Question

Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?

Answer 1

The moment of inertia for the hollow cylinder=$I_1=m{r}^2$

The moment of inertia for the solid sphere=$I_2=\frac{2}{5}m{r}^2$

For hollow sphere we have $\tau =I_1{\alpha }_1$

For solid sphere we have $\tau =I_2{\alpha }_2$

$⟹\frac{{\alpha }_2}{{\alpha }_1}=\frac{I_1}{I_2}=\frac{5}{2}>1$

Thus ${\alpha }_2>{\alpha }_1$

$\omega \left(t\right)={\omega }_0+\alpha t$

The angular velocity($\omega$) at a certain time will be greater for solid sphere.