Establish the relation $\theta=\omega_0\text{t}+\frac{1}{2}\alpha\text{t}^2$ where the letters have their usual meanings.
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We know that angular velocity is defined as $\omega=\frac{\text{d}\theta}{\text{dt}}$
$\therefore\text{ d}\theta=\omega\text{dt}=(\omega_0+\alpha\text{t})\text{dt},$ where $w_0$ is the value of initial angular velocity at time t = 0 and $\alpha=$ uniform angular acceleration. On integrating, we get $\int\limits^\theta_0\text{d}\theta=\int\limits^\text{t}_0(\omega_\circ+\alpha\text{t})\ \text{dt}$
$\therefore\ [\theta]^\theta_0=\Big[\omega_\circ\text{t}+\frac{1}{2}\alpha\text{t}^2\Big]^\text{t}_0$
$\text{or }\theta-0=\omega_0(\text{t)}-0+\frac{1}{2}\alpha\ (\text{t}^2-0)$
$\Rightarrow\theta=\omega_0\text{t}+\frac{1}{2}\alpha\text{t}^2,$ which is the requisite relation.
$\therefore\text{ d}\theta=\omega\text{dt}=(\omega_0+\alpha\text{t})\text{dt},$ where $w_0$ is the value of initial angular velocity at time t = 0 and $\alpha=$ uniform angular acceleration. On integrating, we get $\int\limits^\theta_0\text{d}\theta=\int\limits^\text{t}_0(\omega_\circ+\alpha\text{t})\ \text{dt}$
$\therefore\ [\theta]^\theta_0=\Big[\omega_\circ\text{t}+\frac{1}{2}\alpha\text{t}^2\Big]^\text{t}_0$
$\text{or }\theta-0=\omega_0(\text{t)}-0+\frac{1}{2}\alpha\ (\text{t}^2-0)$
$\Rightarrow\theta=\omega_0\text{t}+\frac{1}{2}\alpha\text{t}^2,$ which is the requisite relation.
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