Question

The moments of inertia of a solid cylinder and a hollow cylinder of the same mass and same radius about the axes of the cylinders are \( I_1 \) and \( I_2 \). The relation between \( I_1 \) and \( I_2 \) is

• A. \( I_1I_2 \)
• B. \( I_1 = I_2 \)
• C. \( I_1I_2 \)
• D. \( I_1 = I_2 = 0 \)

Hint:

For objects with the same mass and radius, a hollow structure always has a greater moment of inertia than a solid one because its mass is distributed farther from the axis of rotation.

Answer 1

The Correct Option is A

The moment of inertia for a solid cylinder about its central axis is given by: \[ I_1 = \frac{1}{2} M R^2 \] where \( M \) is the mass and \( R \) is the radius. For a hollow cylinder (assuming a thin-walled structure), the moment of inertia is: \[ I_2 = M R^2 \] Clearly, \[ I_1 = \frac{1}{2} I_2 \Rightarrow I_1I_2. \] This shows that the moment of inertia of a hollow cylinder is greater than that of a solid cylinder of the same mass and radius.