Topic: Moment of inertia, radius of gyration, MI of simple objects· Rotational Motion

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 an angular speed twice that of the sphere. The ratio of their kinetic energies of…

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 an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation (\(E_{sphere}/E_{cylinder}\)) will be

1. 1 : 5

2. 1 : 4

3. 3 : 1

4. 2 : 3

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Answer: 1. 1 : 5

Rotational kinetic energy is \(E=\tfrac12 I\omega^2\), so find each moment of inertia first.

Sphere about a diameter: \(I_s=\tfrac25mR^2\), so \(E_s=\tfrac12\cdot\tfrac25mR^2\omega^2=\tfrac15mR^2\omega^2\).
Cylinder about its axis: \(I_c=\tfrac12mR^2\) with angular speed \(2\omega\), so \(E_c=\tfrac12\cdot\tfrac12mR^2(2\omega)^2=mR^2\omega^2\).

\(E_s/E_c=\tfrac15 : 1 = 1:5\). Hence option (1).

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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 an angular speed twice that of the sphere. The ratio of their kinetic energies of…

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