Question

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

• A. $2\pi \, mk^{2}r^{2}$
• B. $mk^{2} \, r^{2}t$
• C. $\frac{\left(m k^{4} \, r^{2} t^{5}\right)}{3}$
• D. Zero

Answer 1

Answer: (B)

$a_{c}=k^{2} \, r t^{2}$
or $\frac{v^{2}}{r}= \, k^{2} \, r t^{2}$
or $v=krt$
Therefore, tangential acceleration, $a_{t} \, =\frac{d v}{d t}= \, kr$
or Tangential force,
$F_{t}=m a_{t}=mkr$
Only tangential force does work.
$Power=F_{t} \, v=\left(m k r\right)\left(k r t\right)$
or $Power=m k^{2} \, r^{2} \, t$