A smaller solid sphere of radius R is removed from a solid sphere of radius 2R as shown. Both have uniform density. Find the ratio of moment of inertia of removed smaller sphere to that of the remaining body about the shown Y-axis. Image
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Question

A smaller solid sphere of radius R is removed from a solid sphere of radius 2R as shown. Both have uniform density. Find the ratio of moment of inertia of removed smaller sphere to that of the remaining body about the shown Y-axis. Image

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G1 Physics · Accepted Answer

Step 1: Let mass of the large sphere be M. The small sphere has radius half as large, so mass m = M(1/2)³ = M/8. Step 2: Large sphere moment of inertia about central Y-axis: Iₗ = (2/5)M(2R)² = 8MR²/5. Step 3: The small sphere’s centre is distance R from the large sphere’s Y-axis (as in figure). Step 4: By parallel-axis theorem, I(small) = (2/5)mR² + mR² = (7/5)(M/8)R² = 7MR²/40. Step 5: Remaining body has I(remainder) = Iₗ − I(small) = 8MR²/5 − 7MR²/40 = (64 − 7)MR²/40 = 57MR²/40. Step 6: Ratio…