Answer
It takes the flywheel 177 seconds to reach the top angular speed of 1200 rpm.
Work Step by Step
We first find the moment of inertia of the flywheel.
$I = \frac{1}{2}MR^2$
$I = \frac{1}{2}(250~kg)(0.75~m)^2$
$I = 70.3~kg~m^2$
We then find the angular acceleration of the flywheel.
$\tau = I~\alpha$
$\alpha = \frac{\tau}{I}$
$\alpha = \frac{50~N~m}{70.3~kg~m^2}$
$\alpha = 0.711~rad/s^2$
Next, we convert the final angular velocity to units of rad/s
$\omega_f = (1200~rpm)(\frac{2\pi~rad}{1~rev})(\frac{1~min}{60~s})$
$\omega_f = 126~rad/s$
We can find the time it takes to reach this final angular velocity.
$t = \frac{\omega_f-\omega_0}{\alpha}$
$t = \frac{126~rad/s-0}{0.711~rad/s^2}$
$t = 177~s$
It takes the flywheel 177 seconds to reach the top angular speed of 1200 rpm.
