Question

A uniform metallic disc of radius r and mass m is spinning with angular speed ${\omega }_0$ about an axis passing through its centre and perpendicular to its plane. If its temperature is increased (slightly) by $\Delta T$ without disturbing the disc, its new angular speed is (The coefficient of linear expansion of the metal is $\alpha$ )

• A. ${\omega }_0(1+2\alpha \Delta T)$
• B. ${\omega }_0(1+\alpha \Delta T)$
• C. ${\omega }_0(1-2\alpha \Delta T)$
• D. ${\omega }_0(1-3\alpha \Delta T)$

Answer 1

The correct option is C ${\omega }_0(1-2\alpha \Delta T)$

From conservation of angular momentum
$\frac{m{R}^2\omega }{2}=constant\Rightarrow 2\frac{\Delta R}{R}+\frac{\Delta \omega }{\omega }=0\Rightarrow \Delta \omega =-2\omega \frac{\Delta R}{R}=-2\omega \alpha \Delta T\Rightarrow {\omega }^{\prime }={\omega }_0+\Delta \omega ={\omega }_0-2{\omega }_0\alpha \Delta T={\omega }_0(1-2\alpha \Delta T)$