352 



Then 



PRACTICAL MATHEMATICS 



?/= Ae 



= e~ 



or 



y = 



- "-a* (C cos dx + D sin dx) 



cos Vb* a 2 x + D sin 



178. Simple Harmonic Motion. This is denned as the motion 

 of a body in which the retardation is proportional to the distance 

 the body is from some given position ; and if x is this distance 



d?x 

 and t is the time, the equation of motion will be -^ = n*x, 



where ri 2 is the constant of proportion. 



The solution will be x = A sin nt + B cos nt, where A and B are 

 constants which can be evaluated if the initial conditions are 

 known. 



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



Since x = A sin nt + B cos nt 



and x = when t = 0. Then B = 0, 



Hence x = A sin nt 



also 

 but 



dx 



v = T- = nA cos nt 

 dt 



v = V Q when t = 0. 



Then nA = v n 



The final solution is x = sin nt. 

 n 



FIG. 115. 



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



n 



an angle nt to the vertical diameter, then P x , the projection of 



P on the horizontal diameter, is such that OPj = sin nt 

 (Fig. 115). 



