Question

A particle of mass m moving in a circular path of constant radius r such that centripetal acceleration $a_c$ is varying with time t as $a_c=krt$ where $k$ is constant, then:

• A. Power delivered by the centripetal force is zero
• B. Power delivered by the tangential force is $\frac{k{r}^2}{2}$
• C. Power delivered by the net force is equal to power delivered by tangential force
• D. Tangential force $=\frac{mr}{2}\sqrt{\frac{k}{t}}$

Answer 1

Answer: (A), (B), (C)

The correct options are
A Power delivered by the centripetal force is zero
B Power delivered by the tangential force is $\frac{k{r}^2}{2}$
C Power delivered by the net force is equal to power delivered by tangential force

$a_c=\left(krt\right)$
power delivered by the centripetal force = 0
$\frac{m{v}^2}{r}=krt\Rightarrow v^2=\frac{k}{m}{r}^{2}t$
$\Rightarrow 2v\frac{dv}{dt}=\frac{k}{m}{r}^2\Rightarrow \left(\frac{dv}{dt}\right)=\left(\frac{k{r}^2}{2mv}\right)$
$F_t=m\frac{dv}{dt}=\frac{mk{r}^2}{2mv}=\frac{k{r}^2}{2v}=\left(\frac{k{r}^2}{2v}\right)$
$P_{F_t}=\frac{k{r}^2}{2v}\times v=\left(\frac{k{r}^2}{2}\right)$ = power delivered by net force $(\because P_{F_c}=0)$
$\therefore F_t=\left(\frac{k{r}^2}{2v}\right)=\left(\frac{k{r}^2}{2}\right)\times \frac{1}{\sqrt{\frac{k{r}^{2}t}{m}}}=\left(\frac{r}{2}\right)\sqrt{\frac{mk}{t}}$