Question

A uniform disc is spinning about geometrical axis in free space. If its temperature is increased by $\Delta T$, the fractional change in its angular velocity is ($\alpha$ is coefficient of linear expansion and $\alpha \Delta T<<1)$

• A. $2\alpha \Delta T$
• B. $-2\alpha \Delta T$
• C. $-\alpha \Delta T$
• D. $\alpha \Delta T$

Answer 1

The correct option is B $-2\alpha \Delta T$

As the disc expands, no external torque acts on the system, thus angular momentum is conserved.
Due to expansion radius increases from $r_0$ to $r$, the moment of inertia increases and angular velocity decreases from ${\omega }_0$ to $\omega$.

$L_i=L_f$
$\frac{m{r}_0^2}{2}{\omega }_0=\frac{m{r}^2}{2}\omega$

$\Rightarrow \omega ={\left(\frac{r_0}{r}\right)}^2{\omega }_0$

$\Rightarrow \omega ={\left(\frac{1}{1+\alpha \Delta T}\right)}^2{\omega }_0(\because r=r_0(1+\alpha \Delta T\left)\right)$

$\Rightarrow \omega =(1+\alpha \Delta T{)}^{-2}{\omega }_0$

Using binomial theorem,

$\omega =(1-2\alpha \Delta T){\omega }_0(\because \alpha \Delta T<<1)$

$\Rightarrow \frac{\omega -{\omega }_0}{{\omega }_0}=-2\alpha \Delta T$