form. It is found by tapping the body so that it will move, and seeing if the velocity increases or diminishes. In the first case the weight in E is too large, in the second too small.

The first law of friction is that the friction is proportional to the pressure. The ratio of these two quantities is called the coefficient of friction. Make the load on C, including its own weight, equal to 1, 2, 3, 4, 5 kgs. in turn, and measure the friction. The latter equals the weight of E plus the load added to it minus the friction of the pulley. If great accuracy is required, a table should be prepared, giving the magnitude of the latter for different loads. Compute the coefficient of friction from the observations, and if the law is correct they should all give the same result. Measure, in each case, the friction of repose and of motion, and notice that the latter is always much the smaller.

Secondly, the friction is independent of the extent of the surfaces in contact. This law follows from the preceding, but it is well to prove it independently by turning on its different sides, so as to vary the areas in contact. The friction will be found to be the same in each case. Finally, measure the coefficients in a number of cases, and compare the results with those given in the tables of friction. Of. Ball "Exp. Mach"

31. ANGLE OF FRICTION.

Apparatus. In Fig. 27, AB is a stand with an upright BC. AD is a board hinged at A, which may be set at any angle by a cord passing over the pulley C. The hinge is best made of soft leather held by a strip of brass, and its distance from the upright should be just one metre. A scale of millimetres is attached to the upright, and a wire parallel to AD serves as an index. Evidently the reading of the scale gives the natural tangent of the angle of inclination DAB. A cord attached to D passes over the pulley C, around the wheel F, and is stretched by the counterpoise G. F may be clamped in any position, or turned by a crank attached to it. AD may thus be set at any angle, and its position is to be determined when so inclined that any given body, as E, is just on the point of sliding. E may be made exactly like the sliding mass in Experiment 30, but to measure its friction of motion a fine wire should be attached to it and wound around the axle of F. When the crank is turned raising AD, the body E' is thus drawn slowly down the inclined plane. In order that it may not move too rapidly this portion of the axle should be much

smaller than that around which the cord CF is wound. EF should be a wire, as if a thread is used it will stretch, giving E an irregular motion, alternately starting and stopping.

E

C

Experiment. To measure the coefficient of friction of repose, turn the crank until the body begins to slide; the reading of the scale gives the tangent of the angle of inclination. But decomposing the weight of E into two parts, parallel and perpendicular to the plane, their ratio will equal the coefficient of friction, and also the tangent of the inclination. Hence the coefficient of friction is given directly by the scale. Meas

Fig. 27.

B

ure again the coefficients found in Experiment 30, and see if the results agree with those then obtained. Ball. p.8%.

32. BREAKING WEIGHT.

Apparatus. In Fig. 28, B is a thumb screw, by which a spring balance A may be drawn back so as to exert a strain on the wire CD. Near C is placed a spring buffer so that when the wire breaks, the jar may be diminished. D may be a simple peg to which the wire is attached, or a spool with a clamp by which it is held at any point. A convenient method of connection is to attach one end of the wire to a chain, either link of which may be passed over the peg D, according to the length employed. To test the accuracy of the balance a cord may be substituted for CD, passed over a pulley E, and stretched by weights. Some cord and fine copper or iron wire of various sizes should be furnished, also a gauge to measure its diameter.

Experiment. Fasten a cord to C, and pulling it over E, record the reading of A, when by attaching weights, the strain is made in turn 0, 5, 10, 15, 20, &c., pounds. To eliminate the friction of the pulley, turn B first in one direction and then in the other, and take the mean of the readings of A in each case. Now construct a residual curve, in which abscissas represent the reading of A,

and ordinates the difference between this reading and the weight applied. From this curve we can readily determine the true strain, knowing the reading of A, however inaccurate the latter may be.

B

Fig. 28.

D

To measure the breaking weight of any body, attach one end of it to C and the other to one link of the chain. Pass the latter over the peg D, so that C shall be a short distance from its spring buffer. Turn B and watch the index of A, until the cord breaks. A small block of wood may be placed in front of the index to show the greatest tension attained, but care must be taken that it is not disturbed by the recoil. Repeat several times with other portions of the same cord and take the mean of the observed maximum tensions. Do the same with some specimens of wire, and compute their tenacity T, or strength per square inch. For this purpose measure their diameter d with accuracy, by the gauge, and calling W the breaking weight, we have, 4 W πα

πα

4

:1= W: T, or T

33. LAWS OF TENSION.

Apparatus. In Fig. 29, A is a cast-iron bracket firmly fastened to the wall. A hook is attached at B, and from it the scale pan E is hung by the wire or rod to be tested. Weights may then be applied so as to give any desired tension to the latter. A brass rod BD hangs by the side of BC, being fastened to it by a small clamp at the top. Fine lines are drawn on both rods, and their relative change in position measures the elongation of BC. G is a reading microscope made of a brass tube about 6" long, with the Microscopical Society's screw cut in one end, so that any microscope objective may be used with it. A positive eyepiece with a scale at its focus is slipped into the other end. This microscope is mounted like a cathetometer, by fastening it to a vertical brass tube screwed into the base of a music stand. It may be raised or lowered, and held at any point by a set screw. The following additional apparatus is also needed. Several wires or rods of various materials, as wood, copper, brass, iron, lead, and some of the same material but different diameters. A millimetre scale to measure their lengths, and a Brown & Sharpe's sheet metal gauge to give their diameters. Also a set of large weights to vary the ten

sion. To prevent too sudden a jar on the wire, a board should be placed under E to support it when a weight is added, then lowering it by means of a screw.

A

G

Experiment. To measure the extension in any case, attach the wire to be tried to the hooks at B and C, and clamp BD to its upper end. Draw a line on BC opposite one of those on BD, and focus the microscope G on them. Read their relative posi tion by means of the scale in G, then apply the weight and read again. The length of BC is thus increased, while that of BD is unaltered; hence the change in their relative distances equals the extension. To reduce this to millimetres, focus the microscope on a standard millimetre, and thus measure the scale in G directly. The distance from F to the clamp is measured by the millimetre scale, and the diameter of BC by the gauge.

Fig. 29.

The laws of tension may now be determined.

1st. The extension is proportional to the length. Use a copper wire about a millimetre in diameter, and mark on it a number of lines at different heights. Measure the extension for each with a load of 20 kgs. It will be found proportional to the distances from the clamp. 2d. The extension is proportional to the weight applied. Measure the deflection for the lower lines, increasing the weight 2 kg. at a time, from 0 to 20 kilogrammes. Removing the latter weight, see if the wire has returned to its original length; any increase is called the permanent set. See if the results agree with the law. 3d. The elongations are inversely proportional to the cross-section. Try the series of wires of the same material, measuring the diameter of each with the gauge, and using the same weight for all. The product of the square of the diameters by the elongation should be constant. The modulus of elasticity is the force which would be required to double the length of a body of cross-section unity, supposing this could be done without breaking it, or changing the law which holds for small weights. To compute it, supposed the diameter, the length, and e the elongation of a wire under a tension T

If the cross-section was

unity, to produce the same elongation we should increase the force

4T

in the same ratio as the two sections, or make it

πα

various substances provided, and compare the results with those given in the books. Finally, with an undue load the wire will take a permanent set, which increases if the wire is stretched for a considerable time. Study the laws regulating this property in the case of lead, in which the set is very marked.

ter.

34. CHANGE OF VOLUME BY TENSION.

Apparatus. A rubber tube AB, Fig. 30, about a metre long, and two or three centimetres in diameter, is closed above and below by plugs. The upper one is perforated, and carries a glass tube with graduated scale attached. A scale is also placed by the side of the rubber tube, and a number of points are marked on the latA cord and friction pin C (like that of a violin) is fastened to the lower plug, by which the tension of the tube may be varied. On the other side of the scale is a square rod DE of elastic rubber, about the size of the tube, and similarly marked. A pair of outside calipers capable of measuring objects as large as the rod to within a tenth of a millimetre, is also needed.

Experiment. The tube should be calibrated by weighing it when empty, and when filled with water to the zero, or beginning of the glass tube, also when filled to some division n, near its top. Call these three weights w', w', w'". Then w"

of a cylinder of water just filling the tube, or the known length, and r the radius.

= r2l, we obtain r =

w'

= the weight

r2l, in which 7 is From the equation w" - 20′

The volume per unit of length w'll

w'

πι

w"

and in the same way for the glass tube it is

Call the ratio of these two, or

b. So much of the

w"

w" l w' n work may be done once for all. Any change in volume of the interior of the tube can be accurately measured by noting the change of level in the glass tube.

Fill the tube with water to the point marked n, and read the position of the marks. Stretch it by turning the pin below so as