Question

What is the work done in $t$ seconds by a body of mass $m$ moving in a circular path of radius $r$ with a constant angular acceleration? Initially the body was at rest.

• A. $m{\alpha }^2{R}^2{t}^2$
• B. $2\pi m{v}^{2}t$
• C. $\frac{1}{2}m{\alpha }^2{R}^2{t}^2$
• D. zero

Answer 1

Answer: (C)

$\frac{1}{2}m{\alpha }^2{R}^2{t}^2$

The angular velocity $\omega$ of mass $m$ after time $t$ is given by
$\omega ={\omega }_o+\alpha t=0+\alpha t=\alpha t$
According to work energy theorem, the work done on an object is equal to change in its kinetic energy i.e. $W=\Delta K.E.$
$\Rightarrow W=K.E_f-K.E_i\Rightarrow W=\frac{1}{2}m{v}^2-0\Rightarrow W=\frac{1}{2}m(\omega R{)}^2=\frac{1}{2}m{\omega }^2{R}^2\Rightarrow W=\frac{1}{2}m(\alpha t{)}^2{R}^2=\frac{1}{2}m{\alpha }^2{R}^2{t}^2$