CHAPTER VI. 



Composition of Periodic Curves of Different Amplitudes but of same Periodic Time. 



GRAPHICAL METHODS. 



3O. Consider a point P moving in a counter-clockwise direc- 

 tion with uniform angular velocity p round a circle of centre 

 (Fig. 9). Suppose that at time t = o, P is at the point A, 

 and that at any subsequent time t it occupies a position such 

 that the angle A OP = pt. Let AOA' and BOB' be two diameters 



at right angles to each 

 other, and let PN be 

 drawn at right angles 

 to AA. Let OP = r. 

 Then 



PN=OPsinAOP 



= r sinpt 



Therefore PN is a 

 sine function of the 

 time and r is its maxi- 

 mum value. 



As P travels round 

 the circle the value of 

 PN changes from zero 

 to its maximum value OB, diminishing to zero when P reaches A', 

 after which it changes in sign, attaining a negative maximum 

 when at B', and completing its cycle at A. 



If we plot a curve having values of pt as abscissse and the 

 corresponding values of PN as ordinates, we obtain the usual curve 

 of sines shown in Fig. 3. 



Suppose now that a second point Q travels round a concentric 

 circle of radius / starting from a t' seconds later than P starts 

 from A ; then at time t the position of Q will be such that the 



FIG. 9. 



