Question

A particle of mass m moves along a circle of radius $R$ with a normal acceleration varying with time as $a_n=b{t}^2$, where b is a constant. Find the mean value of power averaged over the first 2 seconds after the beginning of motion. ($m=1,b=2,R=1$)

Answer 1

$v=\sqrt{bR}t$
$\Rightarrow \frac{dv}{dt}=\sqrt{bR}$
For circular motion work done by normal force is zero. For
tangential forces.
$F_t=m\frac{dv}{dt}=m\sqrt{bR};P=F_{t}vcos\theta ;P=F_{t}v=mbRt$
Average power $=\frac{{\int }_0^{T}P\left(t\right)dt}{{\int }_0^{T}dt}={\int }_0^T\frac{mbRtdt}{T}$
$\frac{mbR{\left(\frac{t^2}{2}\right)}_0^T}{T}=\frac{mbRT}{2}=2s$