Question

A solid cylinder of length \( L \) and diameter \( d \), made of a material with specific heat \( s \), is rotating about its axis with \( n \) rotations per second. When stopped, 50% of its energy is used in rising its temperature. The change in its temperature is:

• A. \( \frac{\pi^2 n^2 d^2}{s} \)
• B. \( \frac{ \ln^2 n d^2 }{16s} \)
• C. \( \frac{\pi^2 n^2 d^2}{2s} \)
• D. \( \frac{\pi^2 n^2 d^2}{8s} \)

Hint:

The temperature change due to the conversion of rotational kinetic energy into heat is related to the moment of inertia and angular velocity.

Answer 1

The Correct Option is D

The energy of a rotating cylinder is \( E = \frac{1}{2} I \omega^2 \), where \( I = \frac{1}{2} m r^2 \) and \( \omega = 2\pi n \). The energy used in heating the cylinder is related to the temperature change, and after simplifying the formula, we obtain: \[ \Delta T = \frac{\pi^2 n^2 d^2}{8s} \]