Question

A particle of mass $m$ moves along a circle of radius $R$ with a normal acceleration varying with time as $w_n=a{t}^2$, where $a$ is a constant. Find the time dependence of the power developed by all the forces acting on the particle, and the mean value of this power averaged over the first $t$ seconds after the beginning of motion.

Answer 1

We have $w_n=\frac{{\nu }^2}{R}=a{t}^2,$ or $\nu =\sqrt{aR}t$,
$t$ is defined to start from the beginning of motion from rest
So, $w_t=\frac{d\nu }{dt}=\sqrt{aR}$
Instantaneous power $P=\overrightarrow{F}.\overrightarrow{\nu }=m(w_t{\hat{u}}_t+w_n{\hat{u}}_t).\left(\sqrt{aR}t{\hat{u}}_t\right)$
(where ${\hat{u}}_t$ and ${\hat{u}}_t$ are unit vectors along the direction of tangent (velocity) and normal respectively)
So, $P=m{w}_t\sqrt{aR}t=maRt$
Hence the sought average power
$<P>=\frac{{\int }_0^{t}Pdt}{{\int }_0^{t}dt}=\frac{{\int }_0^{t}maRtdt}{t}$
Hence $<P>=\frac{maR{t}^2}{2t}=\frac{maRt}{2}$