A paricle of mass m moves along a circle of radius R with a normal acceleration varying with time as `w_n=at^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.

Explanation

We have `w_n=v^2/R=at^2`, or, `v=sqrt(aR)t`,
t is defined to start from the beginning of motion from rest.
So, `w_t=(dv)/(dt)=sqrt(aR)`
Instantaneous power, `P=vecF*vecv=m(w_thatu_t+w_nhatu_t)*(sqrt(aR)thatu_t)`,
(where `hatu_t` and `hatu_t` are unit vectors along the direction of tangent (velocity) and normal respectively)
So, `P=mw_tsqrt(aR)t=maRt`
Hence the sought average power
`lt P gt =(underset(0)overset(t)intPdt)/(underset(0)overset(t)intdt)=(underset(0)overset(t)intmaRtdt)/(t)`
Hence `lt P ge(maRt^2)/(2t)=(maRt)/(2)`