A solid cylinder of mass M and radius R is rotating about its geometrical axis. A solid sphere of the same mass and same radius is also rotating about its diameter with an angular speed half that of t

Question

A solid cylinder of mass M and radius R is rotating about its geometrical axis. A solid sphere of the same mass and same radius is also rotating about its diameter with an angular speed half that of the cylinder. The ratio of the kinetic energy of rotation of the sphere to that of the cylinder will be

TestHub · Accepted Answer

Moment of inertia of the solid cylinder about its geometrical axis is $I_{c} = \frac{1}{2} M R^{2}$

Moment of inertia of the solid sphere about its diameter is $I_{s} = \frac{2}{5} M R^{2}$

Given that the angular speed of the cylinder is $ω_{c} = ω$ and that of the sphere is $ω_{s} = \frac{ω}{2}$

Rotational kinetic energy of the cylinder is $K_{c} = \frac{1}{2} I_{c} ω_{c}^{2} = \frac{1}{2} (\frac{1}{2} M R^{2}) ω^{2} = \frac{1}{4} M R^{2} ω^{2}$

Rotational kinetic energy of the sphere is $K_{s} = \frac{1}{2} I_{s} ω_{s}^{2} = \frac{1}{2} (\frac{2}{5} M R^{2}) (\frac{ω}{2})^{2} = \frac{1}{20} M R^{2} ω^{2}$

The ratio of their kinetic energies is $\frac{K _{s}}{K _{c}} = \frac{\frac{1}{20} M R ^{2} ω ^{2}}{\frac{1}{4} M R ^{2} ω ^{2}} = \frac{4}{20} = \frac{1}{5}$

Answer: $1 : 5$

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