Question

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

• A. zero
• B. $m{k}^2{r}^2{t}^2$
• C. $m{k}^2{r}^{2}t$
• D. $m{k}^{2}rt$

Answer 1

Answer: (C)

Centripetal acceleration, $a_c=k^{2}r{t}^2$
where, $a_c=\frac{v^2}{r}$
$\begin{array}{l}\Rightarrow \frac{v^2}{r}=k^{2}r{t}^2\\ \Rightarrow v=krt\dots \dots ..(1)\\ \text{ Tangential acceleration, }{a}_t=\frac{dv}{dt}=kr\end{array}$
Tangential force acting on the particle, $F=m{a}_t=mkr$
Power delivered, $P=\overrightarrow{F}\cdot \overrightarrow{V}=Fv\cos \theta$
$\begin{array}{l}\therefore P=FV=(mkr)\times \mathrm{krt}\left(\because \theta =0^{\circ }\right)\\ ⟹P=m{k}^2{r}^{2}t\end{array}$