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 :

$\textit{I} \omega _{0} = \textit{I} \omega + \textit{mRv}$
or $\textit{I} \omega _{0} = \textit{I} \frac{\textit{v}}{\textit{R}} + \textit{mRv}$
or $\textit{v} = \frac{\textit{I} \omega _{0}}{\frac{\textit{I}}{\textit{R}} + \textit{mR}}$
or
or $\textit{v} = \frac{\omega _{0}}{\frac{1}{\text{R}} + \frac{\textit{mR}}{\textit{I}}}$
Now $\textit{I}_{\text{solid} \text{sphere < }} \textit{I}_{\text{hollow}}$
∴ $\textit{v}_{\text{solid} < } \textit{v}_{\text{hollow}}$
∴ $\textit{v}_{1} < \textit{v}_{2}$