Question

A rigid body rotates about a fixed axis with variable angular velocity equal to (a-bt) at time t where a and b are constants. The angle through which it rotates before it comes to rest is___?

• A. $\frac{a^2}{b}$
• B. $\frac{a^2}{2b}$
• C. $\frac{a^2}{4b}$
• D. $\frac{a^2}{2b^2}$

Answer 1

The correct option is B $\frac{a^2}{2b}$

Let the angle rotated be $\theta$

Given,

$\omega =a-bt$

$\frac{d\theta }{dt}=a-bt$

Also,

It will come to rest when, $\omega =a-bt=0$

$t_f=\frac{a}{b}$

${\int }_0^{\theta }d\theta ={\int }_0^{\frac{a}{b}}a-bt$

$\theta =\frac{a^2}{b}-\frac{a^2}{2b}=\frac{a^2}{2b}$

Option $\text{B}$ is the correct answer