Answer
Ans:
$$
\begin{aligned}
\omega_B =156 \mathrm{rad} / \mathrm{s}
\end{aligned}
$$
Work Step by Step
Angular velocity can be determine:
$\omega_s=$ Angular velocity of the shaft
$$
\begin{aligned}
& \int d t=\int \frac{d \omega_S}{\alpha_S} \\
& \int_0^t d t=\int_1^{\omega_S} \frac{d \omega_S}{4 \omega_S^{3 / 4}} \\
& t_0^t=\left.\omega_S{ }^{1 / 4}\right|_1 ^{\mid \omega_s} \\
& t=\omega_S^{1 / 4}-1 \\
& \omega_S=(t+1)^4
\end{aligned}
$$
$$
\begin{aligned}
& t=4 \mathrm{~s}, \omega_s=? \\
& \omega_s=5^4=625 \mathrm{rad} / \mathrm{s}
\end{aligned}
$$
The brush is connected to the shaft so:
$\omega_B=$ Angular velocity of the brush
$$
\begin{aligned}
& \omega_B r_B=\omega_s r_s \\
& \omega_B=\left(\frac{r_s}{r_B}\right) \omega_s=\left(\frac{0.25}{1}\right)(625)=156 \mathrm{rad} / \mathrm{s}
\end{aligned}
$$
