Question

A particle of mass $m$ moves along a circular path of radius $r$ with a centripetal acceleration $a_{n}$ changing with time $t$ as $a_{n} = kt^{2}$, where $k$ is a positive constant. The average power developed by all the forces acting on the particle during the first $t_{0}$ seconds is

• A. $mkrt_{0}$
• B. $\frac{mkrt^{2}_{0}}{2}$
• C. $\frac{mkrt_{0}}{2}$
• D. $\frac{mkrt_{0}}{4}$

Answer 1

Answer: (B)

Given $a_{n}=k t^{2}$
or $\frac{v^{2}}{r}=k t^{2}$ or $v^{2}=k r t^{2}$
Therefore, average power delivered $=$ Total work done/Total time elapsed $=$ Increase in KE/Total time elapsed
or $\langle P\rangle \frac{\frac{1}{2} m\left(v^{2}-0^{2}\right)}{t_{0}}=\frac{m}{2} \frac{k r t_{0}^{2}}{t_{0}}=\frac{m k r t_{0}^{2}}{2}$ Alternative: $v=\sqrt{k r t} \Rightarrow a_{1}=\sqrt{k r}$
$P=F_{t} v=m a_{t} v=m k r t$
$\begin{aligned} P=F_{t} v &=m a_{t} v=m k r t \\ &\langle P\rangle=\frac{\int_{0}^{t_{0}} P d t}{t_{0}}=\frac{1}{2} m k r t_{0}^{2} \end{aligned}$