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?

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):

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.