The moment of inertia of a rod of mass M and length L about an axis through its centre and perpendicular to its length is α. The rod is cut into two equal parts and the two parts are joined symmetrically to form a cross. The moment of inertia of the cross about an axis through its centre and perpendicular to the plane of the cross is:
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Question

The moment of inertia of a rod of mass M and length L about an axis through its centre and perpendicular to its length is α. The rod is cut into two equal parts and the two parts are joined symmetrically to form a cross. The moment of inertia of the cross about an axis through its centre and perpendicular to the plane of the cross is:

Options

G1 Physics · Accepted Answer

For the original rod, α = ML²/12. After cutting, each rod has mass M/2 and length L/2. The axis passes through the centre of each small rod and is perpendicular to the plane. Moment of inertia of one part: I₁ = (1/12)(M/2)(L/2)² = ML²/96. There are two identical perpendicular parts, so I = 2I₁ = ML²/48. Since α = ML²/12, I = α/4. Correct option: B.