Explain angular velocity and angular acceleration about a fixed axis,derive the equations of rotational motion,and write the analogy between the equations of linear motion and rotational motion.

(N/A) Consider a rigid body rotating about a fixed $Z$-axis in a Cartesian coordinate system. Any particle $P$ of the body moves in a circular path in a plane perpendicular to the $Z$-axis.
Let the angular position of particle $P$ at time $t=0$ be $\theta_{0}$ and at time $t$ be $\theta_{0}+\theta$. Thus,the angular displacement in time $t$ is $\theta$.
$1$. Angular Velocity $(\omega)$: It is defined as the time rate of change of angular displacement.
$\omega = \frac{d\theta}{dt}$. Since the direction is along the fixed $Z$-axis,it can be treated as a scalar.
$2$. Angular Acceleration $(\alpha)$: It is defined as the time rate of change of angular velocity.
$\alpha = \frac{d\omega}{dt}$. Similarly,it can be treated as a scalar.
Equations of Linear Motion (for constant acceleration $a$):
$v = v_{0} + at$
$x = x_{0} + v_{0}t + \frac{1}{2}at^{2}$
$v^{2} = v_{0}^{2} + 2a(x - x_{0})$
Equations of Rotational Motion (for constant angular acceleration $\alpha$):
$\omega = \omega_{0} + \alpha t$
$\theta = \theta_{0} + \omega_{0}t + \frac{1}{2}\alpha t^{2}$
$\omega^{2} = \omega_{0}^{2} + 2\alpha(\theta - \theta_{0})$
Analogy Table:
| Linear Motion | Rotational Motion |
| :--- | :--- |
| Displacement $(x)$ | Angular displacement $(\theta)$ |
| Initial velocity $(v_{0})$ | Initial angular velocity $(\omega_{0})$ |
| Final velocity $(v)$ | Final angular velocity $(\omega)$ |
| Acceleration $(a)$ | Angular acceleration $(\alpha)$ |