIQUIDS BETWEEN PLATES. [17 plates are farthest apart, to a point high above this level, on the other side where the plates are in touch with each other. The general effect is similar to that obtained by a row of small tubes of gradually decreasing bore. We may consider that at some point along the curve a thin vertical slice or rectangular prism of the liquid is taken. Let d, the distance between the plates at the point chosen, be the width of the prism; .r (very small), the thickness of the prism in a direction parallel to the plates and to the surface of the liquid in the dish ; and // the height of the prism above the surface of the liquid in the dish. The surface tension which acts upon this prism evidently has a vertical component upward equal to 2 T x, where T is the value of the surface tension in dynes per cm. This force must equal the weight of the prism of liquid which is h x d D g, where D is the density of the liquid and g the acceleration due to gravity. 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 on the paper the surface of the water between the plates, the surface of the water in the dish, the outline of the piece of metal, and the edge where the plates touch each other. (If the water-curve between the plates is not a smooth one, it will be necessary to raise and lower the plates in the dish until the surfaces of the glass be- tween them is thoroughly wet.) Removing the sheet of pa- per, draw a line on the paper to show the position of the inner edge of the piece of brass. This line, as well as the line show- ing the position of the edge where the plates were in touch with each other, should be perpendicular to the line representing the surface of the water in the dish. -Select any point P on the curve representing the surface of the water between l8] VISCOSITY. FLOW OF LIQUIDS IN TUBES. 41 the plates. From this point draw a line perpendicu- lar to the line representing the surface of the water in the dish, and call its length h. Let x be an infinitesimal dis- tance through P at right angles to this last line. To determine the width d between the plates at P, proceed as follows: Draw a line through P parallel to the line representing the surface of the water in the dish and let the length along this line from P to the line showing where the plates were in touch with each other be /. Let the whole distance from the inner edge of the piece of brass to this same line be L. Measure the thickness d l of the piece of brass with a micrometer caliper. Then, at the point P, Derive this equation. From the values of d and h thus found, calculate the surface tension of water in dynes per cm. Repeat the measurements and calculation for one or two other points on the curve. (fr) Repeat (a), using alcohol instead of water, and find the surface tension of alcohol. 18. VISCOSITY. FLOW OF LIQUIDS IN TUBES. Reference. Duff, p. 137. 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 42 VISCOSITY. FLOW OF LIQUIDS IN TUBES. [l8 each of these cylindrical shells will be moving more slowly than in the shell immediately inside, and more rapidly than in the shell immediately outside. This relative motion of adjacent layers of the liquid is determined by the internal friction or viscosity of the liquid. Viscosity varies greatly with the kind of liquid used, this dependence upon the character of the li- quid being indicated by the coefficient of viscosity. If a liquid is very viscous, like syrup, its coefficient of viscosity 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 /, and the coefficient of vis- cosity c of the liquid. These quantities are connected by the relation To compare the coefficients of viscosity of two different li- quids, it is evident, if the above relation be accepted, that, for the same tube and equal times of flowing, the coefficients will be in inverse proportion to the volumes, or c l : c 2 == V 2 : V y . 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 reservojr 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 l8] VISCOSITY. FLOW OF LIQUIDS IN TUBES. 43 means of this arrangement a constant head of pressure may be obtained. The tube, which carries the liquid from the res- ervoir to the small-bore tube, is quite large, so that the fric- tional resistance which it offers to the flow will be negligible as compared to that offered by the small tube. This makes it reasonable to assume (as is done in the experiment) that the head of pressure is all employed against the frictional resis- tance offered by the small tube. (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 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. (&) 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 ? Employing the C. G. S. system of units, calculate the coefficient of vis- cosity of the water for the three cases, and take the average value. /| /| EFFLUX OF GASES. RELATIVE DENSITIES. [19 (c) With one of the tubes, use water at 5o-6oC. in the reservoir, and compare with previous results to determine 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 (&), noting the effect upon viscosity of different sorts of dissolved sub- stances. 19. EFFLUX OF GASES. RELATIVE DENSITIES. Reference. Duff, p. 166. The object of this experiment is to find the relative densities of certain gases from the observation of the relative times of efflux of equal volumes of these gases through a small aper- ture. 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 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 efflux 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 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 2O] ABSOLUTE CALIBRATION OF A THERMOMETER. 45 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 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. ABSOLUTE CALIBRATION OF A THERMOM- ETER. References. Watson's Practical Physics, p. 162; Edser, p. 23. . The object of this experiment is to plot a curve from which the true temperature may be obtained corresponding to each 46 ABSOLUTE CALIBRATION OF A THERMOMETER. [2O scale-reading of a given mercurial thermometer. Such a curve is called the calibration curve of the thermometer. The process of obtaining it is absolute since it does not involve comparison with a standard thermometer. (a) Correction near the Lower Fixed Point. Put the thermometer through the cork in a test-tube, having rilled 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. Let us suppose that the mean of these readings is + 0.2 C. Since the true temperature of freezing water is oC., the correction corres- ponding to the given scale-reading of the thermometer is 0.2, for this when added to the reading gives the true temperature. (b) Correction Near the Upper Fixed Point. Place the thermometer through the cork in the tube at the top of the boiler, 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 ther- mometer when it becomes steady. Allow the boiler to cool slightly, and repeat, making three readings in all. If the in- strument be provided with a water-manometer, take the man- ometer-reading simultaneously with the temperature-reading. Read the barometer and determine the pressure of the steam, and find from the Tables the true boiling-point temperature for this pressure. Let us suppose that the mean of the readings of the thermometer is 99.iC., while the true temperature is 99.8 C. Then the correction corresponding to this scale-read- ing of the thermometer is +0.7, for this when added to the reading gives the true temperature. (c) Let the thermometer cool slowly to about the tempera- ture of the room, and repeat (a). If the freezing point ob- 2O] ABSOLUTE CALIBRATION OF A THERMOMETER. 47 served now is different from that observed in (a), use the mean of the two values in the calibration that follows. As- suming the temperature of freezing water to be oC., write the corrections of the thermometer for the scale-readings observed in (a) and (&). Record these two corrections by points on coordinate paper, having as abscissae the scale-readings of the given thermometer from o to 110, and as ordinates the corresponding corrections 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 scale-reading 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 diameter 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. Ask for as- sistance, if necessary. Place the lower end of the thread, ap- proximately 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 position of the upper end. Repeat with the lower end at the successive points 20, 30, 40, etc., up to 90 ; then come down again with up- per end at 100, 90, 80, etc., reading the position of the lower end each time. (e) Record the observations and Calculations/ in tabular form in six columns as follows : 1 i ) The reading of the lower end of the thread. (2) The corresponding reading of the upper end. (3) The length of the thread in each position. (4) The mean length / for each interval. 48 ABSOLUTE CALIBRATION OF A THERMOMETER. [2O By the mean length for each interval is meant the mean of the reading 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 mean lengths throughout the whole range and record this as the mean length L of the thread for the whole range. (5) The correction, L /, for the length of each interval, that is, the difference between the mean length for all intervals and the observed length for 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 read- ing. 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 (e, 6). This can best be done by plotting. On the plat made in (c), plot points whose abscissae are 10, 20, 30, etc., and whose corresponding ordinates are found by measuring from the slanting line, already drawn, distances equal to the corres- ponding corrections found in (e, 6) measuring up or down from this line according as the corrections are plus or minus. The smooth curve, which should now be drawn through these plotted points, is the calibration curve of the thermometer. What are the temperatures corresponding to readings of o, 2 5> 5> 75 and 100 on the given thermometer? 21 ] RELATIVE CALIBRATION OF A THERMOMETER. 49 21. RELATIVE CALIBRATION OF A THERMOM- ETER. 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, hence the name "relative calibration." If the calibration curve of the standard thermometer can be relied upon, all ir- regularities of any other thermometer can be corrected. The thermometer to be calibrated in this experiment is a 50 thermometer reading to tenths of a degree. Tie the ther- mometer to the standard thermometer with soft cotton twine, winding it between the stems so as to separate them slightly. Put the bulbs nearly opposite each other ; and see that cor- responding 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 catch- ing 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 hundredths of a degree, at intervals of 2 or 3 from about 5 to 45. Keep the water well stirred, and keep the temperature fairly cori- 5 then V" = V [i -f ft (t 2 fj]. Dividing both members by w, and substituting for m/V\ and m/V" their equivalents some of it would be withdrawn with the thermometer and relatively large changes in mass introduced. If the thermometer were placed in the alloy and left there, there would be great danger of its breaking on the solidification of the metal. After the melting point of the alloy has been found, bring the water to the boiling point, taking care that no water gets inside the crucible. Remove the crucible, noting the time, and quickly wipe the outside ; then quickly an I carefully lower it into the calorimeter, right side up, with its contained alloy, letting the water run into the crucible and thus more quickly cool the alloy. Only a few seconds should elapse between the re- moval of the alloy from the hot water-bath and its immer- sion in the calorimeter. Stir the mixture continuously, not- ing the time and temperature when the mixture becomes uniform and starts to cool, and then again 5, 10 and 15 min- utes later. By plotting times as abscissae ani j temperatures as ordin- ates, find the temperature which the mixture should have if its temperature could be made uniform the instant the alloy enters it. The effect of radiation is thus accounted for. A little consideration will make it evident that the cooling curve 78 HEAT OF VAPORIZATION AT BOILING POINT. [3$ in the plot, prolonged, backward to intersection with the axis of ordinates will give the correct temperature. The specific heats of the solid and liquid alloy will be given in the Tables, or if time permits they may be found by the method of mixtures. The specific heat of nickel and copper may be found in the 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 point of the alloy necessary? Discuss the relative accuracy with which the different masses used must be determined in order that the precision of measurement of the result may be 2 per cent. Point out the principal sources of error in the experi- ment. 35. HEAT OF VAPORIZATION AT BOILING POINT. References. Edser, p. 152; Watson's Practical Physics, p. 237. In this experiment Kahlenberg's modification of Berthelot'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, journal of Physical Chemistry, 1901, Vol. 5, p. 215. 35] HEAT OF VAPORIZATION AT BOILING POINT. 79 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 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 wacer-equivalent of the calor- imeter and contents, including the stirrer, thermometer, empty worm, and water. The necessary specific heats may be ob- tained from the Tables. Calculate the amount of heat gained by the calorimeter. Knowing the mass of the vapor con- densed, the change in temperature of the liquid, and the spe- 80 HEAT OF VAPORIZATION AT ROOM TEMPERATURE. [ 36 cific heat of the liquid (see the Tables for the specific heat), calculate the heat transferred to the calorimeter, and deter- mine the heat of vaporization of the liquid at its boiling point 36. HEAT OF VAPORIZATION AT ROOM TEM- PERATURE. Reference. Duff, p. 233. 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 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 in- creasing 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 calori- meter, 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 alcohol. Place the cover on the calorimeter, with 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 temperature of the liquid 2 or 3 above the room temperature. Pass the dry air gently through the liquid, al- lowing ample room for the vapor-charged air to escape, un- til the temperature is as much below room temperature as the initial temperature was above it. If the air is forced too rapidly through the liquid, not all of the liquid which is car- ried away will be vaporized. Weigh the calorimeter and re- maining liquid. A 50 thermometer graduated in tenths of 37] FREEZING POINT OF SOLUTIONS. Ri a degree should be used. Wet the thermometer, with the li- quid used, about as high as the depth to which it will be placed in the liquid in the calorimeter, 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 (a) two or three times. When finished, empty and dry the calorimeter. If a liquid other than water was used, it 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 (fr), taking the mean as the final value. It will be necessary to assume that the heat, used up in vapor- izing the liquid, all came from the calorimeter and its contents. The mean of the initial and final amounts of liquid in the calor- imeter should be taken as the amount of liquid cooled. (d) Point out the chief sources of error. Give a reason why the value of the heat of vaporization of a liquid increases, when the temperature, at which the vapori- zation of that "liquid occurs, is lowered. Evaporation takes place from dry ice at temperatures be- low the freezing point. This change from solid directly to vapor is called sublimation. By what amount would you ex- pect the heat of sublimation of ice at oC. to exceed the heat of vaporization of water at oC.? 37. FREEZING POINT OF SOLUTIONS. References. Watson, p. 268; Watson's Practical Physics, p. 258; Edser, p. 167. The object of this experiment is (i) to observe the lowering of the freezing point of water caused by dissolving table salt and sugar in it to form solutions of different concentrations. 82 FREEZING POINT OF SOLUTIONS. [37 and (2) to determine the molecular weights of the salt and the sugar by means of this lowering. (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. Choose such amounts of these solutions in the three cases as will con- tain the same mass of the solvent, for example, 100 gm. of the 4 per cent, 104.3 S m - f the 8 P er cent > an d 109.1 gm. of the 12 per cent solution. (b) Repeat (a) with aqueous solutions of sugar of 6, 12. and 1 8 per cent concentration. (c) Tabulate the results of (a) and of (b), and for each of the six cases calculate the lowering, per gram of dissolved substance, of the freezing point of a .given mass oi water. What relation seems to hold between the change of freezing point of a given mass of water and the mass of dissolved sub- stance ? Note the difference of freezing points for 12 per cent solu- tions of table salt and sugar. (d) Calculate the molecular weights (M) of table salt and sugar from the relation M = Ks/St, where s is the number of grams of dissolved substance, 5 is the number of grams of the solvent, t is the depression of the freezing point, and K is a constant, whose value is 1850 for aqueous solutions. Water in dilute solutions is thought to have the power of breaking up the molecules of some dissolved substances into ions, each ion having the same effect in lowering the freezing point as a molecule has. The result of this is that the observed lowering of the freezing point of water is three times as great in the case of some dissolved substances and twice as great in 38] HEAT OF SOLUTION. 83 the case of others as the value calculated by the formula. What ionizing effect, if any, has water on the table salt and sugar ? What relative lowering of freezing point of equal masses of water would you expect (a) if equal numbers of molecules of table salt and sugar were brought into solution, (&) if equal masses ? 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) 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. (b) Set up the calorimeter, with the jacket filled with 84 HEAT OF SOLUTION. [38 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. (c) Make a similar trial with a second portion of salt, having the cup at about 4OC. Make sure that there is the proper difference between cup and jacket at the time the salt is poured in. To determine the water-equivalent of the calor- imeter cup and stirrer, it will be necessary to know their masses and the specific heats of the metals of which they are made. The water-equivalent of the thermometer may be calculated from the number of cc. which it displaces when immersed in a graduate to the proper depth. (d) Call the specific heat of the salt x, and its heat of so- lution in water y. Write for each of the cases (b) and (c) an equation involving the following quantities : 1. Heat lost by water in cup. 2. Heat lost by cup, stirrer, and thermometer. 3. Heat gained or lost by salt in changing temperature. 4. Heat absorbed during solution of salt. It will be well to assume that the salt initially is at the room temperature in the two cases. Solve the two equations for .1- and y. Why should the value of the heat of solution as obtained by this method be proportionately more accurate than that for the specific heat ? Caution : Do not leave the solution standing in the cup. Wash it out as soon as possible. 39] HEAT OF NEUTRALIZATION. 85 39. HEAT OF NEUTRALIZATION. When an aqueous solution of a strong acid is poured into an aqueous solution of a strong alkali until a neutral mixture is formed, the essential chemical reaction which occurs is the formation of water. For instance, if aqueous solutions of hy- drochloric acid and sodium hydroxide are made to form a neutral mixture, although the mixture is a solution of sodium chloride (table salt), the only chemical reaction occurring is the formation of water. The heat generated is called the heat of neutralization. The object of the present experiment is to determine the heat of neutralization corresponding to the formation of a gram-molecular weight of water. In the case just mentioned, this will occur when 1000 gm. each of normal solutions of the acid and the alkali are mixed. A 0.5 normal solution of each of the above compounds is furnished. By a normal solution is meant one which, in 1000 cubic centimeters of the solution, contains a mass of the com- pound (which is to enter into the new combination) equal in grams to its molecular weight. Thus the normal solution of sodium hydroxide is a solution which contains, in 1000 cc. of the solution, 40 gm. (23+16+1) of sodium hydroxide, or 23 gm. of sodium, 16 gm. of oxygen, and I gm. of hydro- gen. The 0.5 normal solution contains one-half as much in the same volume of solution. It is evident that if equal volumes 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. 86 COEFFICIENT OF EXPANSION OF AIR. [40 (a) Measure out 100 cc. of the sodium hydroxide solution in the cup, and the same volume of the hydrochloric acid solu- tion in the beaker. Wet the inside of the beaker with the acid solution before pouring the measured amount into it. This is to compensate for the liquid which remains in the beaker when later it is emptied. 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. It may safely be 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 cc. of normal so- lution had been used in each case. Will this cause the forma- tion of one gram-molecular weight of water? (b) Repeat the work, if there is time, with solutions of potassium hydroxide and sulphuric acid, and compare the re- sults with that in (a). 40. COEFFICIENT OF EXPANSION OF AIR AT CONSTANT PRESSURE BY FLASK METHOD. The purpose of this experiment is to determine the coefficient of expansion of air by observing the contraction (inside of a glass flask or bulb) of a given mass of air when its tempera- ture is lowered a measured amount. 40] COEFFICIENT OF EXPANSION OF AIR. 87 A glass bulb with a long tubular neck closed by a stop-cock is suspended in a steam bath to bring the enclosed air to the temperature of boiling water. The stop-cock is then closed, imprisoning in the bulb a given mass of air at atmospheric pressure. The bulb is then inverted and plunged into a bath of ice-water, the stop-cock is opened, and the enclosed air is brought again to atmospheric pressure. The apparent volume of the given mass of air when contracted is found by determin- ing the difference between the masses of ice-water filling the whole bulb and that part of the bulb not occupied by the con- tracted air. Let V 2 and V represent the volumes of the enclosed air at the temperature t 2 of the steam and the temperature / t of the ice-water, respectively. Let V 2 be the volume of the whole bulb as determined by filling with ice-water. If a is the coefficient of expansion of air at constant pressure and y the coefficient of cubical expansion of glass, we will have, by the definitions of a and y, (1) V,= V, [I +a (*,--.*.) ], (2) V,=- V' t [I + y (V--*,) ]. Solving for a, we get *V- v, _^ r ~^iU-'i) vj' If the temperature t l of the ice- water is not oC., the value of a obtained from (3) will not be referred to standard initial temperature and the result cannot, therefore, be compared with the value given in the Tables. (a) Thoroughly dry the bulb by rinsing out ten times or more with dry air. This is done by alternately exhausting by means of the jet-pump and admitting dry air from the room. Then hang it in the boiler with the bulb down and with the 88 COEFFICIENT OF EXPANSION OF AIR. [40 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 can not enter. Boil the water, causing the steam to pass around the bulb until the air inside it is at the temperature of the steam. Then introduce the stop- cock thinly coated with grease, close the cock, remove from the boiler, and allow to cool. (Care is necessary here to keep the bulb air-tight and at the same time to keep the stop-cock from breaking when it is cooled.) Next place it under the surface of the ice-water, and open the stop-cock, under water, allowing the water to enter but not the air to escape. After allowing time enough for the bulb to come to the temperature of the ice-water, raise the bulb so that the level of the water inside is the same as that without, thus assuring the same pressure in the enclosed air as before. Close the stop-cock, remove and dry, and then carefully weigh. In order to obtain the volume, the bulb 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 bulb great care should be taken not to break the stop-cock by the heat. These weighings will enable you to determine V 2 , the inside volume of the whole bulb at oC., and V r , the volume of the air in the bulb when under atmos- pheric pressure and at oC. From the results of the above measurements and the coefficient of expansion of glass (see the Tables) find the coefficient of expansion of air. (b) If time permits, repeat with some available gas other than air, and compare the result with that of air. 41 ] CONSTANT-PRESSURE AIR-THERMOMETER. 89 41. EXPANSION OF AIR. CONSTANT-PRESSURE AIR-THERMOMETER. References. Edser, p. 108; Duff, p. 187. The object of this experiment is (i) to determine the mean coefficient of expansion, between oC. and iooC., of Air at constant pressure, by means of the constant-pressure air- thermometer; and (2) to test the gas-lazvs. In the form of air-thermometer used dry air is contained in a glass tube graduated in cc. and closed at one end. The graduated tube is connected to an open glass tube by rubber tubing, the whole forming a "U" containing mercury. The pressure on the enclosed air can be regulated to any desired constant value by raising or lowering the open glass tube. Surrounding the graduated tube con- taining the air is a vessel, covered by an asbestos jacket, in which a water-bath 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. Coefficient of Expansion. (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, adjust the pressure so that it is 10 cm. less than atmospheric pressure, and read the volume. Fill the vessel with water at ioC., adjust the pressure to the same value as before, 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 minutes or more) for the enclosed air to come to the same temperature as the bath, and each time adjusting the pressure so that it is 10 cm. less than at- mospheric pressure. The mercury meniscus on the closed- tube side should always be as close to the bottom of the jacket 90 CONSTANT-PRESSURE AIR-THERMOMETER. [4! as will just permit of reading the meniscus. This is done so that the enclosed air will not be outside the water-bath. Empty the vessel, place a cover over the top, and pass steam through the vessel, in at the bottom and out at the top. Use two Bunsen burners, if necessary, to obtain an abundant flow of steam. After waiting five or ten minutes for the enclosed air to reach the temperature of the steam, take another read- ing of the volume, the pressure conditions being the same as before. (b) Make another and similar series of observations at a pressure 10 or 20 cm. above atmospheric pressure. (c) 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 TOOC., 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 coeffi- cient of expansion of air. Gas Laws Tested. (d) From the two curves in (e) find out by proportion if the volume of a gas varies directly as the absolute temperature when the mass and pressure of the gas remain constant. By taking some particular volume and noting the tempera- ture and pressure -on each curve corresponding to that volume find out by proportion if the pressure of a gas varies directly as the absolute temperature when the mass and volume of the gas remain constant. Similarly, by taking some particular temperature on the two curves and noting the volume and pressure on each curve cor- responding to that temperature, find out if the volume of a gas varies inversely as its pressure when the mass and tempera- ture of the gas remain constant. 42] CONSTANT-VOLUME AIR-THERMOMETER. C)t 42. CONSTANT-VOLUME AIR-THERMOMETER. References. Duff, p. 186; Millikan, p. 125. The object of this experiment is to study the law of varia- tion of the pressure of a given mass of enclosed air whose volume is kept constant while its temperature is changed. The air is enclosed in a glass bulb mounted on the frame used in Exps. 4 and 5. The frame is placed near a table so that the bulb may be surrounded by a water-bath, by shaved ice, or by a steam-bath, the table and an iron stand being made use of to support each bath in turn. A thermometer is placed in the bath to give its temperature. The pressure on the enclosed gas is regulated by raising or lowering the open tube, as is done in the experiment on Boyle's law. The value of this pressure may be determined from the barometer-read- ing and the difference in the levels of the mercury on the two sides of the frame. Each time before taking the readings, the volume of the air in the bulb is made the same by bring- ing the mercury meniscus to the level of the wire point inside the glass tube attached to the bulb. Caution : The mercury on the bulb side should always be lowered some distance before changing to a lower temperature. Be especially careful to do this before removing the steam- bath 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 changing the tempera- ture, but wait until the meniscus set at the wire-point remains stationary. (a) Without any bath in the reservoir, while all is at the room-temperature, bring the mercury to the wire point ''anil determine the difference in level of the mercury columns. Re- cord the room-temperature, and the barometer-reading. (b) After having lowered the mercury on the bulb side, surround the bulb with shaved ice, and then determine the Q2 CONSTANT- VOLUME AIR-THERMOMETER. [42 pressure with the meniscus at the wire point. The tempera ture may be taken as oC. Melt the ice with warm water, and then make a series of de- terminations of the pressure when the water in the vessel is successively at a temperature of 10, 20, 30, 45, 60, and 8oC. ( approximately )X Remove the water-bath, substitute a steam-bath in its place, and make another determination. The temperature of the steam-bath may be found by determining the boiling point of water from the known atmospheric pressure (see Tables). 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 which will best represent the average position of the points of the plot. Calculate the mean increase of pressure per degree increase in temperature from oC. to iooC., and divide the result by the pressure at oC., using values taken from the plot. This is the temperature coefficient (ft) of pressure of a gas. U& Write it(-is)a decimal and find its reciprocal. The negative of this represents what point on the absolute scale of tempera- tures ? (d) Write an equation connecting P , the pressure at o ; P, the pressure at t ; t ; and (3. Using this equation and the pressure obtained in (a), cal- culate the temperature of the room, thus using the apparatus as a thermometer. Compare the result with the room tem- perature 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] VAPOR-PRESSURE AND VOLUME. 93 43. VAPOR-PRESSURE AND VOLUME. References. Duff, p. 227; Millikan, p. 152. The purpose of this experiment is to study the relation be- tween the vapor-pressure of a saturated vapor and the vol- ume of the vapor, when its temperature is kept constant. An ordinary barometer tube, about a meter long and filled with mercury, is inverted in a cistern of mercury. A small amount of the liquid, whose vapor-pressure- is to be studied, is introduced into the tube, rising to the top of the tube and vaporizing. By raising or lowering the tube in the cistern, the volume of the vapor can be changed. The corresponding vapor-pressure is found by determining the difference between the barometer-reading and the height of the mercury column above the mercury in the cistern. Experience has shown that the saturated vapor above the surface of a liquid exerts a pressure which depends only on the nature and temperature of the liquid. It should follow from this, if the temperature is kept constant, that the vapor- pressure of any given liquid is independent of the volume of the vapor, as long as the vapor remains saturated. (a) Unsaturated and Saturated Vapors. Fill a barometer tube with clean mercury to within I cm. of the end, and plac- ing the thumb over the end, invert the tube slowly so as to make the large air-bubble pass up along the tube. If this is done a number of times and along different sides of the tube, most of the air-bubbles will be washed out of the tube. Now fill the tube brimming full and invert in the cistern, tak- ing care not to allow any air to enter. Clamp the tube in a vertical position and measure the height of the mercury col- umn above the surface of the mercury in the cistern. Com- pare with the reading of the laboratory barometer, and if the difference is greater than i cm., remove the tube and fill again more carefully. 94 VAPOR-PRESSURE AND VOLUME. [43 With a medicine dropper introduce a little ether into the tube, taking care not to force any air in with it. Observe if the ether all vaporizes or not; note the height of the mercury column ; and continue to add more and more ether until the mercury stops falling. Finally record the height of the mer- cury column and the length of the vapor-column above it. As- suming that the temperature has remained equal to room-tem- perature, what can you conclude about the relative pressures of saturated and unsaturated ether-vapor at the same tempera- ture? (b) Vapor-Pressure and Volume. Unclamp the tube and gradually lower it in the cistern, 20 cm. at a time. After waiting a minute each time for equilibrium to be established between the vapor and the liquid, read the height of the mer- cury column and the length of the vapor column. From the height of the mercury column and the barometer-reading the vapor-pressure can be found, while the length of the vapor - column may be taken to represent the volume of the vapor. Continue until all of the vapor is condensed. The gas which remains is air, and if present in considerable amount, its pressure will constitute an appreciable part of the measured pressure, especially in the later measurements. (c) Mixture of Vapor and Gas. With the medicine drop- per force more air into the tube and repeat the measurements of (b). Remove the barometer tube from the cistern, fill again with clean mercury, introduce enough air to cause the mercury col- umn to drop 20 cm. or so, and repeat the measurements of (&\ What effect does the presence of a gas in the vapor have on the results? (e) Vapor-Pressure and Nature of the Liquid. Repeat (a) with alcohol, and if time permits, with water also. Does the vapor-pressure depend upon the nature of the liquid? (f) Plot the results of (a), (b), (c), and (d) for ether on a 44] VAPOR-PRESSURE AND TEMPERATURE. 95 -is single sheet of coordinate paper, with pressures as ordinates and volumes as abscissae. From the results and the curves determine (i) how the pressure of a vapor at a given temperature depends upon the degree of saturation, (2) how the pressure of a saturated vapor at a given temperature depends upon the volume of the vapor, and (3) whether the vapor pressure depends upon the nature of the liquid or not. 44. VAPOR-PRESSURE AND TEMPERATURE. References. Duff, p. 227; Millikan, p. 152. The object of this experiment is to study the relation be- tween the temperature and the pressure of saturated water- vapor. The method employed is that referred to in Exp. 22 as the "static" method of determining the boiling point of a liquid at different pressures. Two barometer tubes, filled with mercury, are inverted and mounted side by side in a vessel of mercury. One of the tubes contains, above the mercury, water-vapor with an excess of water present, while the other tube is left to be used as a barometer. By means of a water- bath surrounding the upper half of the tubes, the temperature of the water-vapor can be brought to any desired point. The bath is connected to a heater and the change in temperature is brought about by circulation. The pressure of the saturated water-vapor at any temperature will be the difference between the heights of the mercury columns in the two tubes. At each temperature the pressure of a saturated vapor of a given liquid has a definite value which depends on the tem- perature and the nature of the liquid, but is independent of the volume of the vapor. When the temperature is raised, not only is the vapor heated and the pressure raised, but more li- quid is vaporized, so there are two influences tending to in- crease the pressure of the vapor. The purpose of the present 96 VAPOR-PRESSURE AND TEMPERATURE. [44 experiment is to plot the curve which shows how rapidly the vapor-pressure increases as the temperature is raised, in the case of water-vapor. (a) Read the heights of the mercury columns in the two tubes for ten different temperatures between room-temperature and 8oC. (approximately). To raise the temperature about 5 or i o at a time, heat the boiler only for two or three min- utes, then remove the burner, and stir the water-bath until a uniform temperature prevails. By this time the water-vapor inside the tube will have reached the temperature of the bath. In taking the temperature-readings hold the bulb of the ther- mometer slightly below the center of the space filled with water-vapor. The mercury-equivalent of the column of water above the mercury in the tube containing the vapor should be taken into account in estimating the pressure. Always wait until conditions have become steady before tak- ing readings at a new temperature. (b) While the temperature is falling by radiation, take as many readings of both mercury columns (at intervals of about 5) as time will allow. (c) Plot a curve from the results of (a) and (b), with the pressures of the saturated water-vapor as ordinates and the temperatures as abscissae. Draw the curve so that it will rep- resent the average positions of all the plotted points. (d) Extend the curve back to intersect with the pressure ordinate corresponding to oC. Assuming that, instead of water-vapor, you are given a perfect gas whose pressure at oC, is the same as that of the saturated water- vapor, calculate what the pressure of the gas would be at 25, 50, 75, and iooC., its mass and volume being kept constant. For this purpose it will be convenient to use the law expressing the relation between the pressure and the absolute temperature of a given mass of a perfect gas kept at constant volume. Plot the results in a curve, and compare with the curve obtained in (<:) 45] HYGROMETRY. 97 Does the pressure of saturated water-vapor increase with the temperature more or less rapidly than does the pressure of a gas kept at constant volume? Would the results be different if the volume of the satu- rated vapor were kept constant? (See Exp. 43.) Determine, from the curve in (d), the boiling point of water at a pressure of 50 cm. 45. HYGROMETRY. References. Duff, p. 240; Millikan, p. 164; 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, and is usually expressed in grams per cubic meter. The relative humidity is the ratio of the amount of water-vapor actually present in the air to the amount required to saturate it at the same temperature^ the latter quantity being the max- imum amount of water-vapor that can be held in suspension at that temperature. The relative humidity is therefore equal to d/D, where D is the maximum density of the water-vapor at the given temperature. 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 pro- portional 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, relative humidity is equal to p/P. 98 HYGROMETRY. [45 (I) Regnaulfs Hygrometer. (a) Partially fill one of the hygrometer tubes with ether, and place a thermometer in the liquid. Force a current of air through the ether with a bicycle pump. The rapid evapor- ation of the liquid causes the temperature to fall. When the tube and the air immediately above 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 tempera- ture at which the dew begins to form. Allow the tube to become warm and record the temperature at which the dew disappears. Take the mean of these two as the dew-point. Make three such determinations of the dew-point. (b) From the 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 D, the number of grams of saturated water-vapor in a cubic meter of air at the room-temperature (see the Tables). II. Wet- and Dry-bulb Hygrometer, or Auguste'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. Swing the hygrometer back and forth in the air so as to increase the circulation of air about the wet bulb. After the two thermometers come to constant temperatures, record the temperature t of the dry bulb, and the tempera- ture ^ of the wet bulb. Read the barometer. The following empirical formula may then be used: /> = />! 0.0008 b (f *!>, where p is the pressure of water- vapor present in the atmos- phere and the value of which is to be found; p^ the pressure of saturated vapor at the (temperature of the wet-bulb (ob- 46] DENSITY OF THE AIR BY THE BARODEIK. 99 tained from the Tables) ; and b is the barometric pressure, all being expressed in millimeters of mercury. Find the pres- sure P of saturated water-vapor at the room-temperature from the Tables, and calculate the relative humidity. Find then the absolute humidity as in (fr). From the Tables and the readings of the wet- and dry-bulb hygrometer^ find the dew-point. Compare the values obtained in I and II for the humidity and the dew-point. 46. DENSITY OF THE AIR BY THE BARODEIK. The barodeik is an ordinary balance, having a hermetically 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. I. To find the difference between the barodeik reading and the true density of the air. (a) Set and read the barometer with great care. Read the wet- and dry-bulb hygrometer. From the Tables calculate the dew-point and also the pressure of the 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 the water- vapor in the air would be the maximum pressure. (b) 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 IOO DENSITY OF THE AIR BY THE BARODEIK. [46 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 Jf 1 = 0.001293 I+g , ?6 , where a is the coefficient of expansion of a gas. The mass of water-vapor in the same volume is 293 IT* 7 -6- 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 in- strument. (d) Record the difference between the reading thus ob- tained and the true density found in (b) ; prefix 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 to the right of the center of the scale, 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 side, 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, 47] COEFFICIENT OF FRICTION. 101 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. (d) 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 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. 47. COEFFICIENT OF FRICTION. Reference. Duff, p. 95. 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- 102 COEFFICIENT OF FRICTION. [47 site to the force necessary just to produce motion (starting friction), or to keep the body moving at constant speed (mov- ing friction). If P is the force between the two surfaces and normal to them, and F is the force of friction, the ratio is called the coefficient of starting or moving friction, as the case may be, and is usually denoted by the Greek letter /x. By 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 i, it can be shown that the coefficient of friction is equal to tan i. (a) The coefficient of friction is to be found between blocks and the surface of a plane whose inclination can be varied. Take one of the blocks and weigh it. Determine the force of starting friction and also of moving friction 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) motion 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, 48] CONSERVATION OF MOMENTUM. 1 03 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. N (e) 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), (d), and (e), stating your conclusions. What do you con- clude from (/) ? From (g) ? Upon what does the friction between two surfaces depend? 48. CONSERVATION OF MOMENTUM. COEF- FICIENT OF RESTITUTION. : ' K Reference. Millikan, p. 58. In any system of bodies, which is not acted upon by outside forces 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 lim- ited to two and 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 drawn aside and then IO4 CONSERVATION OF MOMENTUM. [48 released. At the lowest point of its swing it strikes the ball B. Let m be the mass of A and t its velocity just as it strikes B. Its momentum then at this instant is m v u^ The ball B will at once start off with a velocity ^ 2 , say, and a mo- mentum m 2 z> 2 , if m 2 is its mass. The ball A may continue on with a diminished velocity, v^ ; or remain at rest, if it loses all of its momentum ; or it may rebound, in which case v 1 is negative. After impact the two balls will move away from each other with a relative velocity which is greater the greater their elas- ticity. 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, V -- V (1) * = ^- l , t U, where the velocities before impact, w x and u 2 , and the veloc- ities after impact, v and v z , are all in the same straight line. One or more of the velocities may be negative, or the par- ticular value of a velocity may be zero, as in the case just outlined for the two balls where 2 = o since the second ball was at rest before the impact. The value of e always lies be- tween zero and unity. For "perfectly elastic" bodies e = i, but for all actual bodies the angle of twist in radians, / the length of the rod, and r its radius, we may write 2MI 2MI (i) = ;, or n= -j . Trnr Trfir* In the above equation, n is constant for a given material and 50] HOOKERS LAW FOR TWISTING. IOO, is called the "coefficient of rigidity," or sometimes the "mod- ulus of torsion." The apparatus consists of two heavy table-clamps, one of them carrying a wheel about a half-foot in diameter. In the hub of the wheel is a socket in which the rod to be tested is centered and rigidly fastened. The other end of the rod is held in a similar socket mounted in the other clamp. A scale- pan, attached to the rim of the wheel, is for the load. Two smaller clamps, supporting graduated arcs, are placed in po- sition at desired points along the rod. A metal pointer, clamped to the rod under each of the arcs, provides a way for determining the relative twist in the rod between the two clamps. For testing, four rods are provided, two of the same diameter but different substance, and two of the same sub- stance but different diameter. (a) Set the rod of smaller diameter in place, clamped firmly at both ends to prevent slipping. Place the pointers exactly 20 cm. apart and adjust the graduated arcs in such a position with reference to the pointers as to avoid errors due to parallax in making the readings. Set both pointers at the zero marks. Place a 2OO-gm. mass in the scale-pan and record the positions of the two pointers. The twist of the rod between them is measured by the difference between the readings of the pointers. (b) Repeat (a), adding masses to the pan, preferably 200 gm. at a time up to about 1.5 kg., or until the "limit of elasticity" of the rod is reached. Whenever this limit is passed the rod will fail to untwist completely upon the removal of the masses in the pan. Record, in tabular form, the masses tised, the corresponding angle of twist, and the increase in the angle for each 2OO-gm. mass added. Measure the diameter of the wheel^ and the diameter of the rod, the latter with great care. IIO FRICTION BRAKE. POWER SUPPLIED BY A MOTOR. [51 (c) Repeat (a) and (&) with the pointers adjusted to include lengths of 40 cm. and 80 cm. of the rod. (d) Replace the rod by one of the same substance but dif- ferent diameter. Measure the diameter as in (b), taking ten or more readings but using only the five smallest of them in averaging for a mean value. Repeat the measurments of (c) for a length of this rod equal to 80 cm. (e) Repeat (d) with a rod of different substance, but hav- ing the same length and radius as that used in (d). (/) From your results show how the angle of twist varies with the twisting moment, with the length of the rod, and with its radius. Expressing all the quantities in equation (i) in the units of the absolute C. G. S. system, calculate the coefficient of rig- idity for each of the cases above. Note if its value is depen- dent only on the substance, or not. Point out how the data af- ford a verification of Hooke's law. If the radius of the wire be measured to an accuracy of o.oi mm., with what accuracy should the length be measured in order that the result may be affected to the same degree by both? 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 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 5i ] FRICTION BRAKE. POWER SUPPLIED BY A MOTOR. in 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 2-rrLnP, 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 the 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 of the six measurements. In what units is the power ex- pressed, if the force of the balances is in dynes and the lever 112 ABSORPTION AND RADIATION. \$2 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 f the angle of contact being the same in the two cc.ses. 52. ABSORPTION AND RADIATION. References. Duff, p. 253; 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 outside of the box. The bulb of one of the thermometers is 53] RATIO OF THE TWO SPECIFIC HEATS OF AIR. 113 bare, another is silvered, and the third is coated with lamp- black. All three thermometers should, initially, 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 thermom- eter-bulbs and condense 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 readings 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 thermom- eters 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 SPECIFIC HEATS OF AIR. Reference. Duff, p. 264. The object of this experiment is to obtain the value of the ratio y of the specific heat of air at constant pressure to it* specific heat at constant volume. The method employed is a modification of that used first by Clement 'and Desormes. A. quantity of the gas, compressed in a large flask, is momentarily 114 RATIO OF THE TWO SPECIFIC HEATS OF AIR. [53 put in communication with the atmosphere to allow its pres- sure to fall adiabatically to atmospheric pressure, its temper- ature simultaneously falling a little. The gas, when shut off again from the atmosphere, gradually warms up to its initial temperature, causing an appreciable rise in its pressure. Let />,be the pressure in the compressed gas at the start, v the volume of unit mass of the gas and f its temperature (the same as that of the room). Let p Q , v 2 , and t z be the corres- ing values of these quantities immediately after communica- tion between the compressed gas and the atmosphere is es- tablished. Then p 2 , v 2 , and x will be the values of these same quantities at the end^ if p 2 is the final pressure. The gas has now been in three conditions, as follows : Condition Pressure Vol. of igm. Temperature I. p, v, t, II. p v 2 t 2 III. p 2 v 2 t, The change from I to II was adiabatic, since no time was al- lowed for heat to pass in or out of the gas by conduction or ra- diation ; hence, by the law for adiabatic changes in a perfect gas, (0 s^A^z'/A. or O 2 /"i/=A/A- The change, from I to III was isothermal; hence, by Boyle's law, (2) "ViA = ^A or (v^/Vif = (A /A/ Hence (p l / p., / := (p l I p Q ); or, taking the logarithm and solving for Y (^ log A -log A logA-logA' The desired ratio may be obtained, experimentally, there- fore, by observing the values of the three pressures. The apparatus consists of a large carboy provided with a 53 RATIO OF THE: TWO SPECIFIC HEATS OF AIR. 115 large-bore stop-cock so that the enclosed space may at pleas- ure be opened to, or shut off from, the atmosphere. The pres- sure of the enclosed air is measured by an oil manometer, whilst air can be forced into or withdrawn through another inlet. To thoroughly dry the enclosed air, some strong sul- phuric acid is poured into the bottom of the carboy. (a) Close the stop-cock, and with a bicycle pump intro- duce enough air in the carboy to give a reasonably large dif- ference of pressure, as indicated by the manometer. Shut off connection between the carboy and the pump, and wait a few minutes until the temperature of the enclosed air is the same as that of the room, which will be when the manometer shows a steady ? constant pressure. Read the manometer and the bar- ometer. To get the value of the pressure-difference recorded by the manometer, it will be necessary to know the density of the oil used. This is posted on the apparatus. (b) Open the stop-cock wide and thus connect the en- closed air with the atmosphere. Leave open only for a second, then close again. Wait some time until the temperature of the enclosed air has risen again to that of the room, as indi- cated by a steady, constant difference in pressure ; then read the manometer. (c) Using the data in (a) and (b), determine from equa- tion (3) the value of y for air. (d) Repeat (a), (&), and (c) two or three times, and take the mean of the results. Obtain from the Tables the values of the two specific heats of air, calculate their ratio, and compare with the result just found by experiment. Point out the principal sources of error, stating how each affects the result. Explain why the specific heat of a gas at constant pressure should be greater than its specific heat at constant volume. n6 LOGARITHMS. 1 2 3 4 .5 6 7 8 9 123 456 789 10 ITT 12 1 13 114 15 |161 0000 0043 0086 0492 0864 1206 0128 0170 O2I2 0253 0294 0334 0374 Use Table on p. 118. 0414 0792 1139 0453 0828 H73 0531 0899 1239 0569 0934 1271 0607 0969 1303 0645 100^ 1335 0682 1038 1367 6719 1072 1399 0755 iro6 T430 4811 3710 3 6 10 15 19 23 14 17 ax 13 16 19 26-30 34 24 28 31 23 26 29 F 4 6l I 7 6l 2041 1492 1790 2068 1523 1818 2095 1553 1847 2122 1584 1875 2148 1614 1903 2175 164^ !93i 2201 1673 1959 2227 1703 1967 2253 1732 2014 2279 36 9 36 8 35 8 12 15 18 ix 14 17 ii 13 16 21 24 27 20 22 25 18 21 24 17 18 19 2304 2553 2 7 88 2330 2577 2810 2355 2601 2833 2380 2625 2856 2405 2648 2878 2430 2672 29OO 2455 2695 2923 2480 2718 2945 2504 2742 2967 2529 2765 2989 5 7 5 7 4 7 IO 12 15 9 ' *4 9 IX *3 17 20 22 16 19 21 16 18 20 20 |2T 22 23 124 25 126 3010 3032 3054 3075 3096 3H8 3139 3160 3181 3201 4 6 8 ii 13 15 17 19 3222 3424 3617 3243 3444 3636 3263 3464 3655 3284 3483 3674 3304 3502 3692 3324 3522 3711 3345 3541 3729 3365 3560 3747 3385 3579 3766 3404 3598 3784 4 6 4 6 4 6 8 10 12 8 10 12 7 9 11 14 16 18 14 15 i-7 3 '5 '7 3802 3979 4^0 3820 3997 4166 3838 4014 4183 3856 4031 4200 3874 4048 4216 3892 4065 4232 39<>9 4082 4249 3927 4099 4265 3945 4116 4281 3962 4133 4298 4 5 3 5 3 5 7 9 ii 7 9 10 7 8 10 12 14 16 12- 14 15 ix 13 15 27 28 29 43M 4472 4624 4330 ^87 4639 4346 4502 4654 4362 45 I8 4669 4378 4533 4683 4393 4548 4698 4409 4564 4713 4425 4579 4728 4440 4594 4742 4456 4609 4757 3 5 3 5 3 4 689 6 8 9 679 II 13 I 4 IX 12 14 IO 12 13 30 4771 4786 4800 4814 4829 4843 4857 4871 4886 4900 3 4 6 7 9 10 ii 13 31 132 133 4914 5051 5185 4928 5065 5198 4942 5079 5211 4955 5092 5224 4969 5105 5237 4983 5"9 5250 4997 5132 5263 5011 5M5 5276 5024 5159 5289 5038 5172 5302 3 4 3 4 3 4 6 7 8 5 7 8 5 6 8 10 II 12 9 11 12 9 10 12 134 35 36 137 38 39 5315 5441 5563 5328 5453 3575 5340 5465 5587 5353 5478 5599 5366 5490 5611 5378 5502 5623 5391 5514 5635 5403 5527 5647 54i6 5539 5658 5428 5551 5670 3 4 4 4 5 6 8 5 6 7 5 6 7 9 10 ii 9_io ii 8 10 ii 5682 5798 59" 5694 S89 5922 5705 5821 5933 5717 5832 5944 5729 5843 5955 5740 5855 5966 5752 5866 5977 5763 5877 5988 5775 5888 5999 5786 5899 6010 3 3 3 5 6 7 5 6 7 457 8 9 10 8 9 to 8 9 10 40 41 42 43 44 45 46 to 48 49 [50 6021 6031 6042 605.3 6064 6075 6085 6096 6107 6117 3 4 5 6 8 9 10 6128 6232 6335 6138 6243 6345 6149 6253 6355 6160 6263 6365 6170 6274 6375 6180 6284 6385 6191 6294 6395 6201 6304 6405 6212 6314 6415 6222 6325 6425 3 3 3 4 S 6 4 5 6 4 5 6 7 8 9 7 8 9 7 8 Q 6435 6532 6628 6444 6542 56 37 6454 6551 6646 6464 6561 6656 ^474 6571 6665 6484 6580 6675 6493 6590 6684 6503 6599 6693 6513 6609 6702 6522 6618 6712 3 3 4 5 6 4 5 6 7 8 9 7 8 9 6721 6812 6902 6730 6821 6911 6739 6830 6920 6749 6839 6928 6758 6848 6937 6767 6857 6946 6776 6866 6955 6785 6875 6964 6794 6884 6972 6803 6893 6981 3 3 3 455 445 445 6 7 8 6 7 fa 6 7 8 6990 ,998 7007 7016 7024 7033 7042 7050 7059 7067 3 345 6 7 8 51 52 53 54 7076 7160 7243 7084 168 251 7093 7177 7259 7101 7185 7267 7110 7193 7275 7118 7202 7284 7126 7210 7292 7135 7218 7300 7143 7226 7308 7152 7235 73i6 3 2 2 345 345 345 6 7 8 6 7 7 667 7324 7332 7340 7348 7356 7364 7372 738o 7388 7396 I 2 2 345 6 6 7 LOGARITHMS. 117 1 2 3 4 5 6 7 8 9 123 456 789 55 740-1 7412 7419 7427 7435 7443 7451 7459 7466 7474 I 2 2 345 5 6 7 56 57 58 7482 7559 7634 7490 7566 7642 7497 7574 7649 7505 7582 7657 7513 7589 7664 7520 7597 7672 7528 7604 7679 7536 7612 7686 7543 7619 7694 755J 7627 7701 I 2 2 I 2 2 f I 2 345 3 4 3 4 567 5 6 7 5 6 7 59 60 61 7709 7732 78S3 7716 7789 7860 7723 7796 7868 773i 7803 7875 7738. 7810 7882 7745 7818 7889 7752 7825 7896 7760 7832 79<>3 7767 7839 7910 7774 7846 7917 112 112 I 1 2 3 4 3 4 3 4 5 6 7 5 6 t 5 6 6 62 63 64 7924 7993 8062 7931 Sooo 8069 7938 8007 8075 7945 8014 8082 7952 8021 8089 7959 8028 8096 7966 8035 8102 7973 8041 8109 7980 8048 8116 7987 8055 812? I I 2 112 I I 2 112 3 3 3 3 3 3 566 5 5 6 5 5 6 65 8129 8136 8142 8149 8156 8162 8169 8176 8182 8189 3 3 5 5 t 66 67 68 69 70 71 8195 8261 8325 8388 8451 8513 8202 8267 8331 8395 8457 8519 8209 8274 8338 8215 8280 8344 8222 8287 8351 8228 8293 8357 8235 8299 8363 8241 8306 8370 8248 8312 8376 8254 8319 8382 8445 8506 8567 112 I I 2 I I 2 33 556 33 556 33 456 8401 8463 8525 8407 8470 853i 8414 8476 8537 8420 8482 8543 8426 8488 8549 8432 8494 8555 8439 8500 8561 112 2 3 455 72 73 74 8573 8633 8692 875i 8579 8639 8698 8585 8645 8704 859 1 8651 8710 8597 8657 8716 8603 8663 8722 8609 8669 8727 8615 8675 8733 8621 8681 8739 8627 8686 8745 I I 2 ri2 112 112 2 3 2 3 2 3 455 5 5 5 5 75 8756 8762 8768 8774 8779 8785 8791 8797 8802 233 ' 5 ^ 4 4 76 77 78 8808 8865 8921 8814 8871 8927 8820 8876 8932 8825 8882 8938 8831 8887 8943 8837 8893 8949 8842 8899 8954 8848 8904 8960 8854 8910 8965 885-9 8915 8971 112 I I 2 I I 2 233 233 233 79 80 81 8976 9031 9085 8982 9036 0090 8987 9042 9096 8993 9047 9101 8998 9053 9106 9004 9058 9112 9009 9063 9117 9015 9069 9122 9020 9074 9128 9025 9079 9133 112 1 I 2 I I 2 233 233 233 4 4 4 4 82 83 84 9138 9191 9243 9143 9196 9248 9149 9201 9 2 53 9154 9206 9258 9*59 9212 9263 9165 9217 9269 9170 9222 9274 9iZ5 9227 9279 9180 9232 9284 9186 9238 9289 112 112 I I 2 233 233 233 4 4 :: 85 "86 87 88 9294 9299 9304 9309 9315 9320 9325 9330 9335 9340 233 4 4 5 9345 9395 9445 9350 9400 9450 9355 9405 9455 9360 9410 9460 9365 9415 9465 9370 9420 9469 95i8 9566 9614 9375 9425 9474 938o 9430 9479 9385 9435 9484 939 9440 9489 I 1 I I 2 A 2 3 3 3 89 90 91 9494 9542 959^ 9499 9547 9595 9504 9552 9600 9509 9557 9605 9513 9562 9609 9523 957i 9619 9528 9576 9624 9533 958i 9628 9538 9586 9 6 33 Oil Oil Oil 2 3 2 3 2 3 3 3 3 92 93 94 9638 9685 9731 9643 9689 9736 9647 9694 974i 9~7S6 9652 9699 9745 9657 9703 9750 9661 9708 9754 9666 97*3 9759 9671 9717 9763 9675 9722 9768 9680 9727 9773 Oil Oil I I 2 3 2 3 2 - 3 3 3 3 4 95 9777 9782 979> 9795 9800 9805 9809 9814 9818 1 I 3 3 3 4 96 97 98 9823 9868 9912 9827 9872 9917 9832 9877 9921 9836 9881 9926 9841 9886 9930 9845 9890 9934 9850 9894 9939 9854 9899 9943 9859 993 9948 9863 9908 9952 I I I I I I 2 3 2 3 2 3 344 344 344 99 9956 9961 9965 9969 9974 9978 9983 9987 999 1 9996 I I 223 334 LOGARITHMS. 1 2 3 4 5 6 7 8 9 100 ooooo 043 087 130 173 217 260 303 346 389 101 102 103 432 860 oi 284 475 903 326 518 945 368 56i 988 410 604 030 452 647 072 494 689 "5 536 732 157 578 775 199 620 817 242 662 104 105 106 703 02 119 531 745 160 572 787 202 612 828 243 653 870 284 694 912 325 735 953 366 776 995 407 816 036 449 857 078 490 898 107 108 109 938 03342 743 979 383 782 OI9 423 822 060 463 862 100 503 902 141 543 941 181 583 981 222 623 O2 1 262 663 060 302 703 100 INDEX. Absolute, expansion of mer- cury, 55. humidity, 97. Acceleration, average linear, 25. uniform, 25. due to gravity, 31. machines, 25, 26. normal, 27. Air-buoyancy, correction for, in weighing, 3. Air, density of, 21, thermometer, constant pressure, 89. constant volume, 91. Apparent expansion, of alcohol, 62. Archimedes' principle, 3, 53. Auguste's psychrometer, 98. Balance, Jolly's, 4. model, 7. sensitive, I. Barodeik, calibration of the, 99. Boyle's law, experimental study of, ii. Brake, Prony, no. Callendar's method for me- chanical equivalent of heat, 69. Capillarity, rise of liquids in tubes, 37. Centripetal force, 28. Charles' Law, 22; tested, 89, 91. Coefficient of expansion, of liquid, by Archimides' prin- ciple, 52. by Regnault's method, 55. by pycnometer, 61. of glass by weight -ther- mometer, 58. Coefficient, of friction, 102. of restitution, 104. of rigidity, 108. Coefficient of expansion, of air at constant pressure; flask method, 86. with air thermometer, 89. Coincidences, method of, 32. Component of force, 38. Composition and resolution of forces, 15, 17. Constant - pressure air -thermom- eter, 89. Constant-volume air-thermom- eter, 91. Contact, angle of, 38. Cooling curve, 74. Cooling, law of, 67. Correction of a mercury ther- mometer, 46. Dalton's Law, 97. Density, of air, determination of, 21. of carbon dioxide, de- termination of, 23. of a cylindrical solid, 3- of solids, with Jolly's balance, 4. Dew-point, 97. hygrometer, 98. Double weighing, 4. Equation, the force, 34. of moments, 30. Expansion, apparent and abso- lute, 62, 64. absolute, of olive oil, 55- Force, centripetal, definition of, 27. formula, 28. equation, the, 34. table, vector sum using the, 15. Forces, experimental study of three, 17. 120 INDEX Freezing point, of solutions, 81. Friction, coefficient of, 102. "G" determination of, with the fall-machine, 24; with pendu- lum, 31. Glass, coefficient of expansion of, 58. Gram-molecular weight, 85. Heat, capacity, 65. of fusion, Wood's Alloy, 76. mechanical equivalent of, 69. of neutralization, 85. of solution, 83. of vaporization, at the boiling point, 78. at room temperature, 80. Hooke's Law, 106, 108. Humidity, absolute, 97. relative, 97. Hygrometer, Regnault's wet- and-dry bulb, 97. Impact, elastic, 104. inelastic, 105. Jolly's balance, 4. Kinetic energy, loss of in im- pact, 104. Lever arm, defined, 29. Liquid, density of, 6. Logarithms, table of, 4-place, 1 16. Mechanical equivalent of heat, by Callendar's method, 69. by Puluj's method, 71. Method of coincidences, 32. of cooling, 67. of mixtures, 76. of heating, 66. of vibrations, I. Modulus, of torsion, 108. Young's, by stretch- ing, 1 06. Moments, principle of, 30. Momentum, conservation of, 103. Motion, study of, uniformly ac- celerated, 24. Normal, acceleration, 28. force, 27. Normal solution, 85. Parallelogram of forces, 16. Pendulum, the simple, 31. Points, fixed, of thermometer, 4.6. Principle of moments, 29. Prony brake, no. Pycnometer, expansion of a liquid by, 61. Radiation, rate of, 112. Ratio of specific heats, 114. Relative humidity, 97. Resolution of forces, 15, 17. Restitution, coefficient of, 104. Resultant, of two forces, 16. Rider, use of the centigram, 2. Rigidity, coefficient of, 108. Sensitiveness, of balance, de- fined, 2. formula for, 8. Solution, heat of, 83. normal, 85. Solutions, freezing point of, 81. Specific heat of a liquid by method of heating, 66. by method of cooling, 67. Surface tension, by Jolly's bal- ance, 36. in capillary tubes, 37. between plates, 39. Tension, surface, by direct measurement, 36. Thermometer, absolute calibra- tion of, 45. comparison of alcohol and water, 54. constant- pressure air-, 89. relative calibration of, 49. Torsion, modulus of, 108. Twisting, Hooke's law for, 108. Uniform, accelerated motion, 24. circular motion, 26. Vapor pressure, and volume, 93. INDEX 121 Variation, of boiling point of water with pressure, 50. Vibrations, method of, 2. Viscosity, coefficient of, 42. Volumenometer, the, 13. Water, equivalent of a body, 65. equivalent of a ther- mometer, 65. expansion curve of, 63. Weighing, by method of vibra- tions, i. method of double, 4. Weight thermometer, 58. Weights, molecular, 82. Wet-and-dry, bulb thermome- ter, 97. Young's modulus, by stretching, 1 06. MEASUREMENT OF PHYSICAL QUANTITIES. Experimental work has one of two objects ; either to find out what kind of a result follows under given conditions, or to find out the numerical relations between different quantities. The first class of experiments is called qualitative, the second quan- titative. In the earlier days of any science qualitative experi- ments are numerous ; when the science is more mature, the ma- jority of the experiments are quantitative. The determination of various quantitative relations is the object of physical meas- urement. In making a physical measurement, the magnitude of each quantity concerned has to be expressed in terms of some unit, and the process of measurement consists essentially in finding how many times this unit is contained in the given quantity. The distance between two points, for example, may be ex- pressed in terms of the number of foot rules which could be laid end to end between these points. Some quantities can thus be measured directly, but others can be measured only indirectly. Thus the density of a solid cylin- der of any substance cannot be experimentally determined by finding how many times the unit of density is contained in the density of the cylinder. It would be determined usually by measuring the mass, length, and diameter of the cylinder, and from them calculating the density. The great majority of physical measurements are indirect measurements. ERRORS. Every measurement is subject to errors. In the simple case of measuring the distance between two points by means of a meter rod, a number of measurements usually give different 124 MEASUREMENT Ol- PHYSICAL QUANTITIES. results, especially if the distance is several meters long and the measurements are made to small fractions of a millimeter. The errors introduced are due in part to (1) Inaccuracy of setting at the starting point. (2) Inaccuracy of setting at intermediate points when the distance exceeds one meter. (3) Inaccuracy in estimating the fraction of a division at the end point. (4) Parallax in reading; that is, the line from the eye to the division line read is not perpendicular to the scale, and, where both eyes are used, the imaginary line joining the two eyes is not parallel to these division lines. (5) The meter rod not being straight. (6) The temperature not being that for which the meter red was graduated. (7) Irregular spacing of divisions. (8) Errors in the standard from which the division of the meter rod was copied. Besides the above there are doubtless other sources of error. It may be well here to note that blunders, such as mistakes due to mental confusion in putting down a wrong reading, or mistakes in making an addition, are not usually classed as errors. Of the above errors, (i), (2), and (3) can be very much decreased by having fine divisions on the scale and by reading with microscopes ; (4) can be made small by bringing the scale on the meter rod close to the object to be measured; (5) can be made very small by using a meter rod of special design, or, in rough work, by holding the meter rod against a straight edge; (6) can be nearly eliminated by using the meter rod only at the proper temperature, or, if its temperature and co- efficient of expansion are known, by calculating a correction to be applied; (7) can be diminished only by a careful com- parison of the lengths of the different divisions; and (8) can only be corrected for when something is known of the relative MEASUREMENT OF PHYSICAL QUANTITIES. 125 accuracy of the standard from which the meter rod was copied. But even with the most refined methods and the most care- ful application of corrections, different measurements of the same distance usually give different results. Errors due to (6), (7), and (8) may be determinate errors, that is., errors for which more or less accurate corrections can be calculated; whereas those due to (i), (2), and (3) are indeterminate errors, that is, errors for which corrections can- not be calculated. Moreover, of those errors for which cor- rections are not applied, some, like those due to (i), (2), and (3), will be variable in amount and will tend to make the value obtained sometimes too large and sometimes too small; while others, like those due to (7) and (8), when corrections for them are not applied, will be constant and will tend to make the value obtained always too large or always too small. Since the average value of those variable errors which tend to make a result too large will after a considerable number of measurements be about the same as the average value of those variable errors which tend to make the result too small, the mean of a large number of measurements is usually nearly free from variable errors. In order as nearly as possible to do away with constant errors, the same quantity should be meas- ured by as many different methods as possible. The results by the different methods will usually differ somewhat, but from them all a value can be calculated which is probably nearer the true value than is any one of the separate results. TRUSTWORTHY AND SIGNIFICANT FIGURES. Since all measurements are subject to errors, it is important to be able to determine how many figures of a result can be trusted. In direct measurements it is usually possible to make a fairly accurate estimate of the extent to which a reading can be trusted. Thus in reading, by means of the unaided eye, the position of a fine line which crosses a meter rod, the reading 126 MEASUREMENT OF PHYSICAL QUANTITIES. will not be in error by so much as a millimeter, but pretty surely will be in error by more than a thousandth of a milli- meter. So the extent to which the reading can be trusted will lie between these limits. A person who is accustomed to estimating fractions of a small division will be rather sure of not making an error so great as the tenth of a millimeter, and he can often trust his reading to a twentieth of a millimeter. It is convenient always to put down all the figures that can be trusted, even if some of them are ciphers. Thus the state- ment that a distance is 50 cm. implies that there is reason for supposing that the distance really lies between 49 cm. and 51 cm., whereas the statement that the distance is 50.00 cm. im- plies that there is reason for supposing that the distance really lies between 49.99 cm. and 50.01 cm. When the distance is said to be 50 cm., the second figure is the last in which any confidence can be placed ; when the distance is said to be 50.00 cm., the fourth figure is the last in which any confidence can be placed. This fact is indicated by saying that in the first case the quantity can be trusted to the second significant figure, and in the other case to the fourth. By the number of signifi- cant figures in a quantity is meant the number of trustworthy figures, counting from the left, irrespective of the decimal point; thus there are two significant figures in 0.000026. If ei distance is about 50000 km. and the third significant figure is the last in which any confidence can be placed, this fact may be indicated by saying that the distance is 50.0 X io 3 km. In indirect measurements the result is usually calculated by some formula. To find out how many figures should be kept in the result consider the following two cases : I. If the result is the algebraic sum of several quantities, such as 214.3, 3641, and 0.506, it is seen that in the sum 251.216, no figure beyond that in the first decimal place can be trusted, because, in the quantity which has the fewest trust- worthy decimal places, namely 214.3, no figure beyond the 3 can be trusted, otherwise it would have been expressed. So MEASUREMENT OF PHYSICAL QUANTITIES. 127 the sum will not be written 251,216, but will be written 251.2. This suggests the following rule : RULE L In sinus and differences no more decimal places should be retained than can be trusted in the quantity having fewest trustworthy decimal places. 2. If the result is the product of two quantities, such as 314.428 and 32.6, then the product cannot be trusted to more significant figures than appear in the quantity having fewest trustworthy figures, irrespective of the decimal place. To make this clear, consider the following products : 314.428 X 32.5 = 10218.9100 314.428 X 32.6 = 10250.3528 314.428 X 327 10281.7956 314. X 32.6 = 10236.4 It is seen from the first, second and third products that if the quantity which is supposed to be 32.6 is really 32.5 or 32.7, then after the first three significant figures the true value of the product differs materially from the value obtained. The sec- ond and fourth of the above products show that if more than three significant figures cannot be trusted in one of two quan- tities which are to be multiplied, it is not worth while to use more than three, or at most four, significant figures of the other. The product in this case would be written 1.02 X io 4 , or at most 1.024 X io 4 . These facts suggest the following rule: RULE IL In products and quotients no more significant figures should be kept than can be trusted in the quantity having fewest trustworthy figures. Until the final result is reached, it is often worth while to keep one more figure than the above rules indicate. For logarithms a safe rule is the following : RULE III. When any of the quantities which are to be mul- tiplied or divided can be trusted no closer than o.oi of one per cent., use a five-place table; when any of them can be trusted 128 MEASUREMENT OF PHYSICAL QUANTITIES. no closer than o.i of one per cent., use a four-place table; and when any of them can be trusted no closer than i per cent., use a slide rule. REQUIRED ACCURACY OF MEASUREMENT. From Rule I it will be seen that if a small quantity is to be added to a large one, the percentage accuracy of the measure- ment of the small quantity need not be so great as that of the large one. Thus if H = a + b, and if a is about 100 cm. and b about i cm., a I per cent, error in a will produce in H no greater effect than a 100 per cent, error in b. When quantities are to be added or subtracted, they should be measured to the same number of decimal places. From Rule II it will be seen that if a small quantity and a large one are to be multiplied, the percentage accuracy of the measurement of the small quantity should be at least as great as that of the large one. Thus if H = ab, a i per cent, error in a will produce in H the same effect as a i per cent, error in b. So that if a is about 100 cm., and b is about i cm., and if b cannot be trusted closer than o.oi cm., there is no gain in ac- curacy by measuring a much closer than i cm. When quanti- ties are to be multiplied or divided they should be measured to within the same fractional part of themselves, for example, all of them within i per cent, and none of them much closer, or all of them within o.oi of one per cent and none of them much closer. The last statement needs modification in the case of a power. If the value found for a quantity a is i per cent too large, that is, is i .010, then the value that will be obtained for a~ is [.020 1 a, which is about 2 per cent too large, and the value ob- tained for a 3 is 1.0303010, which is about 3 per cent, too large. In general, if the value found for a is k per cent too large, the value that will be obtained for o n will be nk per cent, too large. So that a quantity which is to be squared, cubed, or raised to MEASUREMENT OF PHYSICAL QUANTITIES. 129 some higher power should be measured with more care than if it entered the formula only to the first power. It is evident, then, that a preliminary study of the required accuracy of measurement will not only save much time, by pointing out those quantities which need to be measured with only a rough accuracy, but will also serve to determine those quantities, usually the smallest, in the measurement of which great care must be taken and sensitive instruments used. (Largely reproduced from Perry & Jones.) I3O MEASUREMENT OF PHYSICAL QUANTITIES. TABLE I. USEFUL NUMERICAL RELATIONS. Mensuration. Circle: circumference = 2^r\ area = Sphere: area = 4^r; volume = 4/3*r 3 . Cylinder: volume = T 2 /. Length. i centimeter (cm.) = 0.3937 i n - i meter (m.) = 3.281 ft. I kilometer (km.) 0,6214 mi. i micron (ut) = o.ooi mm. = 0.00394 i n I sq. cm. =0.1550 sq. in. I sq. m. = 10.674 sq. ft. Area. i inch (in.) = 2.540 cm. i foot (ft.) = 0.3048 m. i mile (mi.) = 1.609 km. i mil = o.ooi in. = 0.00254 cm. i sq. in. = 6.451 sq. cm. i sq. ft. = 0.09290 sq. m. Volume. i cc. = 0.06103 cu. in. i cu. m. = 35-317 cu. ft. i liter (1000 cc.) = 1.7608 pints. i cu. in. i cu. ft. i quart 16.386 cc. 0.02832 cu. m. 1.1359 liters. Mass. i gram (gm.) : 15.43 gr. i kilogram (kg.) = 2.2046 Ib. i grain (gr.) = 0.06480 gm. i pound (lb.)= 0.45359 kg. Density. i gm. per cc. = 62.425 Ib. per cu. ft. i Ib, per cu. ft. == 0.01602 gm. per cc. Force. i gram's weight (gm. wt.) = 980.6 dynes (go = 980.6 cm./sec. 2 .) i pound's weight (Ib. wt.) = 0.4448 megadynes (go = 980.6.) (The "gm. wt.' is here denned as the force of gravity acting on a gram of matter at sea-level and 45 latitude. The "Ib. wt." is similarly defined.) MEASUREMENT OF PHYSICAL QUANTITIES. TABLE II. 131 USEFUL NUMERICAL RELATIONS. Pressure and Stress. i cm. of mercury at oC. = J 3-596 gm. wt. per sq. cm. = 0.19338 Ib. wt. per sq. in. i in. of mercury at oC. = 34-533 gm. wt. per sq. cm. = 0.49118 Ib. wt. per sq. in. Work and Energy. kilogram-meter (kg. m.) = 7.233 ft. Ib. foot-pound (ft. Ib.) = 0.13826 kg. m. joule = io 7 ergs. foot-pound = 1-3557 X io 7 ergs. (go = 980.6 cm./sec. 2 .) foot-pound = 1-3557 joules (go = 980.6.) joule = 0.7376 ft. Ib (go = 980.6.) Power (or Activity). I horse-power (H. P.) = 3 3000 ft. Ib. per min. i watt = i joule per sec. = IO T ergs per sec. i horse-power = 745.64 watts (go = 980.6 cm./sec. 2 ) i wa,tt = 44.28 ft. Ib. per min. (g = 980.6) Thermometric Scales. C = 5/9(F- 3 2) | (C = centigrade temperature; F = 9/5C + 32. F = Fahrenheit temperature.) Mechanical Equivalent. I gm. -calorie = 4.187 X io 7 ergs. = 0.4269 kg. m. (g = 980.6 cm./sec. 2 ) = 3-088 ft. Ib. (g = 980.6.) 132 MEASUREMENT OF PHYSICAL QUANTITIES. TABLE III. DENSITY OF DRY AIR. (Values are given in gms. per cc.) Temp. Barometric Pressure (Centimeters of Mercury) C. 72 73 74 75 76 77 .001225 .001242 .001259 .001276 .001293 .001310 i 220 237 254 271 288 305 2 216 233 250 267 283 300 3 212 228 245 262 279 296 4 207 224 241 257 274 290 5 .OOI2O3 .001219 .001236 .001253 .001270 .001286 6 198 215 232 248 265 282 7 194 211 227 244 260 277 8 190 206 223 239 256 272 9 186 202 219 235 251 268 10 .001181 .001198 .001214 .001231 .001247 .001263 ii 177 194 210 226 243 259 12 173 l8 9 2O6 222 238 255 13 169 202 218 234 250 14 165 181 197 214 230 246 15 .001161 .001177 .001193 .001209 .001225 .001242 16 i57 173 I8 9 205 221 237 17 169 185 201 217 233 18 149 165 181 197 213 229 19 145 161 177 I 93 209 224 20 .001141 .001157 .001173 .0011 89 .001204 .001220 21 137 i53 169 185 200 216 22 133 149 165 I & I 9 6 212 23 130 145 161 177 208 24 126 141 157 173 ~~^88 204 25 .001122 .001138 .001153 .001169 .001184 .001200 26 118 i34 149 165 1 80 196 27 114 130 145 161 176 192 28 I IO 126 142 157 172 188 29 107 122 138 153 169 184 30 .001103 .001119 .001134 .001149 .001165 .001180 Corrections for Moisture in the Atmosphere. Dew-point Subtract Dew-point. Subtract. Dew-point. Subtract. 10 .00000 i +2 .000003 + 14 . 000007 g 2 + 4 4 + 16 8 6 2 + 6 4 + 18 9 - 4 2 + 8 5 + 20 .000010 2 3 + 10 6 +24 13 3 + 12 6 +28 16 MEASUREMENT OF PHYSICAL QUANTITIES. 133 TABLE IV. DENSITIES AND THERMAL PROPERTIES OF GASES. The densities are given at oC. and 76 cm. pressure, and the specific heats at ordinary temperatures. The coefficients of cubical expansion (at constant pressure) of the gases listed below are not given in this Table; they are about the same for all the permanent gases, being approximately 1/273 or 0.003663, if referred in each case to the volume of the gas at oC. Gas or Vapor. Formula Density (gms. per cc.) Molecular Weight Cp-Cv C P (cals. pergm.) Air _ 0.001293 1.41 0.237 Ammonia NH 3 O.OOO77O 17.06 33 530 Carbon dioxide CO, O.OOI974 44.00 .29 .203 Carbon monoxide CO O.OOI234 28.00 .40 243 Chlorine CU 0.003133 70.90 32 .124 Hydrochloric acid HC1 0.001616 36.46 40 .194 Hydrogen H 2 0.0000896 2.016 41 3410 Hydrogen sulphide Nitrogen, pure H 2 S N, 0.001476 0.001254 34-08 28.08 34 .41 ^245 .244 Nitrogen, atmospheric 0.001257 Oxygen 0, 0.001430 32.00 .41 .218 Steam (iooC.) H 2 O 0.000581 18.02 .28 .421 Sulphur dioxide SO, 0.002785 64.06 .26 154 TABLE V. DENSITY AND SPECIFIC VOLUME OF WATER. Temp. C. Density (gms. per cc.) Specific Volume (cc. per gm.) Temp. C. Density (gms. per cc.) Specific Volume (cc. per gm.) 0.999868 I.OOOI32-' 20 0.99823 I.OOI77 i 927 073 25 777 294 2 968 032 30 567 435 3 992 008 35 406 598 3.98 I.OOOOOO f. 000 40 224 782 5 .999992 008 50 .98807 1.01207 6 9 68 032 60 324 705 7 929 071 / 70 .97781 1.02270 8 876 124. rf 80 183 902 9 808 192 90 .96534 1.03590 10 727 273 IOO .95838 1-04343 15 126 874 102 693 5oi 134 MEASUREMENT OP PHYSICAL .QUANTITIES. TABLE VI. DENSITIES AND THERMAL PROPERTIES OF LIQUIDS. The values given in this Table are mostly for pure specimens of the liquids listed. The student should not expect the properties of the average laboratory specimen to correspond exactly in value with them. With a few exceptions the densities are given for ordi- nary atmospheric temperature and pressure. The specific heats and coefficients of expansion are in most cases the average values between o and iooC. The boiling points are given for atmos- pheric pressure, and. the heats of vaporization are given at these boiling points. d w w o hjW <3 CD n* n 8 o 2. ^ f& T3 P C rl- ~ P g. 3j rt- 5' O ~ 1 ' Liquid * O C/) O *~*" 1 *^' CKJ ra ^3 r^- H*'- O calories 3 pergm. degrees cals. gms. per cc. per deg. per degree C. C. per gm. Alcohol (ethyl) 0.794 .68 .00111 78 205* (methyl) 796 .60 .00143 66 262f Benzene .880 .42 .00123 80 93-2 Carbon bisulphide 1.29 .24 .OOI2O 46.6 8 4 Ether Glycerine 74 (o) 1.26 .00162 .000534 35 90 Hydrochloric acid 1.27 75 .000455 no Mercury 13.596 (o) 033, .OOOl8l5 357 67 Olive oil .918 .0007,21 Nitric acid 1.56 :66 .00125 86 US Sea-water 1.025 .938 Sulphuric acid 33 .00056 338 122 Turpentine Water See Tab.V. 47 I.OO .OOIO5 See Tab. V. 159 TOO 70 537 * The heat- of vaporization of ethyl alcohol at oC. is 236.5. t The heat of vaporization of methyl alcohol at oC. is 289.2. "MEASUREMENT OF PHYSICAL- QUANTITIES. -135 TABLE VII. DENSITIES AND THERMAL PROPERTIES OF SOLIDS. The values given in this Table are mostly for pure specimens of the substances listed. The student should not expect the prop- erties of the average laboratory specimen to correspond exactly in value with them. As a rule the densities are given for ordi- nary atmospheric temperature and pressure. The specific heats and coefficients of expansion are in most cases the average values between o and iooC. The melting points and heats of fusion are given for atmospheric pressure. 8 01 p 11. Ir ? ^3 2L 2.*"* 3 . "-K rD Solid. *<" I" 1 " 3"> O r|'| O !IL. 3 3 3 cals. per cals. per gms per cc. gm. per degree C. degrees C gm. Aluminum 2.70 O.2I9 .0000231 658 Brass, cast 8.44 .092 .0000188 , drawn 8.2P .092 .OOOOI93 Copper German-silver 8.92 8.62 094 .0946 .0000172 .OOOOlS IO90 860 43-0 Glass, crown 2.6 .OOOOC90- " , flint 3-9 .117 .0000079 Gold Hyposul. of soda 19-3 1,71 .0316 445 .0000144 1065 4 8 Ice .918 502 . .000051 .80. Iron, cast 7-4 113 .0000106 1 100 23-33 , wrought 7.8 US .000012 1600 Lead 0315 .000029 326 5-4 Mercury .0319 39 2.8 Nickel 8.90 .109 .OOOOI28 1480 4-6 Paraffin, wax .90 .560 .OOOOO8-23 52 35-1 , liquid .710 Platinum 21.50 .0324 .OOOOC90 1760 27.2 Rubber, hard 1.22 331 .000064 Silver 10-53 .056 .0000193 960 2' 1. 1 Sodium chloride 2.17 .214 .000040 "800 Steel 7-8 .118 .OOOO 1 1 1375 Wood's alloy, solid 9.78 0352 75-5 8.40 " , liquid .0426 1 ,,o1> 136 MEASUREMENT OF PHYSICAL QUANTITIES. TABLE VIII . SURFACE TENSION OF PURE WATER IN CONTACT WITH AIR. Temp. C. Tension (dynes per cm.) Temp. C. Tension (dynes per cm.) Temp. C. Tension (dynes per cm.) 5 75-5 74-8 30 35 71.0 70.3 60 65 66.0 65.1 10 74-0 40 69.5 70 64.2 15 73-3 45 68.6 80 62.3 20 72-5 50 67.8 100 56.0 5 71.8 55 66.9 Crit. Temp. o.o TABLE IX. SURFACE TENSIONS OF SOME LIQUIDS IN CONTACT WITH AIR. Dynes per cm. Dynes per cm. Alcohol (ethyl) Alcohol (methyl) Benzene Glycerine at 20 at 20 at 15 at 18 22-24 22-24 28-30 63-65 Mercury Olive oil Petroleum Water (pure) at 20 at 20 at 20 at 20 470-500 32-36 24-26 72-74 TABLE X. VISCOSITY OF WATER. Temp. C. Coeff. of Vise. (C.G.S. Units) Temp. C. Coeff. of Vise. (C.G.S. Units) Temp. C. Coeff. of Vise. (C. G.S. Units) 5 10 15 20 O.OI78 .0151 .0131 .0113 .0100 25 30 35 40 50 0.0089 .0080 .0072 .0066 0055 60 70 80 90 100 O.OO47 .0041 .0036 .0032 .0028 MEASUREMENT OF PHYSICAL QUANTITIES. TABLE XL VISCOSITY OF AQUEOUS SOLUTIONS OF SUGAR. 137 % Sugar Coeff. at 20C. (C. G. S. Units.) Coeff. at 3 oC. (C. G. S. Units.) 5 10 20 4 O.OIOO .0117 .0132 .0191 .0600 0.0080 .0089 .0104 .0145 .0423 TABLE XII. COEFFICIENTS OF FRICTION. Substances. Static Coefficient. Kinetic Coefficient. Metals on metals (dry) from 0.2 to 0.4 from 0.18 to 0.35 (wet) " 0.15 0-3 ' 0.14 ' 0.28 " (oiled) 0.15 0.2 ' 0.14 ' 0.18 Wood on wood (dry) (a) direction of fiber " 0.5 0.7 < 0.2 ' 0-3 (b) normal to fiber 0.4 0.6 1 0.18 ' 0-3 Leather belt on wood pulley 0.45 0.6 ' 0.3 ' 0.5 " " iron " " 0.25 0-35 0.2 ' 0.3 TABLE XIII. ELASTIC CONSTANTS OF SOLIDS. (Approximate Values.) Substance. Bulk-Modulus. (C. G, S. Units.) Simple Rigidity. (C. G. S. Units.) Young's Modulus. (C. G. S. Units.) Aluminum Brass, drawn Copper 5-5 x ic" 10.8 x " 16.8 x " 2.5 x IO U 37 x 4-5 x 6.5 x 10 u 10.8 x 12.3 x T0 o t fMocc 5 x ( Iron, wrought Steel 14.6 x " 18.4 x 2.4 x 77 x 8.2 X 7.0 x 19.6 x 21.4 x 138 MEASUREMENT OF PHYSICAL QUANTITIES. TABLE XIV. (a) BOILING POINT OF WATER AT DIFFERENT BARO- METRIC PRESSURES. (b) VAPOR-TENSION OF SATURATED WATER- VAPOR. (This table may be used either (a) to find the boiling point t of water under the barometric pressure P, or (b) to .find the vapor-tension P of water-vapor saturated at the temperature t, the dew-point.) t Cent. P cm. D gm.|cc. t Cent. P cm. D gm.|cc. t Cent. P cm. D gm.|cc. 10 .22 2.3XIO' 6 30 3-15 30.ixio~ 6 88? 5 49.62 9 23 2.5x 1 35 4.18 39-3X " 89 50.58 8 25 40 5-49 50-9X " 89-5 51-55 7 27 2.9X ' 7.14 65-3x " 90 52-54 428.4xio~" 6 .29 3-2X ' 50 9.20 83-Ox " 90-5 53-55 5 32 3-4x ' 55 1 1 *y$ I04.6x " 91 54-57 4 34 3-7x " 60 14.88 I3o.7x " 91-5 55-6i 3 -37 4.ox " 65 18.70 I62.IX " 92 56.67 2 39 4.2X " 70 23-31 I99-5X " 92-5 57-74 I .42 4-5x ' 24.36 93 58-83 O 46 4-9x ' 72 25-43 93-5 59.96 I 49 5-2x ' 73 26.54 94 6 1. 06 2 -53 5-6x ' 74 27.69 94-5 62.20 3 57 6.ox ' 75 28.88 243-7 " 95 63-36 511.1 4 .61 6.4x ' 75-5 29.49 95-5 64.54 5 .65 6.8x ' 76 30.11 96 6574 6 .70 7-3x ' 76.5 30-74 96.5 66.95 7 -75 7-7x ] 77 31.38 97 68.18 8 .8q 77-5 32.04 97-5 69.42 Wo 85. ,.91 87x ' 9-3x ' 78 78-5 32.71 33-38 98 98.2 70.71 71.23 n .98 IO.QX ' 79 34-07 98.4 71-74 12 .04 io.6x ' 79-5 34-77 98.6 72.26 13 .n II. 2X ' 80 35-49 295-9 " 98.8 72-79 14 .19 12. OX " 80.5 36.21 99 73-32 15 2? I2.8X " 81 36.95 99-2 73-85 16 -35 13. 5x " 81.5 37-70 99-4 74.38 17 44 I4-4X " 82 38.46 99-6 74.92 18 -53 I5.2x " 82.5 39-24 99-8 75-47 19 -63 i6.2x " 83 40.03 IOO 76.00 606.2 " 20 *74 17. 2X " 83.5 40.83 100.2 76.55 21 -85 I8.2X ' 84 41.65 100.4 77.10 22 -96 I 9-3 X ' 84-5 42.47 100.6 77.65 23 2.09 20.4X ' 85 43-32 357-1 " 100.8 78.21 24 2.22 21. 6x ' 85-5 44.13* 101 78.77 25 2-35 22.9X ' 86 45-05 102 8 1. 60 26 2-50 24.2X ' 86.5 45-93 103 84.53 27 2.6 5 25. 6x ' 87 46-83 105 90.64 715.4 " 28 2.8l 27. ox ' 87-5 47-74 107 97.11 29 2-97 28.5X ' 88 48.68 no 107-54 840.1 " MEASUREMENT OF PHYSICAL QUANTITIES. 139 TABLE XV. THE WET- AND DRY-BULB HYGROMETER. DEW-POINT. This Table gives the vapor-pressure, in mercurial centimeters, of the water-vapor in the atmosphere corresponding to the dry-bulb reading tC. (first column) and the difference (first row) between the dry-bulb and wet-bulb readings of the hygrometer. Having obtained from this Table the value of the vapor-pressure for a given case, the dew-point can be found by consulting Table XIV. The data given below are calculated for a barometric pressure equal to 76 cm. tc. Difference between Dry-bulb and Wet-bulb Readings. 0- 1 2 S- 4 6 6 7- 8 9 10* cm. cm. cm. cm. cm. cm. cm. cm. cm. cm. cm. 10 92 .81 .70 .60 50 40 31 .22 13 ii .98 87 76 65 7C .26 17 ^ 12 .v^w 05 *~>/ 93 */ w .82 V -'O .71 !6o 50 oo .40 30 r / .21 .12 P3 13 .12 I.OO .89 .76 65 55 45 35 25 .16 .07 14 .19 1.07 94 .83 7i .61 50 .40 30 .20 .11 15 .27 1.14 I.OI .90 78 .66 55 45 34 25 15 16 35 1.22 1.09 97 84 73 .60 50 .40 30 .19 i? 44 1-30 1. 17 1.04 .91 .80 67 56 45 35 24 18 54 i-39 1-25 1. 12 99 .86 74 63 51 .40 30 ig 63 1.49 i-34 1.20 1.07 94 .81 69 57 46 35 20 74 1-59 i-43 1.29 i. 02 .88 76 . 64 52 .41 21 85 1.69 1-53 1.38 1.24 1. 10 96 84 71 47 22 1.97 i. 80 1.64 1.48 1-33 1.19 05 ?9iJ .66 54 23 2.09 1.92 1-75 1-59 1-43 1.28 13 I.OO .86 73 .61 24 2.22 2.04 1.86 1.70 i. .53 1.38 23 1.09 94 .81 .68 25 2-35 2.17 1.99 1.81 1.64 1.48 33 1.18 1.03 .90 -76 26 2.50 2.31 2. II 1.94 1.76 1-59 43 1.28 1.13 98 84 27 2.65 2-45 2.25 2.07 1.88 1.71 1.38 1-23 i. 08 93 28 2.81 2.60 2.40 2.20 2.01 1.83 i 66 1.49 1-33 1.18 i. 02 29 2.98 ' 2.76 2-55 2-35 2.15 1.96 1.78 1.61 1.44 1.28 1. 12 30 3-15 2-93 2.71 2.50 2.29 2.IO 1.91 1-73 1-55 1-39 1.23 TABLE XVI. Miscellaneous. (i). Heat of Neutralization. Any strong acid with any strong alkali evolves ( + ) about 761 calories for every gm. of water formed. (2.) Heat of Solution. For Calcium oxide (CaO), + 327 cals. per gm. " Sodium chloride (NaCl), 21 " " " " hydroxide (NaOH), + 248 " " " " hyposulphite (NaSO + 5H 2 O), 44.8 " (3). Lowering of Freezing Point of Water. For dilute aqueous solutions of any salt the lowering is proportional to the mass of salt dissolved in the same mass of water. If ionization does not occur, each gram- molecular weight of the salt in TOGO gm. of water will lower the freezing point 1.86. If ionization occurs, how- ever, the lowering is increased two, three, or more times, depending upon the number of separate parts or ions into which the molecule of the salt is divided by the water. I4O MEASUREMENT OF PHYSICAL QUANTITIES. TABLE XVII. Natural Sines. 0' 6' 12 18 24 30' 36 42 48' 54' 123 4 5 0000 0017 0035 0052 0070 0087 0105 0122 0140 oi57 369 12 15 1 2 3 0175 0349 0523 0192 0366 0541 0209 0227 0384^0401 05580576 0244 0419 0593 0262 0436 0610 0279 0454 0628 029703140332 0471 04880506 0645 0663 O68O 369 369 369 12 15 12 15 12- 15 4 5 6 0698 0872 1045 0715107320750 0889 0906 0924 1063! io8o| 1097 0767 0785 0941 0958 1115 1132 0802 0976 "49 0819 0993 1167 0837 ion 1184 0854 1028 I2OI 369 369 369 12 15 12 14 12 14 7 8 9 1219 1392 1564 1736 1236 1409 1582 1253 1426 1599 1271 1444 1616 1288 1461 1633 1305 1478 1650 1323 1495 1668 1340 1513 1685 1357 1530 1702 1374 1547 1719 369 369 369 12 14 12 14 12 14 10 1754 1771 1788 1805 1822 1840 i8 57 |i874 1891 369 12 14 11 12 13 1908 2079 2250 1925 2096 2267 2436 2605 2773 I942JI959 2113 2130 2284 2300 1977 2147 2317 1994 2164 2334 2OII 2181 2351 2028 2198 2368 2045 2215 2385 2O62 2232 2402 369 369 368 II 14 II 14 II 14 14 15 16 2419 2588 2756 2453 2622 2790 2470 2639 2807 2487 2656 2823 2504 2672 2840 2521 2689 2857 2538 2706 2874 2554 2723 2890 2571 2740 2907 368 368 3 6 8 II 14 II 14 II 14 17 18 19 2924 3090 3256 2940 3107 3272 3437 2957 3123 3289 3453 2974 3MO 3305 2990 3156 3322 3486 3007 3173 3338 3024 3190 3355 3040 3206 337i 3057 3223 3387 3074 3239 3404 368 368 3 5 8 II 14 II 14 II 14 20 3420 3469 3502 35i8 3535 3551 3567 3 5 8 II 14 21 22 23 3584 3746 397 3600 3762 3923 3616 3778 3939 3633 3795 3955 3649 3811 397i 3665 3827 3987 3681 3843 4003 3697 3859 4019 3714 3875 4035 3730 3891 4051 3 5 8 3 5 8 3 5 8 II 14 II 14 II 14 24 25 26 4067 4226 4384 4083 4242 4399 4099 4258 4415 4H5 4274 443i 4131 4289 4446 4147 4305 4462 4163 4321 4478 4179 4337 4493 4195 4352 4509 42IO 4368 4524 3 5 8 3 5 8 3 5 8 II 13 II 13 10 13 27 28 29 4540 4695 4848 4555 4710 4863 4571 4726 4879 4586 474i 4894 4602 4756 4909 4617 4772 4924 4633 4787 4939 4648 4802 4955 4664 4818 4970 4679 4833 4985 3 5 8 3 5 8 3 5 8 10 13 10 13 10 13 30 5000 5150 5299 5446 5015 5030 5045 5060 5075 5090 5105 5120 5135 3 5 8 10 13 31 32 33 5165 53M 546i 5180 5329 5476 5195 5344 5490 5210 5358 5505 5225 5373 5519 5240 5388 5534 5255 5402 5548 5270 5417 5563 5284 5432 5577 2 5 7 2 5 7 2 5 7 IO 12 IO 12 10 12 34 35 36 5592 5736 5878 5606 5750 5892 5621 5764 5906 5635 5779 5920 5650 5793 5934 5664 5807 5948 5678 5821 5962 5693 5707 572i 5835 5850 5864 5976159906004 2 5 7 2 5 7 2 5 7 IO 12 10 12 9 12 37 38 39 6018 6157 6293 6428 6032 6170 6307 6441 6046 6184 6320 6060 6198 6334 6074 6211 6347 6088 6225 6361 6494 6101 6239 6374 6ii5 l 6i296i43 6252 6266 6280 6388 6401 6414 2 5 7 2 5 7 247 9 12 9 ii 9 u 40 6455 6468 6481 6508 6521 6534 6547 247 9 ii 41 42 43 6561 6691 6820 65746587 67046717 68336845 6600 6730 6858 6613 6743 6871 6626 6639 67566769 688416896 6652 6665 6678 6782 6794 6807 69096921 6934 247 246 246 9 ii 9 n 8 u 44 6947 6959 6972 6984 6997 7009 7022 7034 7046 7059 246 8 10 MEASUREMENT OF PHYSICAL QUANTITIES. TABLE XVII. (Coned.) Natural Sines. 141 7071 6' 7083 12 7096 18 24 30 36 42 7157 48 54 718: 123 4 5 45 7108 7120 7133 7M5 7169 2 4 6 8 10 46 47 48 7193 73M 743i 7206 7325 7443 7218 7337 7455 7230 7349 7466 7242 7361 7478 7254 7373 7490 726f 7385 75oi 7278 7396 7513 7290 7408 7524 7302 7420 7536 246 246 246 8 10 8 10 8 10 49 50 51 7547 7660 777i 7558 7672 7782 7570758i 7683 7694 7793 7804 7593 7705 7815 7604 7716 7826 7615 7727 7837 7627 7738 7848 7638 7749 7859 7649 7760 7869 246 246 2 4 5 8 9 7 9 7 9 52 53 54 7880 7986 Sogo 7891 7997 8100 7902 7912 8007! 80 1 8 8nij8i2i 7923 8028 8131 7934 8039 8141 7944 8049 8151 7955 8059 8161 7965 8070 8171 7976 8080 8181 245 2 3 5 2 3 5 7 9 7 9 7 8 55 8192 8202 8211 8221 8231 8241 8339 8434 8526 8251 8348 8443 8536 8261 8271 828) 235 7 8 56 57 58 8290 8387 8480 8572 8660 8746 8829 8910 8988 8300 8396 8490 S 5 Si 8669 8755 8838 8918 8996 83108320 84068415 84998508 8329 8425 8517 8358 8453 8545 8368 8462 8554 8377 8471 8563 235 2 3 5 235 6 8 6 8 6 8 59 60 61 8590 8678 8763 8846 8926 9003 8599 8686 8771 8607 8695 8780 8616 8704 8788 8625 8712 8796 8634 8721 8805 8643 8729 8813 8652 8738 8821 3 4 3 4 3 4 6 7 6 7 6 7 62 63 64 8854 8934 901 1 8862 8942 9018 8870 8949 9026 8878 8957 9033 9107 8886 8965 9041 8894 8973 9048 8902 8980 9056 3 4 3 4 3 4 5 7 5 6 5 6 65 9063 9070 9078 9 J 50 9219 9285 9348 9409 9466 9085 9092 9100 9114 9121 9191 9259 9323 9128 2 4 5 6 66 67 68 9135 9205 9272 9*43 9212 9278 9157 9225 9291 9164 9232 9298 9171 9239 9304 9178 9245 93U 9184 9252 9317 919^ 9265 9330 2 3 2 3 2 3 5 6 4 6 4 5 69 70 71 9336 9397 9455 9342 H03 9461 9354 9415 9472 936i 9421 )478 9367 9426 9483 9373 9432 9489 9379 9438 9494 9385 9444 939 1 9449 9505 2 3 2 3 2 3 4 5 4 5 4 5 9500 72 73 74 95ii 9563 9613 )Vt> 9568 9617 9521 9573 9622 9527 9578 9627 9532 9583 9632 9537 9588 9636 9542 9593 9641 9686 954 9598 9646 9553 9603 9650 9558 9608 9655 9699 2 3 2 2 2 2 4 4 3 4 3 4 75 9659 9703 9744 9781 9664 9668 9673 9677 9681 9690 9694 9736 9774 9810 I 2 3 4 76 77 78 9707 9748 9785 9711 975i 9789 9715 9755 9792 9720 9759 9796 9724 9763 9799 9728 9767 9803 9732 9770 9806 9740 9778 9813 2 2 2 3 3 3 3 2 3 79 80 81 9816 9848 9877 9820 )35i 9880 9823 9854 9882 9826 9857 9885 9829 )S6o )888 )S33 9863 9890 9836 9866 9893 9839 9869 9895 9842 9871 9898 9845 9874 9900 I 2 O O 2 3 2 2 2 2 82 83 84 9903 9925 9945 9905 9928 9947 997 9930 9949 9910 9932 995i WI2 9934 9952 9968 9914 9936 9954 9917 9938 9956 9919 9940 9957 9921 9942 9959 99 2 3 9943 9960 O O O 2 2 I 2 I I 85 9962 9963 9965 9966 9969 9971 9972 9973 9974 O O I I I 86 87 88 9976 9986 9994 )977 9987 9995 9978 9988 9995 9979 9989 9996 9980 9990 9996 9981 9990 9997 9982 999 i 9997 9983 999 2 9997 9984 9993 9998 9985 9993 9998 I 000 O O O I I I I O O 89 9998 9999 9999 9999 9999 I.OOO nearly I.OOO nearly I.OOO nearlv I.OOO nearly I.OOO nearly O O O O 142 MEASUREMENT OF PHYSICAL QUANTITIES. TABLE XVIII. Natural Tangents. 0' 6 12' 18 24 30 36 42 48' 54' 123 4 5 12 I 4 - .0000 0017 0035 0052 0070 0087 0105 0122 0140 oi57 369 1 2 3 .0175 0349 .0524 0192 0367 0542 0209 0384 0559 0227 0402 0577 0244 0419 0594 0262 0437 0612 0279 0454 0629 0297 0472 06 47 0314 0489 0664 0332 0507 0682 369 369 369 12 I 5 12 I 5 12 15 4 5 6 .0699 .0875 .1051 0717 0892 1069 0734 0910 1086 0752 0928 1104 0769 0945 1122 0787 0963 "39 0805 0981 "57 0822 0998 "75 0840 1016 1192 0857 1033 I2IO 369 369 369 12 I 5 12 I 5 12 I 5 7 8 9 .1228 .1405 .1584 1246 1423 1602 1263 1441 1620 1281 1459 1638 1299 1477 1655 1317 1495 1673 1334 1512 1691 1352 1530 1709 1370 1548 1727 1908 1388 1566 1745 1926 369 369 369 12 I 5 12 J 5 12 I 5 10 .1763 1781 1799 1817 1835 1853 1871 1890 369 12 I 5 11 12 13 .1944 .2126 .2309 1962 2144 2327 1980 2162 2345 1998 2180 2364 2016 2199 2382 2035 2217 2401 2053 2235 2419 2071 2254 2438 2089 2272 2456 2107 2290 247S 3 6 9 369 369 12 15 12 15 12 15 14 15 16 2493 .2679 .2867 2512 2698 2886 2530 2717 2905 2549 2736 2924 2568 2754 2943 2586 2773 2962 2605 2792 2981 2623 28ll 3000 2642 2830 3019 3211 3404 3600 2661 2849 3p_38 3230 3424 3620 369 369 369 12 l6 13 l6 13 16 17 18 19 3057 3249 3443 3076 3269 3463 3096 3288 3482 3"5 3307 3502 3134 3327 3522 3153 3346 354i 3172 3365 356i 3I9 1 3385 3581 3 6 10 3 6 10 3610 13 .16 13 l6 13 17 20 .3640 3659 3679 3699 3719 3739 3759 3779 3799 3819 3 7 10 I 3 17 21 22 23 3839 .4040 4245 3859 4061 4265 3879 4081 4286 3899 4101 4307 39 J 9 4122 4327 3939 4142 4348 3959 4163 4369 3978 4183 4390 4000 4204 44" 4020 4224 4431 4642 4856 5073 3 7 I0 3 7 10 3 7 10 13 17 >4 17 14 17 24 25 26 4452 .4663 .4877 4473 4684 4899 4494 4706 4921 4515 4727 4942 4536 4748 4964 4557 4770 4986 4578 479' 5008 45994621 48134834 50295051 7 10 7 ii 7 " 14 18 14 18 15 18 27 28 29 5095 5317 5543 5U7 5340 5566 5139 5362 5589 5161 5384 5612 5184 5407 5635 5206 5430 5658 5228 5452 5681 52505272 5475 5498 5704 5727 5295 5520 5750 7 >i 8 ii 8 12 15 18 "5 J9 '5 19 30 5774 5797 5820 5844 5867 5890 59M 5938 6176 6420 6669 596i 5985 4 8 12 16 20 31 32 33 .6009 .6249 .6494 6032 6273 6519 60566080 6297 6322 6544 6569 6104 6346 6594 6128 6371 6619 6152 6395 6644 6200 6445 6694 6224 6469 6720 4 8 12 4 8 12 4 S 13 16 20 l6 20 17 21 34 35 36 6745 .7002 .7265 6771 7028 7292 6796 7054 7319 6822 7080 7346 6847 7107 7373 6873 7133 7400 6899 7159 7427 6924 7186 7454 6950 7212 748i 6976 7239 7508 4 9 '3 4 9 '3 5 9 H 1 7 21 l8 22 l8 23 37 38 39 7536 7813 .8098 7563 7841 8127 7590 7869 8156 7618 7898 8185 7646 7926 8214 7673 7954 8243 7701 7983 8273 7729 8012 8302 7757 8040 8332 7785 8069 8361 8662 5 9 '4 5 1 14 5 i i5 l8 23 I 9 2 4 20 24 40 .8391 8421 8451 8481 8511 8541 857i S6oi 8910 9228 9556 8632 5 I0 J 5 20 25 41 42 43 .8693 .9004 9325 8724 9036 9358 87548785 9067 9099 9391 9424 8816 9131 9457 8847 9163 9490 8878 9195 9523 8941 9260 959<> 8972 9293 9623 5 10 16 5 ii 16 6 ii 17 21 26 xi 27 22 28 44 9^57 9691 9725 9759 9793 9827 9861 9896 9930 9965 6 ii 17 2 3 2 9 MEASUREMENT OF PHYSICAL QUANTITIES. TABLE XVIII. (Confd.) Natural Tangents. 143 6' 12 18 24' 30 36 42' 48 54 123 4 5 45 I.OOOO 0035 0070 0105 0141 0176 0212 0575 0951 1343 0247 0283 0319 6 12 18 24 3 46 47 48 1.0355 1.0724 1. 1 1 06 0392 0761 H45 0428 0799 1184 0464 0837 1224 0501 0875 1263 0538 0913 1303 0612 0990 1383 0649 1028 1423 0686 1067 1463 6 12 18 6 .13 19 7 13 20 25 3 25 3 2 26 33 49 50 51 1.1504 1.1918 1-2349 1544 1960 2393 1585 2OO2 2437 1626 2045 2482 1667 2088 2527 1708 2131 2572 I75C 2174 2617 1792 2218 2662 1833 2261 2708 1875 2305 2753 7 '4 21 7 '4 22 8 15 23 28 34 29 36 30 38 52 53 54 1.2799 1.3270 1.3764 2846 3319 3814 2892 3367 3865 2938 34i6 3916 2985 3465 3968 3032 35H 4019 3079 3564 4071 3127 3613 4124 3175 3663 4176 3222 3713 4229^ 18 16 23 8 16 25 9 17 26 3' 39 33 4 34 43 55 1.4281 4335 4388 4442 4496 4550 4605 4659 4715 4770 9 18 27 36 45 56 57 58 59 60 61 1.4826 1-5399 1.6003 4882 5458 6066 4938 5517 6128 4994 5577 6191 5051 5637 6255 5108 5697 6319 5166 5757 6383 5224 5818 6447 5282 5880 6512 5340 594i 6577 ip 19 29 10 20 30 II 21 3 2 38 48 40 50 43 53 1.6643 1.7321 1.8040 6709 739 1 8115 6775 7461 8190 6842 7532 8265 6909 7603 8341 6977 7675 8418 7045 7747 8495 7H3 7820 8572 7182 7893 8650 725J 7966 8728 " 23 34 12 2 4 3 6 13 26 38 45 56 48 60 5' 64 62 63 64 1.8807 1.9626 2.0503 8887 97" 0594 8967 9797 0686 947 9883 0778 9128 9210 9970 6057 0872 0965 9292 0145 1060 9375 0233 "55 9458 6323 1251 9542 0413 1348 14 27 41 IS 29 44 |i6 31 47 55 68 58 73 63 78 65 2.1445 1543 1642 1742 1842 1943 2045 2148 2251 2355 \'7 34 5 68 85 66 67 68 2.2460 2-3559 2-4751 2566 3673 4876 2673 3789 5002 2781 396 5129 2889 4023 5257 2998 4142 5386 3109 4262 5517 3220 4383 5649 3332 4504 5782 3445 4627 59 I(: 18 37 55 1 20 40 60 22 43 65 74 9 2 79 99 87 108 69 70 71 2.6051 2-7475 2.9042 6187 7625 9208 6325 7776 9375 6464 7929 9544 6605 8083 97U 6746 8239 9887 6889 8397 0061 7034 8556 0237 7179 8716 0415 7326 8878 0595 24 47 7' 26 52 78 29 58 87 95 "8 104 130 "5 M4 129 t6i 144 180 162 203 72 73 74 3-0777 32709 3-4874 0961 2914 5105 1146 3122 5339 1334 3332 5576 1524 3544 5816 1716 3759 6059 1910 3977 6305 2106 4197 6554 2305 4420 6806 2506 464^ 7062 ] 3 2 6 4 9 6 I 36 72 108 1 41 82 122 75 3.7321 7583 7848 8118 8391 8667 8947 9232 9520 9812 1 46 94 39 i 86 232 76 77 78 4.0108 4-33I5 4.7046 0408 3662 7453 0713 4015 7867 J022 4374 8288 1335 4737 8716 1653 5107 9152 1976 5483 9594 2303 5864 0045 2635 6252 0504 2972 664* 097C I 53 107 160 J 62 124 186 )J73 146 219 214 267 248 310 292 365 79 80 81 5.1446 5-67I3 6.313^ 1929 7297 3859 2422 7894 4596 292^ 8502 3435 9124 6122 3955 9758 6912 4486 0405 5026 1066 8548 5578 1742 9395 6i4C 2432 187175262 350 437 5350 7920 026^ ice - col- se to be wing to ity with ie value inge n t 82 83 84 7-II54 8.144.. 9-5I44 2066 2636 9.677 3002 3863 9-845 3962 5126 IO.O2 4947 6427 IO.2O 5958 7769 10.39 6996 9152 10.58 8062 0579 10.78 9!58 2052 10.99 3j>2 028' 357^ II. 2C 1 1 Differei 1 umns cea 1 useful, o I the rapic 1 which tl 1 of the t 1 changes. 85 11-43 11.66 11.91 I2.I6 1243 12.71 13.00 13-30 i3-9f 86 87 88 14.30 19.08 28.64 14.67 19.74 30.14 15-06 20.45 31.82 15.46 21.20 33.69 15.89 22.02 35-80 16.35 22.90 38-19 16.83 23.86 40.92 17-34 24.90 44.07 17.89 26.03 47-74 i8.4( 27.2: 52.0* 89 57-29 63.66 71.62 81.85 95-49 114.6 143-2 [91.0 286,5 573-< I >?c: ... *. - ] 7*~ t, . . *^- I I -4 1- y* / ,^U- ^fc-frj THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW RENEWED BOOKS ARE SUBJECT TO IMMEDIATE RECALL i ' b la/I LIBRARY, UNIVERSITY OF CALIFORNIA, DAVIS BookSlip-50TO-5,'70(N6725s8)458 - A-31/5 U8086 Minor, R.S. Physical measurements in the properties of Call Number: QC220 1*8086 Minor, R.S. Physical measurements in the properties of matter and in heat. QC220 M5 LIBRARY UNIVERSITY OF CALIFORNIA DAVIS