Question

A hollow sphere of radius $R$ and mass $m$ is fully filled with a non viscous liquid of mass $m$. It is rolled down a horizontal plane such that its centre of mass moves with a velocity $v$. If it purely rolls then :

• A. Kinetic energy of the sphere is $\frac{5}{6}m{v}^2$
• B. Kinetic energy of the sphere is $\frac{4}{5}m{v}^2$
• C. Angular momentum of the sphere about a fixed point on ground is $\frac{8}{3}mvR$
• D. Angular momentum of the sphere about a fixed point on ground is $\frac{14}{5}mvR$

Answer 1

The correct option is C Angular momentum of the sphere about a fixed point on ground is $\frac{8}{3}mvR$

The total angular momentum will be given by the sum of angular momentum of the hollow sphere and the translational momentum of both the hollow sphere and sphere of liquid mass. Thus we have
$L_{Total}=I_{hollowsphere}\omega +m_{hollowsphere}vR+m_{solidsphere}vR$ (momentum is about the fixed point on the ground)
or
$L_{Total}=\frac{2}{3}m{R}^2\omega +2mvR$
or
$L_{Total}=\frac{2}{3}m{R}^2\frac{v}{R}+2mvR$
or
$L_{Total}=(\frac{2}{3}+2)mvR$
or
$L_{Total}=\frac{8}{3}mvR$