Question

A hollow sphere is set into motion on a rough horizontal surface with a speed $v$ in the forward direction and an rotational speed $v/R$ in the anticlockwise direction as shown in figure. Find the translational speed of the sphere (a) when it stops rotating and (b) when slipping finally ceases and pure rolling starts.

• A. The angular momentum of the sphere about its centre of mass is conserved
• B. The speed of the sphere at the instant it stops rotating momentarily is $\frac{v}{3}$
• C. The speed of the sphere after pure rolling starts is $\frac{v}{5}$
• D. Work done by friction upto pure rolling starts is zero

Answer 1

Answer: (A), (B)

The correct options are
A The angular momentum of the sphere about its centre of mass is conserved
B The speed of the sphere at the instant it stops rotating momentarily is $\frac{v}{3}$

$a=\frac{f}{m}\alpha =\frac{5FR}{2mR}=\frac{5F}{2m}\omega ={\omega }_o-\alpha t$

When it stops rotating ,

$\omega =0\therefore t=\frac{{\omega }_o}{\alpha }=\frac{v_{o}2mR}{R5F}=\frac{v_{o}2m}{5F}v=v_o-at=v_o-\frac{f}{m}\times \frac{v_{o}2m}{5F}=v_o-\frac{2}{5}{v}_o=\frac{3v_o}{5}$

When it is in pure rolling

$-r\omega =v\therefore -(r{\omega }_0-r\alpha t)=v_o-at\Rightarrow -v_{o}t+\frac{5F}{2m}t=v_o-\frac{ft}{m}\Rightarrow t=\frac{4m{v}_o}{7f}\Rightarrow v=v_o-\frac{4m{v}_o}{7f}\times \frac{f}{m}=\frac{3v_o}{7}$