The following figure depicts two circular motions. The radius of the circle, the period of revolutin the initial position and the sense of revolution are indicated on the figure. Obtain the simple harmoic motion of the x-projection of the radius vector of the rotating particle P in each case.

Explanation

Given that , time period , `T= 4s`
angular frequency , `omega = ( 2pi)/(T)= ( 2pi)/(4) = (pi)/(2) rad//s`
Angle traversed by paritcle in time, ` t, theta = omegat = (pi)/(2) t`
the radial position of the particle at time t makes an angle `( 45^(@) + theta)`
Displacement of particle `=`Projection of P on x-axis ` = A cos ( 45^(@) + theta) =A cos ( 45^(@) + ( pi)/(2)t)`
(ii)
Given that , time period ` T= 30 s`
`:.` angular frequency `omega = ( 2pi)/(T)`
`+ ( 2pi)/( 30)`
`= (pi)/(15)rad //s`
angle traversed by particle in time ` t , theta = omega t = (pi)/( 15) t`
The radial position of the particle at any time t makes an angle
The displacement of the particle `=` projection of P on x-axis `((pi)/(2)-theta)`
`= B cos ((pi)/(2) - theta)`
`= B cos ((pi)/(2) - omegat)`
`= B cos ( (pi)/(2) - (pi)/(15))`
`=Bcos ((pi)/(15)t-(pi)/(2))` `( :' cos ( - theta) = cos theta)`