Question

A hollow sphere of radius $R$ and mass $m$ is fully filled with water 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 :

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

Answer 1

Answer: (C)

Total kinetic energy = Rotational KE of sphere + Translational KE of sphere + Translation KE of water inside it.
$K=\frac{1}{2} I \omega^{2}+\frac{1}{2} m v^{2}+\frac{1}{2} m v^{2}$
$\Rightarrow K=\left(\frac{1}{2} \times \frac{2}{3} m R^{2} \times \frac{v^{2}}{R^{2}}\right) +\frac{1}{2} m v^{2}+\frac{1}{2} m v^{2}$
$\Rightarrow K=\frac{1}{3} m v^{2}+m v^{2}$
$\Rightarrow K=\frac{4}{3} m v^{2}$
Angular momentum of sphere about a fixed point on ground,
$L=(I \omega+m v R)+m v R$
$\Rightarrow L=\frac{2}{3} m R^{2} \times \frac{v}{R}+2 m v R$
$\Rightarrow L=\frac{2}{3} m v R+2 m v R$
$\Rightarrow L=\frac{8 m v R}{3}$