264 



MOTION UNDER A VARIABLE FORCE 



PN be drawn from P to a fixed diameter AA'. Then we shall find 

 that the point N moves backwards and forwards on the line A A ' 



with simple harmonic motion. 



The acceleration of P is, by 12, 

 an acceleration k*a along PO. This 

 can be regarded as compounded of 

 the acceleration of P relative to N t 

 which must be along NP, and the 

 acceleration of N relative to 0, 

 which must be along ON. Thus 

 the acceleration of N is that com- 

 Fia - 131 ponent of the acceleration of P 



which is in the direction AA 1 . This, however, is known to be 

 k z a cos 0, or k 2 ON, along NO. Putting ON = s, we have an 

 acceleration J<?s in the direction in which s is measured, namely 

 ON. Thus the point N moves with simple harmonic motion. 



This geometrical interpretation of simple harmonic motion 

 enables us to obtain expressions for v and s directly. The value of 

 s is ON, or a cos 0. Let t = e be an instant at which the point P 

 was passing through the point A! in its motion round the circle, 

 then, at any subsequent instant t, the tune since P was at A' will 

 be t e, so that the angle described by OP will be 6 = k(t e). 



Thus we have 



s = ON= a cosk(t - e). (98) 



This is the same result as is contained in equation (97). We 

 notice that the amplitude s of the motion is the same as the 

 radius a of the circle, and that the frequency k is identical with 

 the angular velocity. On differentiating equation (98), we obtain 

 at once for the velocity 



v = = ka sin k(t e) 

 at 



This result can also be obtained by resolving the velocity ka of 

 the moving point P into two components, along and perpendicular 



