Answer
$a=14.3$ in/s$^2$
Work Step by Step
When $\theta=\frac{\pi}{6}$ rad, $\frac{\pi}{6}=\cos 2t$, $t=0.5099$ s
$\dot{\theta}=\frac{d\theta}{dt}=-2\sin 2t|_{t=0.5099}=-1.7039$ rad/s
$\ddot{\theta}=\frac{d^2\theta}{dt^2}=-4\cos 2t|_{t=0.5099}=-2.0944 rad/s^2$
$r=4$
$\dot{r}=0$
$\ddot{r}=0$
$a_r=\ddot{r}-r\dot{\theta}^2=0-4(-1.7039)^2=-11.6135 in/s^2$
$a_\theta=r\ddot{\theta}+2\dot{r}\dot{\theta}=4(-2.0944)+0=-8.3776 in/s^2$
$a=\sqrt{a_r^2+a_\theta^2}=\sqrt{(-11.6135)^2+(-8.3776)^2}=14.3 in/s^2$
