A rigid body rotates about a stationary axis with an angular velocity that varies with time as $\ome

Question

A rigid body rotates about a stationary axis with an angular velocity that varies with time as $\omega = \omega_0 (1 - t^3/	au^3)$ for $0 \le t \le 	au$, where $\omega_0$ and $	au$ are positive constants. The body comes to rest at $t = 	au$. Which of the following statements is/are CORRECT?

Professor PC Thomas Classes · Answer

$\phi(	au) = \int_0^	au \omega_0 (1 - t^3/	au^3) dt = \omega_0 [	au - 	au/4] = \frac{3}{4} \omega_0 	au$ (A is correct). $\langle \omega angle = \frac{\phi(	au)}{	au} = \frac{3}{4}\omega_0$ (B is incorrect). Angular acceleration $\alpha(t) = \frac{d\omega}{dt} = -\frac{3\omega_0 t^2}{	au^3}$. At stop, $\alpha(	au) = -\frac{3\omega_0}{	au}$ (C is correct). Mean angular acc $\langle \alpha angle = \frac{\Delta \omega}{\Delta t} = \frac{0 - \omega_0}{	au} = -\frac{\omega_0}{	au}$. Since $\alpha(	au) = -\frac{3\omega_0}{	au}$, we have $\langle \alpha angle = \frac{1}{3}\alpha(	au)$ (D is correct).