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and 7, to G, which is an electric bell arranged so as to strike whenever the circuit is closed.

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ing.

Experiment. Connect the battery B with the wires attached to Cand G. The bell will instantly strike. Start the pendulum DC. Whenever it passes through the mercury cup, that is, with every swing, the electric current passes through G and makes the bell strike. Stop CD and start the clock. The strokes now occur at intervals of exactly one second. Now set both pendulums goThe bell will strike only when both are vertical at the same instant. This will occur at regular intervals, equal to the time required by the longer pendulum to lose just one vibration. Record the minute and second of each stroke for ten or fifteen minutes consecutively. The first differences give the intervals, and from the mean of the latter the time of vibration may be computed with great accuracy. For example, if the interval is 47 seconds it denotes that in this time CD made one less, or 46 vibrations, hence the time of a single vibration would be 7=1.0217 seconds. An error of one second in the mean of the interval would make the time 4=1.0213 seconds, or alter the result less than a two-thousandth of a second. The method, therefore, is one of extreme precision. Sometimes, especially when the pendulum is swinging through a small arc, the bell will strike for several consecutive seconds, owing to the considerable interval of time during which contact is made at C, so that for several seconds the circuit is closed at F before it is broken at C. In this case the time of the first stroke should be recorded and their number; the true time being taken as the mean of the first and last. To make CD vibrate in one plane instead of describing an ellipse, attach a fine thread to the ball C; draw it to one side about ten inches; let it come to rest, and then burn the thread. Finally measure the length of the pendulum, or the distance from the knife-edge to the centre of the

Fig. 37.

ball, and compute the force of gravity g from the formula; t = in which t equals the time of vibration, and π = 3.1416.

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This experiment may be repeated with a different length of pendulum, or it may be varied so as to prove that the time increases with the amplitude. In the latter case the are through which the ball swings should be as large as possible, and it should be measured as it progressively diminishes. To compute the theoretical time of swinging through any arc a, divide versin a, or the vertical distance through which the ball moves, by its length, and call the quotient x. Then the time t' for any value of x may be found from the equation = (1 + x + z} 6 x2 + &c.) t, in which t is the time when the arc is very small. When a 180°, or the ball swings through a semicircle, t' 1.180 t, when a 30°, ť′ = 1.0063 t, when a = 10°, t' 10°, t' = 1.00067 t, hence for small arcs the correction for this cause is very small. If great accuracy is required in this experiment the suspending wire should be very light, and with the knife-edge should vibrate in about one second when the ball is removed, or a correction may be applied for them as described in Experiment 40.

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256

42. TORSION PENDULUM.

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Apparatus. AB, Fig. 38, is a vertical wire with an index C, which moves over a graduated circle. Weights of a cylindrical form, as D, may be attached below in such a manner that the wire cannot twist without turning them. To vary the length of the wire it is passed around several small brass tubes E, F, G, placed at different heights, so that it may be clamped at these points by inserting a pin G passing into a hole bored behind them. A scale and clock beating seconds are also needed for this experiment.

Experiment. 1st. The time is independent of the amplitude of the vibration. Use the whole length of the pendulum, and apply such a weight that the time of a single vibration shall be about one second. Twist the index through a small arc, and take the time of one hundred oscillations by noting the position of the index at the beginning of a minute, and the exact time, when after making one hundred single oscillations, it

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Fig. 38.

again reaches the same point. Dividing the interval by one hundred gives the time of a single oscillation. Repeat two or three times with arcs of different magnitudes, and compare the results. 2d. The time is proportionate to the length of the wire. Make the same experiment, first with the wire of its full length, then, passing the pin through the different tubes E, F, clamping it at these points. Measure their distances from B, and compare with the law. In the same way the relation of the time to the diameter of the weight, or to its length, may be tested and compared with theory.

MECHANICS OF LIQUIDS AND GASES.

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43. PRINCIPLE OF ARCHIMEDES.

Apparatus. An inverted receiver A, Fig. 39, with a stopcock, or better, an !" gas valve below. Near the top is placed a hook C with a sharp point, which is used to mark the level of the liquid. The whole may be hung from the scale-pan D of a large balance, EF, which has a counterpoise attached to the other end. G is a beaker to collect the water drawn off, and H a stand by which A may be supported if necessary. A set of weights is needed, also two bodies M and N, one heavier, the other lighter than water. They may be made of metal and wood, or, if preferred, of glass, and loaded so that one shall float, the other sink.

Experiment. 1st. A heavy body when immersed is buoyed up by a force equal to the weight of the displaced liquid. Place the receiver on the stand, fill it with water and

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draw out the latter until the point of the hook
just touches the surface, observing the point
of contact, as in Experiment 13. Place the
beaker G on the scale-pan D, suspend M be-
low it, and add weights to the other side so
as to bring the beam into equilibrium.
now the receiver is brought up under M the
water will rise, and the equilibrium will be
destroyed. Open B and draw off the water
into G until M, being immersed, the level ist
again exactly at C. Now replacing G on D
it will be found that the equilibrium is re-
stored. Hence the loss of weight of Mequals
the gain of G, or the weight of the displaced
liquid, since the level is unchanged.

Fig. 29.

2d. Since action and reaction are equal, the vessel appears to

90

RELATION OF WEIGHTS AND MEASURES.

gain in weight by an amount just equal to the loss of M. Suspend A from the scale-pan, and M from the stand. Bring the water-level to C and counterpoise as before. Immerse M, when the water will rise, and the weight apparently increase. Open B therefore, and draw out the water until the level is restored, when it will be found that the beam is again balanced, showing that it was necessary to draw out a volume of liquid equal to that of M. 3d. A floating body displaces a weight of liquid just equal to its own. Rest A on its stand and restore the water level to C. Place G and N on the scale-pan and counterpoise. Let N float in A, open B until the proper level is attained, collect the water in G, and replacing the latter on the scale it will be found that the equilibrium is restored. That is, the weight of the displaced water, or the increase of G, equals the weight of N.

44. RELATION OF WEIGHTS AND MEasures.

Apparatus. A delicate balance with a counterpoise on one side and scale-pan on the other, below which a small cube of brass is suspended by a very fine platinum wire. In addition, a beaker containing distilled water, a thermometer and a set of weights, must be provided.

Experiment. By definition a gramme is the weight in vacuo of a cubic centimetre of distilled water, at the temperature of maximum density, that is, about 4° C.; the object of the present experiment is to test this relation. Add weights to the scale-pan until equilibrium is established; then immerse the cube in the distilled water, first washing it with caustic potash to remove the air, then very thoroughly with common water to remove the potash, and finally with distilled water. The weight now required to counterpoise it will be greater than that previously taken, by an amount equal to the weight of the displaced water. Record the height of the barometer and the temperature of the water. Next, to determine the volume of the cube, measure the twelve edges very carefully to tenths of a millimetre, and take the mean of each set of four which are parallel. The product of these three means equals the volume. The dividing engine should be used to attain sufficient accuracy in this measurement. Add to this the volume of the wire found by multiplying its cross section by the length

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