Question

A particle of mass m is moving in a circular path of radius R such that its centripetal acceleration $a_c$ is varying with time as $a_c=k{r}^2{t}^3$, where k$=$constant. The power delivered to the particle by the force acting on it is?

• A. $\frac{1}{2}m{k}^2{r}^2{t}^3$
• B. $2mk{r}^3{t}^2$
• C. $\frac{3}{2}mk{r}^3{t}^2$
• D. $mk{r}^3{t}^2$

Answer 1

The correct option is C $\frac{3}{2}mk{r}^3{t}^2$

The centripetal acceleration is given by
$a_c=\frac{V^2}{r}$
Also, $a_c=k{r}^2{t}^3$
$\therefore \frac{v^2}{r}=k{r}^2{t}^3$
$\therefore v^2=k{r}^3{t}^3$
The power delivered to the particle will be
$p=\frac{dW}{dt}=\frac{d}{dt}\left(\frac{1}{2}m{v}^2\right)=\frac{d}{dt}\left(\frac{1}{2}mk{r}^3{t}^3\right)$
$\frac{1}{2}mk{r}^3\frac{d}{dt}\left(t^3\right)=\frac{3}{2}mk{r}^3{t}^2$.