Q. 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 as $a_{c} = k^{2}rt^{4},$ where $k$ is a constant. The power delivered to the particle by the forces acting on its is
A) Zero
B) $mk^{2}r^{2}t^{2}$
C) $\frac{1}{3}mk^{2}r^{2}t^{2}$
D) $2mk^{2}r^{2}t^{3}$
Explanation
(d) $a_{c} = k^{2}rt^{4} = \frac{v^{2}}{r}$ or $v = krt^{2}$
The tangential acceleration is $a_{T} = \frac{dv}{dt} = 2krt$
The tangential force on the particle, $F_{T} = ma_{T} = 2mkrt$
Power delivered to the particle
$= F_{T} = ma_{T} = 2mkrt = F_{T}v = (2mkrt)(krt)^{2} = 2mk^{2}r^{2}t^{3}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.