Question

A solid sphere and hollow sphere of the same mass and radius are given a spin about their centre of mass and then, they are placed on a rough horizontal surface. The spin angular velocity is the same for both the spheres and it is equal to ${\omega }_0$ . Once the pure rolling starts; let $v_1$ and $v_2$ be the linear speeds of their centres of mass, then

• A. $v_1=v_2$
• B. $v_1>v_2$
• C. $v_1<v_2$
• D. Data is insufficient

Answer 1

Answer: (C)

From conservation of angular momentum about point of contact :

$\text{I}{\omega }_0=\text{I}\omega +\text{mRv}$
or $\text{I}{\omega }_0=\text{I}\frac{\text{v}}{\text{R}}+\text{mRv}$
or $\text{v}=\frac{\text{I}{\omega }_0}{\frac{\text{I}}{\text{R}}+\text{mR}}$
or
or $\text{v}=\frac{{\omega }_0}{\frac{1}{\text{R}}+\frac{\text{mR}}{\text{I}}}$
Now ${\text{I}}_{\text{solid}\text{sphere < }}{\text{I}}_{\text{hollow}}$
∴ ${\text{v}}_{\text{solid}<}{\text{v}}_{\text{hollow}}$
∴ ${\text{v}}_1<{\text{v}}_2$