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. $\frac{{\alpha }^2-{\beta }^2}{2\beta }$
• D. $\frac{(\alpha -\beta )\alpha }{2}$

Answer 1

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

Given,
$\omega =\alpha -\beta t...........\left(1\right)$
The body will stop when $\omega =0$
Putting $\omega =0$ in the equation $1$ above,
$t=\frac{\alpha }{\beta }$
Now, we know that $\omega =\frac{d\theta }{dt}..........\left(2\right)$
Putting equation $2$ in equation $1$ and integrating,
${\int }_0^{\theta }d\theta ={\int }_0^t(\alpha -\beta t)dt.......\left(3\right)$
$\theta =\alpha t-\frac{\beta t^2}{2}$,
at $t=\frac{\alpha }{\beta }$
$\theta =\frac{{\alpha }^2}{2\beta }$