1770 



PLANE MOTION. 



95 



Conversely, it can be shown that if the acceleration be pro- 

 portional to the distance from a fixed point in the direction of 

 the initial velocity, and if it be directed towards this point, the 

 motion is simply harmonic. For we then have 



ar 



dt* 



where fi is constant. The general integral of this differential 

 equation is (compare Art. 122) 



x= 



sn 



C 2 cos 



Differentiating, we find for the velocity 



v= C^i cos i^t C^fi sin fit. 



To determine the constants of integration C v C^ let s=s^ 

 and V = VQ at the time /=o. Substituting these values, we find 

 r =6* 2 and v^fiC^ hence 



x= ^ sin fit+ s cos fit. 

 1* 



Putting v Q /fi=acose, s Q =a sin e, which is always allowable,. 



|ve obtain 



x=a (sin fit cos e+ cos fit sin e) 



a sn 

 This represents a simple harmonic motion whose amplitude 



2 /yLt, and whose epoch-angle is e= tan' 1 ^/^). 

 is the angular velocity of the corresponding uniform circular 

 potion is //,, the period is T 2 TT//^. 



177. If the uniform circular motion of P be projected on the 



[ameter BB', which is at right angles to the diameter A A' 



'ig. 44), we have OP y =y = a sin (o)/+e). Writing this in the 



liuivalent form 



f Tr\ 



y a cos f w/H-eH J, 



