Question

deduce an expression for angular velocity and angular acceleration in terms of SHM

Answer 1

The speed of a particle v in uniform circular motion is its angular speed $\omega$ times the radius of the circle A.
i.e, v =$\omega$A
now the velocity of the projection particle p' at time t is
v(t) = -$\omega$Asin($\omega$t+$\varphi$) where the negative sign indicates that v(t) has a direction opposite to the positive direction of x-axis.
differentiating teh equation w.rt to time we get acceleration of moving particle
$$\begin{gathered} a=\frac{dv}{dt}=\frac{-\omega Ad\sin (\omega t+\varphi )}{dt}=-{\omega }^{2}A\cos (\omega t+\varphi )=-{\omega }^{2}x \\ a=-{\omega }^{2}x \end{gathered}$$