Question

A solid body starts rotating about a stationary axis with an angular acceleration $\alpha ={\alpha }_{0}cos\varphi$, $\varphi$ is

the angle of rotation from the initial position. Find the angular velocity of the body as a function of the angle $\varphi$.

• A. $\omega =\alpha {\alpha }_0$
• B. $\omega =\frac{{\alpha }^2}{R}$
• C. $\omega =\sqrt{2{\alpha }_{0}sin\varphi }$
• D. $\omega =\sqrt{2{\alpha }_{0}cos\varphi }$

Answer 1

The correct option is C

$\omega =\sqrt{2{\alpha }_{0}sin\varphi }$

Angular acceleration = $\alpha$ = rate of change of angular velocity with time.

$\alpha =\frac{dw}{dt}=\frac{dw}{d\theta }\times \frac{d\theta }{dt}$

$\Rightarrow \alpha =\frac{wdw}{d\theta }$

$\Rightarrow {\alpha }_{0}cos\varphi =\omega \frac{dw}{d\varphi }$

$\underset{0}{\overset{w}{\int }}wdw$ = $\underset{0}{\overset{\varphi }{\int }}{\alpha }_{0}cos\varphi d\varphi$

$\Rightarrow \frac{w^2}{2}{|}_0^w={\alpha }_{0}sin\varphi {|}_0^{\varphi }$

$\frac{w^2}{2}={\alpha }_0[sin\varphi -sin0]$

$\Rightarrow w^2=2{\alpha }_{0}sin\varphi$

$\Rightarrow w=\sqrt{2{\alpha }_{0}sin\varphi }$