Question

A body rotates about a fixed axis with an angular acceleration of one radian/second. Through what angle does it rotate during the time in which its angular velocity increases from 5 rad/s to 15 rad/s.

Answer 1

Given:
Angular acceleration of the body = $\alpha =2\mathrm{rad}/{\mathrm{s}}^2$
Initial angular velocity of the body = ${\omega }_0=5\mathrm{rad}/\mathrm{s}$
Final angular velocity of the body = $\omega =15\mathrm{rad}/\mathrm{s}$
$$\begin{gathered} \mathrm{We}\mathrm{know}\mathrm{that}: \\ \omega ={\omega }_0+\alpha t \\ \Rightarrow t=\frac{\left(\omega -{\omega }_0\right)}{\alpha }=\frac{\left(15-5\right)}{1}=10\mathrm{s} \\ \mathrm{Also}, \\ \theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2 \\ \Rightarrow \theta =5\times 10+\frac{1}{2}\times 1\times 100 \\ \Rightarrow \theta =50+50=100\mathrm{rad} \end{gathered}$$