SIMPLE HARMONIC MOTION.

5. We know that a particle moving with uniform velocity in a circle is acted on by a constant force directed towards the centre.

Let p be the particle, and c the centre. Draw PX and P y perpendiculars on two fixed diameters A A', B B'. The force acting on P can be resolved into two components, one parallel to x c, and the other to y c, and these are represented by the lines x c and y c on. the same .scale on which the whole force is represented by PC. Hence the component force parallel to A A' is proportional to x c. But the P parallel to A A' is the same as the Hence the point x moves as if urged towards c by a force proportional to x c. The point x therefore executes simple harmonic motion.

FIG. 1.

This construction shows that, when a particle executes simple harmonic motion, if a circle be described upon the path of the particle as diameter, and a perpendicular to the path be drawn from the particle, the point in which this perpendicular meets the circle will travel round the circle with uniform velocity. The circle so described is called the auxiliary circle.

6. The greatest velocity of the vibrating particle will be at c, and will be equal to the velocity of the revolving point.1 In the extreme positions A and A', the velocity of the vibrating particle vanishes. The whole motion naturally divides itself into four parts, corresponding to the four quadrants in the figure, and these four parts are reversed copies of each other.

The distance CA or CA' from either of the extreme positions of the particle to the central position is called the amplitude of the vibration. It is the same as the radius of the auxiliary circle.

The period of the vibration is the time of moving from A to A and back, and is the time of a complete revolution in the auxiliary circle.

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7. Let the force acting on the vibrating particle be such as to produce an acceleration μx when the distance from the centre is x. The factor will evidently be constant, since, from our definition of simple harmonic motion, the force is proportional to x. Now it is proved,

in treatises on dynamics, that the acceleration of p is

(T denoting the period), and is directed along PC. Its component along x c is

1 The velocity of P, and the two components of this velocity parallel to AA and B B', are perpendicular to the sides of the triangle C Y P, which may therefore be taken as the triangle of velocities.

the velocity of x.

Hence Cy represents

and, since P and x have identical motions in this direction,

this is also the acceleration of x.

That is, we have

The period therefore depends only on μ and is independent of the amplitude. In other words, the vibrating particle will make the same number of vibrations in a given time whether its excursions be large or small. This equality of period for all amplitudes is called isochronism. It is a general law that the small vibrations of any elastic body are isochronous; and the physical cause of this isochronism is found in the fact that the elastic resistance is proportional to the displacement. This latter fact is known as Hooke's law. It was experimentally discovered by Hooke in the case of the extension of elastic strings, and was expressed by him in the formula Ut tensio sic vis (As the extension, so is the force).

8. It is evident that the motion of the point y in Fig. 1 is similar to that of x, and that the motion of p is the resultant of the two. If p moves round the circle in the direction A B A' B', then y will come to its extreme position B a quarter of a period later than x comes to A. Hence we have the following proposition:

Two equal simple harmonic motions, at right angles to each other, differing in phase by a quarter of a period, compound into uniform circular motion.

9. If 0 denote the angle which c p has swept out since coinciding with CA, and a the radius of the circle, we have

Hence, simple harmonic motion may be defined as motion in which the displacement from the mean position is proportional to the sine or cosine of an angle which varies as the time.

10. If is measured from any fixed radius c E (Fig. 2), we shall have

x=a cos (0-€),

y=a sin (0-e),

where e denotes the angle described in moving from C E to CA; either of these formulæ may, therefore, be employed as the general expression for simple har

monic motion, it being always understood that is directly proportional to the time. If t denote the time occupied in describing the angle 0, and T the period, we have

FIG. 2.

since is described in time t, and 27 in time T.

(2)

In accordance with the usage of the best modern authorities, we shall adopt as the standard formula

0-e, or its equivalent in time, the phase, and a (as already stated) the amplitude.

Substituting for its value from (2), the formula

denotes simple harmonic motion, whose period T is deter

II. From the analogy of (5), which is the general equation of simple harmonic motion, one variable y is said to be a simple harmonic function of another x, when the connection between them can be expressed by the equation

y=a cos (nx—e€).

(7)

12. The curve of which (7) is the equation, is called the simple harmonic curve. It is the curve which is traced, by a point executing simple harmonic vibrations, upon a sheet of paper travelling uniformly in a direction at right angles to the line of vibration.

For if the paper travels with velocity v in the direction of the negative axis of x, the tracing point will travel relatively to the paper in the direction of the positive axis