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A semicircular ring has mass 2 kg and radius 2 m. Let I_O be the moment of inertia about an axis perpendicular to the plane of the ring and passing through the geometric center O of the full circle (midpoint of the diameter), and let I_C be the moment of inertia about a parallel axis through the midpoint C of the arc (the point on the arc that lies on the axis of symmetry, at distance R from O). Which of the following statements is/are correct?

1. I_O = 8 kg-m²
2. The ratio I_C / I_O equals 2.
3. I_C - I_O = 8 kg-m²
4. All of the above

Correct answer: All of the above

Solution

I_O = MR² = 2*4 = 8 kg-m²; applying the parallel axis theorem with d = R gives I_C = 2MR² = 16 kg-m², so the ratio is 2 and the difference is 8 kg-m² — all three sub-statements are true.

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