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

Options

A. (a) `mkrt_0`
B. (b) `(mkrt_0^2)/(2)`
C. (c) `(mkrt_0)/(2)`
D. (d) `(mkrt_0)/(4)`

Answer

C

Explanation

Given `a_n=kt^2`
or `(v^2)/(r)=kt^2` or `v^2=krt^2`
Therefore, average power delivered `=` Total work done/Total time elapsed
`=`Increase in KE/Total time elapsed
or `ltPgt(1/2m(v^2-0^2))/(t_0)=m/2(krt_0^2)/(t_0)=(mkrt_0^2)/(2)`
Alternative: `v=sqrt(krt)impliesa_1=sqrt(kr)`
`P=F_tv=ma_tv=mkrt`
`ltPgt =(underset0overset(t_0)intPdt)/(t_0)=1/2mkrt_0^2`