Answer
The wheel rotates through an angle of $~~1500~rad~~$ in the interval $t = 0$ to $t = 40~s$
Work Step by Step
We can find the angular acceleration:
$\omega = \omega_0+\alpha~t$
$\alpha = \frac{\omega - \omega_0}{t}$
$\alpha = \frac{5.0~rad/s - 0}{2.0~s}$
$\alpha = 2.5~rad/s^2$
We can find the angular speed at $t = 20~s$:
$\omega = \omega_0+\alpha~t$
$\omega = 0+(2.5~rad/s)(20~s)$
$\omega = 50~rad/s$
We can find the angular displacement $\theta_1$ in the first $20~s$:
$\theta_1 = \frac{1}{2}\alpha~t^2$
$\theta_1 = \frac{1}{2}(2.5~rad/s^2)(20~s)^2$
$\theta_1 = 500~rad$
We can find the angular displacement $\theta_2$ between $20~s$ and $40~s$:
$\theta_2 = \omega~t$
$\theta_2 = (50~rad/s)(20~s)$
$\theta_2 = 1000~rad$
The total angular displacement in the first $40~s$ is $1500~rad$
The wheel rotates through an angle of $~~1500~rad~~$ in the interval $t = 0$ to $t = 40~s$
