Answer
(a) $25s$
(b) $25s$
Work Step by Step
(a) We can find the required time as follows:
$\Delta \theta=120rev$
$\Delta \theta=120\times (2\pi rad)=240\pi rad$
We know that
$\omega^2=\omega_{\circ}^2+2\alpha\Delta \theta$
$\implies \alpha=\frac{\omega^2-\omega_{\circ}^2}{2\Delta \theta}$
We plug in the known values to obtain:
$\alpha=\frac{(25rad/s)^2-(35rad/s)^2}{2\times 240\pi rad}$
$\alpha=-0.4rad/s^2$
Now $\omega=\omega_{\circ}+\alpha t$
This can be rearranged as:
$t=\frac{\omega-\omega_{\circ}}{\alpha}$
We plug in the known values to obtain:
$t=\frac{25rad/s-35rad/s}{-0.4rad/s^2}$
$t=25s$
(b) We can find the required time as follows:
$t=\frac{\omega-\omega_{\circ}}{\alpha}$
We plug in the known values to obtain:
$t=\frac{15rad/s-25rad/s}{-0.4rad/s^2}$
$t=25s$
