Obtain the equation $\omega = \omega_{0} + \alpha t$ from first principles.

(N/A) The angular acceleration $\alpha$ is uniform,hence the rate of change of angular velocity $\omega$ with respect to time $t$ is constant:
$\frac{d\omega}{dt} = \alpha$
Rearranging the terms for integration:
$d\omega = \alpha dt$
Integrating both sides:
$\int d\omega = \int \alpha dt$
$\omega = \alpha t + c$,where $c$ is the constant of integration.
Applying the initial condition: at $t = 0$,$\omega = \omega_{0}$.
Substituting these values into the equation:
$\omega_{0} = \alpha(0) + c \implies c = \omega_{0}$
Substituting the value of $c$ back into the integrated equation:
$\omega = \omega_{0} + \alpha t$.