A particle of mass m is moving in a circular path of constant radius r such that its centripetal acceleration `a_(c)` is varying with time t as `a_(c) = k^(2)rt^(2)`, where k is a constant. The power delivered to the particle by the forces acting on it is :

Options

A. `2 pi m k^2r^2t`
B. `m k^2r^2t`
C. `((m k^4 r^2 t^5))/(3)`
D. zero

Answer

B

Explanation

The centripetal acceleration, `ac = k^2rt^2`
`implies (v^2)/r = k^2 rt^2 " " implies " " 1/2 mv^2 = m/2 k^2 r^2 t^2 implies KE= m/2 k^2 r^2 t^2`
`implies d/(dt) (KE) = m k^2 r^2 t implies "Power" = mk^2r^2t`
Alternative method:
Since v = krt and tangential acceleration `a_t = (dv)/(dt) = kr` Tangential force = mkr Instantaneous power, `P = F xx v = m k r xx krt = m k^2 r^2 t`.