Question

A rigid body rotates about a fixed axis with a variable angular velocity equal to $\alpha -\beta t$ at time ${}^{\prime }{t}^{\prime }$ where $\alpha$ and $\beta$ are constants. The angle through which it rotates before it comes to 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. $\frac{{\alpha }^3}{6{\beta }^2}$

Answer 1

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

We have $\omega =\frac{d\theta }{dt}=\alpha -\beta t$
or
$d\theta =(\alpha -\beta t)dt$
integrating
$\theta =\int (\alpha -\beta t)dt$
or
$\theta =\alpha t-\frac{\beta t^2}{2}$
for $t=\frac{\alpha }{\beta }$ we have
$\omega =0$ and $\theta =\alpha (\frac{\alpha }{\beta })-\frac{\beta .{\alpha }^2}{2{\beta }^2}$
or
$\theta =\frac{{\alpha }^2}{2\beta }$