Question 3: Work, Energy and Power & Circular Motion

Question

The 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 particle by the forces acting on it is

Accepted Answer

If v is instantaneous velocity, centripetal
acceleration
$$\begin{aligned}
a_c & =\frac{v^2}{r} \
	herefore \quad \frac{v^2}{r} & =k^2 r t^2 \
\Rightarrow \quad v & =k r t \quad\left[\because a_c=k^2 r t^2 	ext { is given }ight]
\end{aligned}$$
In circular motion, work done by centripetal force is always zero and work is done only by tangential force.
$$	herefore$$ Tangential acceleration
$$a_t=\frac{d v}{d t}=\frac{d}{d t}(k r t)=k r$$
$$	herefore$$ Tangential force $$F_t=m a_t=m k r$$
Power $$P=F_t v=(m k r)(k r t)=m k^2 r^2 t$$

Answer

$$2 \pi m k^2 r^2 t$$

Answer

$$m k^2 r^2 t$$

Answer

$$1 / 3 m k^4 r^2 t^5$$

Answer

zero

Work, Energy and Power & Circular Motion

The 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 particle by the forces acting on it is

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