7.
Which one of the geometries of the same mass and radius has the maximum moment of
inertia?
a. a thin circular disk
b. a thin circular ring
c. a thin spherical shell
d. a solid sphere

Moment of Inertia and Rotational Resistance

Moment of inertia measures rotational resistance, depending on mass distribution relative to the axis of rotation.
For objects of identical mass (M) and radius (R), the thin spherical shell exhibits the highest value
about its diameter. This is consistent with standard physics comparisons used in competitive exams.

Option Analysis

Each option assumes rotation about a diameter (a common axis for fair comparison of planar and 3D shapes).

• Thin circular disk: Mass spreads from center to edge. Formula:
  I = (1/4)MR² = 0.25MR². Derived from the perpendicular axis theorem where the perpendicular-to-plane moment I = (1/2)MR²,
  so diameter value is half.
• Thin circular ring: All mass concentrates at radius R. About diameter:
  I = (1/2)MR² = 0.5MR². Uses perpendicular axis theorem on central perpendicular I = MR².
• Thin spherical shell: Mass entirely at surface, maximizing average distance squared from the axis.
  I = (2/3)MR² ≈ 0.667MR². Obtained via surface integration.
• Solid sphere: Mass fills volume, closer to the axis on average.
  I = (2/5)MR² = 0.4MR². Derived by integrating thin shells.

Comparison Table

Correct Answer and Insight

The thin spherical shell has the maximum moment of inertia.
Mass distributed farthest from the rotation axis yields the largest I, explaining why hollow objects resist rotation
more than solid ones of equal mass and size.