Question

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

Answer 1

Given $\omega =\alpha -\beta t$

$\Rightarrow w=\alpha -\beta t=0$ $\Rightarrow t=\frac{\alpha }{\beta }$

$\omega =\frac{d\theta }{dt}$ $\Rightarrow \frac{d\theta }{dt}=\alpha -\beta t$

$\Rightarrow d\theta =(\alpha -\beta t)dt$ ...1

Integrate the equation 1 $\Rightarrow \int d\theta ={\int }_0^{\frac{\alpha }{\beta }}(\alpha -\beta t)dt$

$\Rightarrow \theta =\frac{{\alpha }^2}{\beta }$

Hence, Angle $=\frac{{\alpha }^2}{\beta }$ $rad$