Answer
(a) $\omega^2 = \omega_0^2 + 2\alpha~(\theta - \theta_0)$
(b) $\alpha = 8.00~rad/s^2$
Work Step by Step
(a) Equation 9.7:
$\omega = \omega_0+\alpha ~t$
$t = \frac{\omega - \omega_0}{\alpha}$
We can replace $t$ in equation 9.11:
$\theta = \theta_0+ \omega_0 ~t + \frac{1}{2}\alpha ~t^2$
$\theta = \theta_0+ \omega_0 (\frac{\omega - \omega_0}{\alpha}) + \frac{1}{2}\alpha (\frac{\omega - \omega_0}{\alpha})^2$
$\alpha~(\theta - \theta_0) = \omega_0 ~\omega - \omega_0^2 + \frac{1}{2}(\omega^2 -2~\omega~\omega_0+ \omega_0^2)$
$2\alpha~(\theta - \theta_0) = 2\omega_0 ~\omega - 2\omega_0^2 + \omega^2 -2~\omega~\omega_0+ \omega_0^2$
$2\alpha~(\theta - \theta_0) = \omega^2 - \omega_0^2$
$\omega^2 = \omega_0^2 + 2\alpha~(\theta - \theta_0)$
(b) $\omega^2 = \omega_0^2 + 2\alpha~(\theta - \theta_0)$
$\alpha = \frac{\omega^2 - \omega_0^2}{2~(\theta - \theta_0)}$
$\alpha = \frac{(16.0~rad/s)^2 - (12.0~rad/s)^2}{2~(7.00~rad)}$
$\alpha = 8.00~rad/s^2$
