A ring is hung from a nail and can make small oscillations without slipping: (i) in its own plane with period T₁, and (ii) back and forth perpendicular to its plane with period T₂. Find T₁/T₂.
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Question

A ring is hung from a nail and can make small oscillations without slipping: (i) in its own plane with period T₁, and (ii) back and forth perpendicular to its plane with period T₂. Find T₁/T₂.

G1 Physics · Accepted Answer

For oscillations in the ring’s plane about a tangent axis, I₁=I about CM+MR²=2MR², so T₁∝√2R. For oscillation about a tangent diameter perpendicular to the ring’s plane, I₂=(1/2)MR²+MR²=(3/2)MR². Since the centre-of-mass distance R is the same, T₁/T₂=√(I₁/I₂)=√(4/3)=2/√3. Correct option: 1.