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 $a_c$ = $k^{2}r{t}^2$, where k is a constant. The power delivered to the particle by the forces acting on it is:

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

Answer 1

The correct option is C $m{k}^2{r}^{2}t$

Centripetal acceleration,

$a_c=k^{2}r{t}^2$. . . . .(1)

where, $r=$ radius

$a_c=\frac{v^2}{r}$

From equation (1)

$\frac{v^2}{r}=k^{2}r{t}^2$

$v^2=k^2{r}^2{t}^2$

$v=krt$. . . . . . . . . . .(2)

Tangential acceleration, $a_t=\frac{dv}{dt}=kr$

Tangential force acting on the particle,

$F=m{a}_t=mkr$

Power delivered to particle by the force will be

$P=\overrightarrow{F}.\overrightarrow{v}=Fvcos\theta$

$P=mkr.krt$ $(cos\theta =1)$

$P=m{k}^2{r}^{2}t$

The correct option is C.