Laws of Oscillation of the Pendulum.

54. The oscillations of the pendulum take place in accordance with the following laws:

1. For pendulums of unequal lengths, the times of oscillution are proportional to the square roots of their lengths.

2. For the same pendulum, the time of oscillation is independent of the amplitude, provided the amplitude be small.

For pendulums of the same length, the time of oscillation is independent of the nature of the material.

Pendulums of wood, iron, copper, glass, all being of the same length, will all oscillate in the same time.

4. For the same pendulum at different places, the times of oscillation are inversely as the square roots of the force of gravity at those places.

These laws are deduced from a course of mathematical reasoning on the theoretical simple pendulum, but they may be verified experimentally by employing a very small ball of platinum, or other heavy metal, and suspending it with a very fine silk thread.

To verify the first law with such a pendulum, we begin by making it vibrate, and then counting the number of vibrations in one minute. Suppose, for example, that it makes seventy-two per minute. Now make the string four times as long as before, and it will be found that the pendulum makes only thirty-six oscillations per minute. If the string is made nine times as long as in the first instance, it will be found that the pendulum makes only twenty-four oscillations per minute, and so on. In the second case the time of oscillation is twice as great, and in the third case it is three times as great as in the first case. Now, because two, three, &c., are the square roots of four, nine, &c, it follows that the law is verified.

To verify the second law, let the same pendulum cscillate, at first

(54) What is the first law of vibration? The second law? trate. The fourth law? How are these laws deduced? verified? How is the second law verified?

The third law? Illus

How is the first law

case.

3.

through an arc, pn, and then through any other arc, rg; it will be found that the number of oscillations per minute is the same in each Hence the law is verified. It is to be observed that the law does not hold true unless the arcs, pn and rg, are very small, that is, not more than three or four degrees.

The property of pendulums, that their times of oscillation are independent of the amplitude of vibration, is designated by the name isochronism, from two Greek words signifying equal times; oscillations performed in equal times are called isochronal.

GALILEO first discovered the fact that small oscillations of a pendulum were isochronal, towards the end of the sixteenth century. It is stated that he was led to the discovery by noticing the oscillations of a chandelier suspended from the ceiling of the Cathedral of Pisa.

Applications of the Pendulum.

55. On account of the isochronism of its vibrations, the pendulum has been applied to regulate the motion of clocks. It was first used for this purpose in 1657, by HUYGHENS, a Dutch philosopher. The motive power of a clock is sometimes a weight acting by a cord wound around a drum, and sometimes a coiled spring similar to a watch spring. These motors act to set a train of wheel-work in motion, which in turn imparts motion to the hands that move round the dial to point out the hour. It is to impart uniformity of motion to this train of wheel-work that the pendulum is used.

Fig. 38 shows the mechanism by means of which the pendulum acts as a regulator. A toothed wheel, R, called a scape wheel, is connected with the train driven by the motor, and this scape wheel is checked by an anchor, mn, which is attached to the pendulum and vibrates with it. The anchor has two projecting points, m and n, called pallets, which engage alternately with the teeth of the scape wheel, in such a manner that only one tooth can pass at each swing

Limitation. What is isochronism? When are vibrations isochronal? Who discovered the pendulum, and when? (55.) What is the principal use of the pendulum? What is the motor in a clock? What is the use of the pendulum? Explain the action of the pendulum as a regulator.

of the pendulum. The motor turns the scape wheel in the direction of the arrow until one of the teeth comes in contact with the pallet, m, which stops the motion of the wheel-work

A

till a swing of the pendulum lifts the pallet, m, from between the two teeth, when a single tooth passes and the wheel-work moves on until again arrested by the pallet, n, falling between two teeth on the other side. second swing of the pendulum lifts out the pallet, n, suffers another tooth to pass, when the wheel-work is again arrested by the pallet, m, and so on indefinitely. The beats of the pendulum being isochronous, the interval of time between the consecutive escape of two teeth is always constant, and thus the motion of the wheel-work is kept uniform. The loss of force which the pendulum continually experiences, is supplied by the motor through the scape wheel and the anchor. This is called the sustaining power of the pendulum. Owing to expansion and contraction from variations of temperature, the length of the pendulum varies, and according to the first law, its time of vibration changes. In nice clocks this change is compensated by a combination of metals. In common clocks, it is rectified by lengthening or shortening the pendulum by a nut and screw, shown at v, by means of which the lenticular bob may be moved up and down. In summer the pendulum elongates and the clock loses time, or runs too slow; this is rectified by screwing up the nut and shortening the pendulum. In winter the pendulum contracts and the clock gains time; this is rectified by unscrewing the nut and lengthening the pendulum.

What effect have variations of temperature on the pendulum? How are these effects.compensated in nice clocks? How in common clocks? Why do clocks lose time in summer and gain time in winter?...

In accordance with the principle enunciated in the fourth law, the pendulum has been used to determine the intensity of gravity at different points on the earth's surface. In this way it has been shown that the velocity acquired by a body falling in vacuum for one second, is 32 feet, in the latitude of the city of New York. It has been found by careful experiment that the length of a pendulum vibrating seconds in New York, is a little over 39 inches.

The length of the seconds pendulum at any place being constant, it has been taken as the basis of the English system of weights and measures, and from the English we have taken our own system.

The pendulum has

been successfully employed by M. FouCAULT, a French physicist of our own day, to demonstrate the daily rotation of our globe. The details. of his experiment are too abstruse to be given in this place.

The Metronome.

56. The METRONOME is a sort of pendulum employed by musicians and others to mark equal intervals of time. It is shown in Fig. 39. It consists of a pen

Fig. 39.

Frig 38.

What principle enables us to measure the force of gravity? How far does a body fall in one second? What is the length of a seconds pendulum in New York Application to weights and measures? What application did FOUCAULT make of the pendulum? (56.) What is a Metronome?

dulum CB, suspended at 0. A weight, A, slides along the rod C, and may be set so as to make the vibrations as slow or as rapid as may be desired. The instrument is set by means of a scale, marked on the rod, so that any number of oscillations may be made in a minute. The pendulum is sustained by a coiled spring which sets, in motion a train. of wheels, somewhat in the manner of a clock. In the drawing the weight is set at 92, which shows that it is to make 92 oscillations per minute.

IV. PRINCIPLES DEPENDENT ON MOLECULAR ACTION.

Molecular Forces.

57. BESIDES the forces which act upon bodies from without and at sensible distances, there is another class of forces continually exerted between the molecules of bodies, and acting only at insensible distances. These forces are called Molecular Forces, and are both attractive and repellent.

The molecules of bodies are held in equilibrium by these forces, and it is to them that are to be attributed many of the most important physical properties. The ultimate particles of bodies do not touch each other, being kept asunder by a force of repulsion, which we have said is in general due to heat; they are prevented from receding from each other too far by a force of attraction, and it is only when these forces just balance each other throughout the body, that it is in equilibrium.

When a body is compressed, the forces of repulsion are called into play, and, acting like coiled springs, they tend to restore the body to its primitive form. In like manner, when a body is elongated, or stretched, the forces of attraction are called into action and tend to restore the body to its primitive form.

Describe it. (57.) What are Molecular Forces? How divided? How are molecules held in place? To what is the repellent force due? Explain the effects of compressing and stretching bodies.