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 $a = k ^{2} rt ^{2}$. where $k$ is a constant. The power delivered to the particle by the force acting on it is given as

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

Answer 1

Answer: (C)

Centripetal acceleration, $a _{ c }= k ^2 rt ^2$
where, $a_c=\frac{v^2}{r}$
$ \begin{array}{l} \Rightarrow \frac{ v ^2}{ r }= k ^2 rt ^2 \\ \Rightarrow v = krt \ldots \ldots . .(1) \\ \text { Tangential acceleration, } a _{ t }=\frac{ dv }{ dt }= kr \end{array} $
Tangential force acting on the particle, $F = ma _{ t }= mkr$
Power delivered, $P =\overrightarrow{ F } \cdot \overrightarrow{ V }= F v \cos \theta$
$ \begin{array}{l} \therefore P=F V=( mkr ) \times \operatorname{krt}\left(\because \theta=0^{\circ}\right) \\ \Longrightarrow P = mk ^2 r ^2 t \end{array} $