Question

A wheel starts rotating from rest at time t = 0 with a angular acceleration of 50 radians/s. 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

• A. 10 rad/s
• B. 250 rad/s
• C. 125 rad/s
• D. 100 rad/s

Answer 1

Answer: (C)

125 rad/s

Answer 2

$\alpha=\alpha_{0}\left(1-\frac{t}{5}\right)$ 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