Rigid Body Dynamics — Solid Sphere What is the mass moment of inertia of a solid sphere of mass m and radius r about any diameter?
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A2 m r^2 / 3
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B2 m r^2 / 5
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Cm r^2
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Dm r^2 / 2
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E—
Answer
Correct Answer: 2 m r^2 / 5
Explanation
Introduction / Context: The mass moment of inertia of standard 3D bodies is fundamental in dynamics and vibration. The solid sphere’s inertia about a diameter is a classic closed-form result used in rolling motion and rotational kinetic energy calculations.
Given Data / Assumptions:
- Uniform solid sphere of mass m, radius r.
- Axis: any diameter through the center (symmetry ensures all diameters are equivalent).
Concept / Approach: For a sphere, I_diameter is obtained by integrating r_perp^2 over the full volume with constant density. The standard result is I = 2/5 m r^2.
Step-by-Step Solution (outline):
Use spherical coordinates and exploit symmetry to evaluate ∫ r_perp^2 dm. The integral yields I = (2/5) m r^2 for a uniform solid sphere.Verification / Alternative check: Compare with a thin spherical shell (hollow sphere): I_shell = (2/3) m r^2, which is larger because more mass is located farther from the center. The solid sphere must have a smaller coefficient than 2/3, and 2/5 satisfies this.
Why Other Options Are Wrong: “2 m r^2 / 3” is for a thin spherical shell, not a solid sphere; “m r^2” and “m r^2 / 2” are not correct for any standard spherical distribution about a diameter.
Common Pitfalls: Mixing up solid vs hollow results; confusing diameter-axis inertia with polar moments of other shapes.
Final Answer: 2 m r^2 / 5.