Rigid-Body Dynamics – Moment of Inertia of a Thin Spherical Shell For a thin spherical shell of mass m and radius r, what is the mass moment of inertia about any diameter?

Introduction / Context:
Standard mass moment of inertia formulas are essential in rotational dynamics. A thin spherical shell concentrates mass at a constant radius, changing its inertia compared with a solid sphere.

Given Data / Assumptions:

• Rigid thin spherical shell (negligible thickness).
• Mass m uniformly distributed over the surface of radius r.
• Axis of rotation: any diameter of the sphere.

Concept / Approach:
Known results: Solid sphere about diameter: I = 2 m r^2 / 5. Thin spherical shell about diameter: I = 2 m r^2 / 3. The shell’s mass is farther from the center compared with a solid sphere, hence the larger coefficient.

Step-by-Step Solution:
Use standard result for a thin shell: I_diameter = 2 m r^2 / 3. No integration needed in an exam setting because it is a tabulated formula.

Verification / Alternative check:
Relative comparison: 2/3 ≈ 0.667 is greater than 2/5 = 0.4, reflecting the mass being distributed at radius r rather than throughout the volume.

Why Other Options Are Wrong:
m r^2 / 3: too small; not a standard result. 2 m r^2 / 5: that is for a solid sphere, not a thin shell. 3 m r^2 / 5 and m r^2: do not correspond to any standard spherical model about a diameter.

Common Pitfalls:
Mixing up solid vs thin-shell formulas; always recall shell has larger inertia for the same m and r.

Final Answer:
2 m r^2 / 3