Which of the following has the highest moment of inertia when each of them has the same mass and the same…

Which of the following has the highest moment of inertia when each of them has the same mass and the same outer radius
  1. a ring about its axis, perpendicular to the plane of the ring
  2. a disc about its axis, perpendicular to the plane of the ring
  3. a solid sphere about one of its diameters
  4. a spherical shell about one of its diameters

Solution

The correct option is (1)
A ring about its axis, perpendicular to the plane of the ring.
(1) Moment of inertia of a ring about its axis and perpendicular to its plane \(=\mathrm{Mr}^2\)
(2) Moment of inertia of a disc about its axis and perpendicular to its plane \(=\frac{1}{2} \mathrm{Mr}^2=0.5 \mathrm{Mr}\)
(3) Moment of inertia of a solid sphere about one of its diameter \(=\frac{2}{5} \mathrm{Mr}^2=0.4 \mathrm{Mr}^2\)
(4) Moment of inertia of a spherical shell about one of its diameter \(=\frac{2}{3} \mathrm{Mr}^2=0.66 \mathrm{Mr}^2\)
Therefore, the moment of inertia of the ring is highest.

Asked in: JEE Mains - Rotational Motion - Test 1

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