Question

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

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

Answer 1

Answer: (C)

$\alpha ={\alpha }_0\left(1-\frac{t}{5}\right)$
At t = 0, $\alpha ={\alpha }_0\therefore {\alpha }_0$ = 50 rad/s${}^2$
$\frac{d\omega }{dt}={\alpha }_0\left(1-\frac{1}{5}\right)$
$\therefore {\int }_0^{\omega }d\omega ={\alpha }_0{\int }_0^5\left(1-\frac{t}{5}\right)dt\Rightarrow \omega ={\alpha }_0{\left[t-\frac{t^2}{10}\right]}_0^5$
$=50\left(5-\frac{25}{10}\right)$ rad/s =125 rad/s