A solid sphere of mass ' m ' and radius ' R ' is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with ang

Question

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

TestHub · Accepted Answer

The moment of inertia of a solid sphere about its diameter is $I_{s} = \frac{2}{5} m R^{2}$ .

The moment of inertia of a solid cylinder about its geometrical axis is $I_{c} = \frac{1}{2} m R^{2}$ .

Let the angular speed of the sphere be $ω_{s} = ω$ .

The angular speed of the cylinder is given as $ω_{c} = 2 ω$ .

The kinetic energy of the solid sphere is:

$K_{s} = \frac{1}{2} I_{s} ω_{s}^{2} = \frac{1}{2} (\frac{2}{5} m R^{2}) ω^{2} = \frac{1}{5} m R^{2} ω^{2}$

The kinetic energy of the solid cylinder is:

$K_{c} = \frac{1}{2} I_{c} ω_{c}^{2} = \frac{1}{2} (\frac{1}{2} m R^{2}) (2 ω)^{2} = m R^{2} ω^{2}$

The ratio of their kinetic energies is:

$\frac{K _{s}}{K _{c}} = \frac{\frac{1}{5} m R ^{2} ω ^{2}}{m R ^{2} ω ^{2}} = \frac{1}{5}$

Answer: $1 : 5$

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