A thin wire of length L and linear mass density m is bent into a circular ring lying in the x–y plane. The y–y′ axis is tangent to the ring as shown. Find its moment of inertia about the tangent y–y′ axis. Image
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Question

A thin wire of length L and linear mass density m is bent into a circular ring lying in the x–y plane. The y–y′ axis is tangent to the ring as shown. Find its moment of inertia about the tangent y–y′ axis. Image

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G1 Physics · Accepted Answer

Step 1: Let M be the total mass of the ring. Linear mass density m = M/L, so M = mL. Step 2: Circumference L = 2πR gives radius R = L/(2π). Step 3: For a thin ring, moment of inertia about a diameter within its plane is I(diameter) = MR²/2 (by symmetry and the perpendicular-axis theorem). Step 4: The shown tangent axis is parallel to the diameter and a distance R from the centre. Apply parallel-axis theorem: I(tangent) = I(diameter) + MR² = (3/2)MR². Step 5: I(tangent) = (3/2)(mL)[L²/(4π²)] = 3…