Mass moment of inertia of a solid sphere Select the correct expression for the mass moment of inertia of a solid sphere of mass M and radius r about any diameter.
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A(2/3) M r^2
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B(2/5) M r^2
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CM r^2
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D(π r^4) / 2
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E(1/2) M r^2
Answer
Correct Answer: (2/5) M r^2
Explanation
Introduction / Context:The mass moment of inertia measures resistance to angular acceleration. For a homogeneous solid sphere, the distribution of mass is spherically symmetric, leading to a canonical result about any diameter.
Given Data / Assumptions:
- Homogeneous solid sphere.
- Mass M, radius r.
- Axis is any diameter (all diameters are equivalent by symmetry).
Concept / Approach:Using standard results from rigid-body dynamics or integration in spherical coordinates, the mass moment of inertia of a solid sphere about a diameter is I = (2/5) M r^2. This is lower than that of a thin spherical shell ((2/3) M r^2) because more mass is concentrated near the axis.
Step-by-Step Solution:
Recall standard formula: Solid sphere about diameter → I = (2/5) M r^2.Thin spherical shell would be I = (2/3) M r^2 (different object).Point mass at radius r would be I = M r^2 (upper bound among listed).Verification / Alternative check:Ratio comparisons: I_solid_sphere / (M r^2) = 0.4, which reflects interior mass distribution inside the radius.
Why Other Options Are Wrong:
- (2/3) M r^2: For a thin spherical shell, not a solid sphere.
- M r^2 and (1/2) M r^2: Do not match the known standard result.
- (π r^4)/2: This is an area moment expression, not a mass moment.
Common Pitfalls:Mixing area second moments with mass moments; confusing solid sphere with hollow shell values.
Final Answer:(2/5) M r^2