Question

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 $a_{c}=k^{2}r~t^{2}$ where k is a constant. The power delivered to the particle by the forces acting on it is ________.

• A. $mk^{2}rt^{2}$
• B. $mk^{2}r^{2}t$
• C. $mkrt$
• D. $mkr^{2}t^{2}$

Hint:

Centripetal force does zero work; only tangential force changes the speed and delivers power.

Answer 1

The Correct Option is B

Step 1: Concept
Power ($P$) = $\vec{F} \cdot \vec{v}$. In circular motion, only tangential force ($F_t$) contributes to power.
Step 2: Analysis
$a_c = v^2/r = k^2 r t^2 \Rightarrow v = krt$. Tangential acceleration ($a_t$) = $dv/dt = kr$. Tangential force ($F_t$) = $m a_t = mkr$.
Step 3: Calculation
$P = F_t \times v = (mkr) \times (krt) = mk^2 r^2 t$.
Step 4: Conclusion
Hence, the power delivered is $mk^{2}r^{2}t$.
Final Answer:(B)