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Question

(C) The total kinetic energy of the system consists of translational kinetic energy and rotational kinetic energy.Total mass of the system $M = m + m

Vedclass · Accepted Answer

$\frac{4}{3} m v^2$

Vedclass · Answer

(C) The total kinetic energy of the system consists of translational kinetic energy and rotational kinetic energy.Total mass of the system $M = m + m = 2m$.Translational kinetic energy $KE_{trans} = \frac{1}{2} M v^2 = \frac{1}{2} (2m) v^2 = m v^2$.Since the sphere is hollow and filled with liquid,the liquid does not rotate with the sphere (it remains stationary relative to the ground). Only the hollow shell rotates.The moment of inertia of a thin hollow sphere is $I = \frac{2}{3} m R^2$.Rotational kinetic energy $KE_{rot} = \frac{1}{2} I \omega^2 = \frac{1}{2} (\frac{2}{3} m R^2) (\frac{v}{R})^2 = \frac{1}{3} m v^2$.Total kinetic energy $KE_{total} = KE_{trans} + KE_{rot} = m v^2 + \frac{1}{3} m v^2 = \frac{4}{3} m v^2$.

$A$ thin hollow sphere of mass $m$ is completely filled with a liquid of mass $m$. When the sphere rolls with a velocity $v$,the kinetic energy of the system is (neglect friction).

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