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CHAPTER I

INTRODUCTION-VIBRATORY MOTION

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1. Sound caused by Motion. If you examine any sounding body you will soon come to the conclusion that the sound is produced by motion of some kind. In many cases the motion is obvious and can be followed by the eye. example, if you take hold of a stretched string between your finger and thumb, pluck it to one side and then let it go, the string will give out a sound and, at the same time, will be seen to swing from side to side. If the rate of motion is very rapid you may not be able to follow each separate swing, but the fact that the string is in motion is shown by its losing its definite outline and presenting the appearance of an indistinct gauzy spindle. By touching the string with the finger the motion can be stopped, and it is then found that the sound ceases at the same time.

The same indistinct appearance is presented by the prongs of a tuning-fork while it is sounding. When any hard object, such as the point of a knife, is held against the prongs, a loud rattling noise is heard; and quite a little shower of spray is thrown up when the points of the fork are dipped into water. If you want further proof of the motion you can easily get it by striking the fork and then making it touch your lips or teeth.

2. The kind of Motion.-A tuning-fork consists essentially of two steel springs united together at one end, and each prong of the fork moves in much the same way as a single straight spring clamped at one end. By taking a long thin

spring (instead of the comparatively short and thick prong of the tuning-fork) we can make the motion slower and examine it at our ease.

EXPT. 1.-Clamp the lower end of the spring in a vice. The spring is elastic: it bends or 'gives.' In order to bend it you have to exert force upon it. In virtue of its elasticity the spring tends to recover its original form, and if you wish to keep it bent you must continue the pressure against it.

Pull the upper end to one side, as at 1, Fig. 1. で

Now let the spring go. It flies back but it does not stop when it has got back to its original position (vertical). It overshoots the mark and moves, with gradually diminishing velocity, to about an equal distance on the opposite side (l). It then begins to return, again overshoots the mark, gradually comes to rest, starts back again, and so goes on swinging from side to side. Owing to the resistance of the air, etc., the amplitude of the excursions gradually diminishes, and finally the spring comes to rest.

Fig. L

3. Vibration. When a body or point moves in the manner above described it is said to vibrate or oscillate. Referring again to Fig. 1, a movement from / to land back again to is called a complete vibration. The amplitude of the vibration is measured by the extent of the motion on either side of the mean position or position of rest, i.e. half the distance ll'.

If you take a watch and examine the rate of vibration of the spring, you will find that this rate is regular and independent of the amplitude of vibration. Whether the excursions made by the end of the spring are great or small, the number of vibrations executed in a given time is constant. Thus the vibration is regular or periodic. The time required to perform

a complete vibration is called the period of vibration. We shall denote this time (measured in seconds or a fraction of a second) by T. If the number of complete vibrations performed in a second be denoted by n, then

I

T=-
n

We may call n the frequency of vibration or the vibrationnumber.

4. Vibration of Pendulum.-The periodic vibration of an elastic spring is very similar to that of a pendulum swinging under the action of the earth's attraction. When the bob of the pendulum is displaced to one side and then let go, it swings to a nearly equal distance on the other side, stops, returns, and goes on swinging from side to side regularly, but with the amplitude of the oscillations gradually diminishing, and finally the pendulum comes to rest. The vibrations are regular the period of vibration is constant and independent of the amplitude. As in the case of the spring, each complete vibration may be divided up into four similar parts.

The pendulum is in its mean position, or position of rest, when the string is vertical and the bob is at its lowest point (N, Fig. 2). Now suppose the bob to be displaced to one side, say to A. The bob not only moves to the left but rises, for it moves in an arc of a circle. In order to raise it through the distance HA you have to exert force and do work in lifting the weight of the bob upwards. You have thus given to the bob the power of doing an equal amount of work in falling through the same distance, i.e. to its position of rest. It possesses energy of position, or potential energy.1 When the pendulum is released it moves with gradually increasing velocity, and this velocity is greatest when the bob is at its lowest point. The energy has now been changed from the potential into the kinetic form. It is this kinetic

1 The energy of a body is its capacity for doing work. A body is said to possess potential energy when it is able to do work in virtue of its position. A raised weight (as in a clock) and a head of water are examples of bodies possessing potential energy. So also are coiled springs, compressed air, etc.; so that position must here be held to include change of form or volume.

A body is said to possess kinetic energy when it is able to do work in virtue of its motion. Thus the wind does (useful) work in turning the sails of a wind-mill; a cannon-ball does (destructive) work in crashing through the walls of a fort.

energy possessed by the bob that prevents it from coming to rest at its lowest point and carries it over to the other side (A'), at the same time lifting it up against the action of gravity. At the end of the first swing (or semi-oscillation) the bob comes to rest for an instant and then retraces its path. In the first quarter-period the energy changes from the potential to the kinetic form, and in the second quarter-period from the kinetic to the potential form. In the third and fourth quarters the same changes are gone through with velocities reversed.

Observe that the velocity is least (zero) at the points where the displacement is greatest (A and A'): the velocity is greatest when the displacement is zero (N).

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5. Simple Straight-line Vibration.-If the distance through which a pendulum swings is small compared with the length of the string, the path of the bob is very nearly a straight line. The following is a better illustration of such

vibration.

EXPT. 2.- -Coil some thin wire round a pencil or corkborer. Hang the spiral up and to its lower end attach a lead bullet (N, Fig. 3). Pull the bullet down to A and then let it go. It flies up to A' and continues for some time to oscillate up and down along a vertical line. This vibration along a straight line may be regarded as a type of the kind of motion which accompanies sound-waves in air.

6. Elasticity.-The force tending to bring the pendulum back to its position of rest is due to the attraction of the earth on the bob. In the case of the spring the force is due to the elasticity of the spring itself. When the spring is bent or pulled out, work is done in overcoming the resistance due to its elasticity, and this work is stored up as potential energy in the spring itself. After being released it moves in the same way and goes through the same changes of energy as the pendulum. Like the latter it is gradually brought to rest by resistances due to the air, etc.

The elasticity of a body may be defined as being that property in virtue of which it requires force to change its form or volume, and recovers its original form or volume when the force is removed.

Solids in general offer resistance to any change either of form or volume, and are therefore said to possess elasticity of form as well as elasticity of volume. Fluids exhibit the latter only, for they do not offer resistance to change of form (see p. 27).

In common language we speak of substances like indiarubber as being very elastic, because they 'give' or stretch readily. In scientific language the term is used in quite a different sense. Without entering into any details of the methods of measuring elasticity, we may here state generally that the elasticity of a substance is measured by the force required to produce a given change of form or volume. If the substance yields readily when force is applied, its elasticity is said to be small: but if it offers great resistance, it is said to be highly elastic. Thus the elasticity of steel and glass is great, for both these substances require the application of considerable force to produce even a small change of form. Again, the elasticity of water is very much greater than that of air if the atmospheric pressure were to be doubled, the volume of any given quantity of air would (according to Boyle's law) be reduced to one-half; whereas such a change of pressure would produce scarcely any appreciable effect upon the volume of a given quantity of water.

7. Graphical Representation of Vibration.—The following experiment shows how the vibration of a tuning-fork may be studied and represented graphically.

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