Question

A hollow sphere of mass M and radius R is rotating with angular frequency $\omega$. It suddenly stops rotating and 75% of kinetic energy is converted to heat. If $s$ is the rise in temperature of the sphere is ________.

(Moment of inertia of hollow sphere $=\frac{2}{3}M{R}^2)$

• A. $\frac{R\omega }{4s}$
• B. $\frac{R^2{\omega }^2}{4s}$
• C. $\frac{R\omega }{2s}$
• D. $\frac{R^2{\omega }^2}{2s}$

Answer 1

The correct option is B $\frac{R^2{\omega }^2}{4s}$

Moment of inertia of the hollow sphere, $I=\frac{2}{3}M{R}^2$

Initial kinetic energy, $E_i=\frac{1}{2}I{w}^2=\frac{1}{3}M{R}^2{w}^2$

Final kinetic energy of the sphere, $E_f=0$

As $75$ % of initial kinetic energy is converted into heat.

$\therefore$ $E_{loss}=E_{heat}$

$0.75\times \frac{1}{3}M{R}^2{w}^2=Ms\left(\Delta T\right)$ where $s$ is the specific heat capacity of the sphere

$⟹$ Rise in temperature, $\Delta T=\frac{R^2{w}^2}{4s}$