Question

A solid sphere of radius r and mass m is spinning about a diameter as axis with a speed ${\omega }_0$. The temperature of the sphere increases by ${100}^{\circ }C$ without any other disturbance. If the coefficient of linear expansion of material of sphere is $2\times {10}^{-4}{/}^{\circ }C$, the ratio of angular speed at ${100}^{\circ }C$ and ${\omega }_0$ is

• A. 1:1
• B. 1:1.04
• C. 1.04:1
• D. 1:1.02

Answer 1

Answer: (B)

1:1.04

Since no external torque is acting on the system, angular momentum will be conserved
So, $I_1{w}_1=I_2{w}_2$
Now, due to thermal expansion, $I_2$ will not be same as that of $I_1$
Moment of inertia $\alpha L^2$
So, change in moment of inertia = $I_o\left(2\alpha \right)\Delta T$
Note that we are writing $2\alpha$ because it is just like area expansion
So change in moment of inertia = $2I\alpha t$
So, $I_2=I_1(1+2\alpha t)$ with t = 100
So, $w_2$ = $\frac{w_1}{1+200\alpha }$
So, ratio = 1 : 1.04