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  4. A wheel starts rotating from rest at time t = 0 with a angular acceleration of 50 radians/s2. The angular acceleration (α) decreases to zero value after 5 seconds. During this interval, α varies according to the equation α=α0(1-(t/5)) The angular velocity at t = 5 s will be

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Q. A wheel starts rotating from rest at time t = 0 with a angular acceleration of 50 radians/s2. The angular acceleration ($\alpha$) decreases to zero value after 5 seconds. During this interval, $\alpha$ varies according to the equation
$\alpha=\alpha_{0}\left(1-\frac{t}{5}\right)$
The angular velocity at t = 5 s will be

AIIMSAIIMS 2017System of Particles and Rotational Motion

Solution:

$\alpha=\alpha_{0}\left(1-\frac{t}{5}\right)$
At t = 0, $\alpha=\alpha_{0}\quad\therefore \quad\alpha_{0}$ = 50 rad/s$^{2}$
$\frac{d\omega}{dt}=\alpha_{0}\left(1-\frac{1}{5}\right)$
$\therefore \,\,\int^{\omega}_{0}\,d\omega=\alpha_{0}\int^{5}_{0}\left(1-\frac{t}{5}\right)dt\,\Rightarrow \,\omega=\alpha_{0}\left[t-\frac{t^{2}}{10}\right]_{0}^{5}$
$=50\left(5-\frac{25}{10}\right)$ rad/s =125 rad/s