Answer
(a) The wheel rotates through an angle of $270^{\circ}$ during the second 1.0-second time interval.
(b) The wheel rotates through an angle of $450^{\circ}$ during the third 1.0-second time interval.
Work Step by Step
(a) In general, $\theta = \frac{1}{2}\alpha~t^2$
Let $\theta_1$ be the angle through which the wheel rotates during the first 1.0-second time interval:
$\theta_1 = \frac{1}{2}\alpha~(1.0~s)^2 = 90^{\circ}$
Let $\theta_2$ be the angle through which the wheel rotates during the second 1.0-second time interval:
$\theta_2 = \frac{1}{2}\alpha~(2.0~s)^2- \theta_1$
$\theta_2 = 4\times \frac{1}{2}\alpha~(1.0~s)^2- \theta_1$
$\theta_2 = 4\theta_1- \theta_1$
$\theta_2 = 3\theta_1$
$\theta_2 = (3)(90^{\circ})$
$\theta_2 = 270^{\circ}$
The wheel rotates through an angle of $270^{\circ}$ during the second 1.0-second time interval.
(b) Let $\theta_3$ be the angle through which the wheel rotates during the third 1.0-second time interval:
$\theta_3 = \frac{1}{2}\alpha~(3.0~s)^2- \theta_1- \theta_2$
$\theta_3 = 9\times \frac{1}{2}\alpha~(1.0~s)^2- \theta_1- \theta_2$
$\theta_3 = 9\theta_1 - \theta_1- \theta_2$
$\theta_3 = 5\theta_1$
$\theta_3 = (5)(90^{\circ})$
$\theta_3 = 450^{\circ}$
The wheel rotates through an angle of $450^{\circ}$ during the third 1.0-second time interval.
