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

A sphere of radius R is cut from a larger solid sphere of radius 2R as shown in the figure. The ratio of the moment of inertia of the smaller sphere to that of the rest part of the sphere about the Y-axis is :

A sphere of radius $R$ is cut from a larger solid sphere of radius $2R$ as shown in the figure. The ratio of the moment of inertia of the smaller sphere to that of the rest part of the sphere about the Y-axis is :

large sphere radius 2R centred on Y-axis with a sphere of radius R cut out on the right, touching the Y-axis (2R diameter marked)

1. $\frac{7}{8}$

2. $\frac{7}{40}$

3. $\frac{7}{57}$

4. $\frac{7}{64}$

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Answer: 3. $\frac{7}{57}$

Key idea: moment of inertia of the remaining part $=$ full sphere $-$ removed sphere, using the parallel-axis theorem for the cut-out.

Let the removed sphere have mass $m$ (radius $R$). The big sphere (radius $2R$) has mass $8m$.

Full sphere about the Y-axis: $\tfrac25(8m)(2R)^2=\tfrac{64}5mR^2$.
Cut sphere (centre at distance $R$ from the axis): $\tfrac25mR^2+mR^2=\tfrac75mR^2$.
Rest: $\tfrac{64}5-\tfrac75=\tfrac{57}5\,mR^2$.

$$\frac{I_{small}}{I_{rest}}=\frac{7/5}{57/5}=\frac7{57}$$

Doubtly AI · 30 Oct 2026 AI-generated · not yet verified Report

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A sphere of radius R is cut from a larger solid sphere of radius 2R as shown in the figure. The ratio of the moment of inertia of the smaller sphere to that of the rest part of the sphere about the Y-axis is :

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