SIMPLE HARMONIC MOTION 



If a particle P describes a circular path of radius -? ft. with 



uniform angular velocity n radians per second, the projection 

 of this circular motion on the horizontal diameter will satisfy 



*U d^ 



the condition that x = sin nt, and consequently -5-5 = n 2 <r. 

 n dt 2 



The body describes a complete oscillation, after passing from 

 O to P 2 , from P 2 through O to P , and from thence backwards 



again to O. This would take the same time as a complete revolu- 

 tion in the corresponding circular motion. 

 The periodic time = time of one complete revolution 



T=^H 



n 



The frequency = number of complete oscillations per second 



(2) Let the initial conditions be x = a and v = when t = 0. 

 Then x = A sin nt + B cos nt 



and x = a when t = 0, then B = a 



Also 



fj'Vt 



-j- = nA. cos nt riB sin nt 

 at 



and v = when t = 0, then A 



The final solution is x = a cos nt. 



FIG. 116. 



If a circle be drawn of radius a and OP is a radius inclined at 

 an angle nt to the horizontal diameter (Fig. 116), then Pj, the 

 projection of P on the horizontal diameter, is such that OP l 



