48 



MOTION IN TERMS OF ACCELERATION. [CHAP. IV. 



circle, and when the moving point is at N on one diameter of 

 this circle, so that ON = x, draw NP at right angles to this 

 diameter, and consider the motion of the point P. 



Fig. 32. 



Let the angle NOP = 0. 



Then x, a cos 6, and y, = a sin 0, are coordinates of P. 



By differentiating we have 



x = a sin 6 6, x = a sin a cos 6 2 

 hence x = - (y0 + x6 2 ) ; 



since x = px, 



we must have = 0, and 2 = /*. 



Hence the point P describes the circle uniformly ; the angular 

 velocity of the radius vector is uniform and equal to tjp, and the 

 angle 6 t \J p. 



The distance of the point N from at time t is given by 

 x = a cos (t Jfi). 



The velocity of the point is directed along xO, and its magni 

 tude is a *J/jL sin (t\lfi). 



The above process shows that the solution of the equation 



00 &quot;&quot;&quot; , 



with the conditions that when t = 0, x - a, x = 0, is x = a cos (t \/fi). 



