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.

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:

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