14.11 Figures 14.25 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.

Fig. 14.25

Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.

As we can see in figure (a):
Timeperiod,T=2s
Amplitude,A=3cm
Attimet=0,phaseangleφ=+π2
Therefore,thedisplacementequationforthex-projectionofOP,attimet:
x=Acos[2πtT+ϕ]=3cos(2πt2+π2)=3sin(2πt2)x=3sinπtcm
Forfigureb:
Timeperiod,T=4s
Amplitude,a=2m
Attimet=0,phaseangle,Φ=+π
Therefore,thedisplacementequationforthex-projectionofOP,attimet:
x=Acos2πtT+ϕ=2cos2πt4+π
x=-2cosπ2tm