Moment of inertia of a solid sphere about a tangential axis Find the area moment of inertia (second moment of mass) of a solid sphere of mass m and radius r about an axis tangent to its surface.

Introduction / Context:
Moments of inertia for standard solids are key in rotational dynamics. The tangent-axis value can be obtained using the parallel-axis theorem from the centroidal value.

Given Data / Assumptions:

• Solid (uniform) sphere of mass m and radius r.
• Known centroidal MOI of a solid sphere: I_center = 2 m r^2 / 5 about any diameter.
• Axis sought is tangent to the sphere (parallel to a diameter and shifted by r).

Concept / Approach:
Apply the parallel-axis theorem: I_tangent = I_center + m d^2, where d is the perpendicular distance between axes. Here, d = r.

Step-by-Step Solution:

Verification / Alternative check:
Dimensional check: units are kg·m^2. Value exceeds the centroidal MOI, as expected when shifting the axis away from the center.

Why Other Options Are Wrong:

• 2 m r^2 / 5: centroidal value only, not tangent.
• 2 m r^2 / 3 and 7 m r^2 / 3: incorrect coefficients; no standard derivation yields these.

Common Pitfalls:
Forgetting to add m r^2 when moving to a tangent axis; mixing up solid versus hollow sphere formulas.

Final Answer:
7 m r^2 / 5