A solid sphere of radius $4$ cm and mass $5$ kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of $1200$ rpm. It is brought to rest in $10$ s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are ______ and ______ respectively.
Answer: (D) $0.0128\pi$ Nm, $100$
Concept. Constant torque gives constant $\alpha=\Delta\omega/t$, and $\tau=I\alpha$. The number of turns is the average angular speed (in rev/s) times the time.
Step 1. Moment of inertia. $I=\tfrac25(5)(0.04)^2=3.2\times10^{-3}$ kg m$^2$.
Step 2. Angular speed. $1200$ rpm $=20$ rev/s $=40\pi$ rad/s; $\alpha=\dfrac{40\pi}{10}=4\pi$ rad s$^{-2}$.
Step 3. Torque. $\tau=3.2\times10^{-3}\times4\pi=0.0128\pi$ N m.
Step 4. Rotations. Average $10$ rev/s for $10$ s: $100$.
Answer. $0.0128\pi$ N m, $100$ (Option D).
Solution by Sreeraj P, M.Sc Physics
