Question

A rigid body rotates about a fixed axis with variable angular velocity equal to $(\alpha -\beta t)$ at the time t where $\alpha ,\beta$ are constants. The angle through which it rotates before it stops is

• A. $\frac{{\alpha }^2}{2\beta }$
• B. $\frac{{\alpha }^2-{\beta }^2}{2\alpha }$
• C. $\frac{3{\alpha }^2}{2\beta }$
• D. $\frac{(\alpha -\beta )\alpha }{2}$

Answer 1

The correct option is A $\frac{{\alpha }^2}{2\beta }$

$\omega =\alpha -\beta t$
when it stops $\omega =0$
$\Rightarrow t=\frac{\alpha }{\beta }$
Also
$\frac{d\theta }{dt}=\alpha -\beta t$
${\int }_0^{\theta }d\theta =\alpha {\int }_0^{\alpha /\beta }dt-\beta {\int }_0^{\alpha /\beta }tdt$
$\theta =\alpha \left(\frac{\alpha }{\beta }\right)-\frac{\beta }{2}\times \frac{{\alpha }^2}{{\beta }^2}$
$=\frac{{\alpha }^2}{2\beta }$