The radius of circle, the period of revolution, initial position and sense of revolution are indicated in the figure. y -projection of the radius vector of rotating particle P is

Given time period, T = 4 s

The relation between time period and angular velocity is $\mathrm{T}=\frac{2\pi }{\omega }.$

$\Rightarrow \mathrm{T}=\frac{2\pi }{\omega }=4\mathrm{s}$

$\Rightarrow \omega =\frac{\pi }{2}\mathrm{rad}/\mathrm{s}$

This angular velocity acts as angular frequency of motion projected on y- axis. Also, the radius of circle will act like
amplitude of simple harmonic motion. Equation of an SHM is $y=A\sin (\omega t+\varphi )$

To fi nd initial phase of SHM, we have to put initial conditions in above equation.

Given at t = 0 ; y = 3

and the amplitude A = 3 m

$\Rightarrow y=3\sin (0+\varphi )=3$

$\Rightarrow \sin \varphi =1$

$\Rightarrow \varphi =\frac{\pi }{2}$

$\therefore \text{The equation}\mathrm{y}=3\sin (\frac{\pi }{2}\mathrm{t}+\frac{\pi }{2})\mathrm{m}$

$=3\cos \left(\frac{\pi }{2}t\right)m$

Using identity,   $\sin \left(\theta +\frac{{\displaystyle \pi }}{{\displaystyle 2}}\right)=\cos \left(\theta \right)$