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

`a_(c)=lambdat^(2)implies(v^(2))/(R )=lambdat^(2)`
`v=sqrt(lambdaR)timpliesa_(1)=(dv)/(dt)=sqrt(lambdaR)`
Tangential force `F_(t)=ma_(t)=msqrt(lambdaR)`
In a circular motion, the work is done only by the tangential force, instantaneous power
`P=vecF_(t).vecv=F_(t)v=msqrt(lambdaR)xxsqr(lambdaR)t`
`P=mlambdaRt`
Average power `barP=(w)/(t)`
work done=`DeltaK` (work-energy theorem)
`W=(1)/(2)mv^(2)-0=(1)/(2)mlambdaRt^(2)`
or
`W=F_(t)s=F_(t)xx(1)/(2)a_(t)^(2)=msqrt(lambdaR)xx(1)/(2)sqrt(lambdaR)t^(2))`
`=(1)/(2)mlambdaRt^(2))`
`barP=(W)/(t)=((1)/(2)mlambdaRt^(2))/(t)=(1)/(2)mlambdaRt`
or
`barP=(int_(0)^(t)Pdt)/(int_(0)^(t)dt)=(int_(0)^(t)mRtdt)/(t)=(mlambdaR(t^(2))/(2))/(t)=(1)/(2)mlambdaRt`