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 particles by the force acting on it is:
A.
$$2\pi m{k^2}{r^2}t$$
B.
$$m{k^2}{r^2}t$$
C.
$$\frac{{\left( {m{k^4}{r^2}{t^5}} \right)}}{3}$$
D.
Zero
Answer :
$$m{k^2}{r^2}t$$
Solution :
The centripetal acceleration
$$\eqalign{
& {a_c} = {k^2}\,r\,{t^2}\,\,\, \Rightarrow \frac{{{v^2}}}{r} = {k^2}\,r\,{t^2} \cr
& \Rightarrow \frac{1}{2}m{v^2} = \frac{m}{2}{k^2}\,{r^2}\,{t^2}.....(i) \cr
& \Rightarrow K.E. = \frac{m}{2}{k^2}\,{r^2}\,{t^2} \cr
& \Rightarrow \frac{d}{{dt}}K.E. = m\,{k^2}\,{r^2}\,t \cr
& \Rightarrow {\text{Power}} = m\,{k^2}\,{r^2}\,t \cr} $$