Figures,. Correspond to two circular motions. The radius of the circle, the peirod of revolution, the initial position, and the sense of revolution (i.e. clockwise or anticlockwise) are indicated on each. Obtain the corresponding equations of simple harmonic motions of the revolving particle P in each case.

Explanation

(i) Time period ,T=2s
Amplitude,A=3cm
At time, t=0, the radius vector OP makes an angle `(pi)/(2)` with the positive x-asix, i.e., phase Angle `phi=+(pi)/(2)`
Therefore, the equation of simple harmonic motion for the x-projection of OP,at time t, is given by the displacement equation:
` x=Acos [(2pit)/(T)+phi]=3cos((2pit)/(2)+(pi)/(2))=-3sin((2pit)/(2))`
`x=-3sinpitcm`
(ii) Time period , T=4s
Amplitude, a=2m
At time t=0, OP makes an angle pi with the x-axis, in the anticlockwise direction. Hence phase angle ,`phi=+pi` Therefore, the equation of simple harmonic motion for the x-proection of Op,at time t,is given as:
`x=acos((2pit)/(T)+phi)=2cos((2pit)/(4)+pi)`
`x=-2cos ((pit)/(2))_(m)`