Answer
(a) t = 6.32 s
(b) The motor turns through an angle of 84.3 radians.
Work Step by Step
(a) At the moment when the motor changes direction, the angular velocity will be zero. We can find the time when the angular velocity is zero.
$\omega(t) = (20-\frac{1}{2}t^2)~rad/s = 0$
$t^2 = 40$
$t = \sqrt{40}$
$t = 6.32~s$
(b) The angle that the motor turns is equal to the area under the omega versus time graph. We can integrate to find this area.
$\Delta \theta = \int_{0}^{\sqrt{40}}\omega(t)~dt$
$\Delta \theta = \int_{0}^{\sqrt{40}}(20-\frac{1}{2}t^2)~dt$
$\Delta \theta = (20t-\frac{1}{6}t^3)\vert_{0}^{\sqrt{40}}$
$\Delta \theta = (20)(\sqrt{40})-\frac{1}{6}(\sqrt{40})^3$
$\Delta \theta = 84.3~rad$
The motor turns through an angle of 84.3 radians.
