A flexible uniform rope is bent into a circular ring of radius R. Its linear mass density is λ. The ring rotates about its centre in its own plane with constant angular speed ω. The tension in the rope is:
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Question

A flexible uniform rope is bent into a circular ring of radius R. Its linear mass density is λ. The ring rotates about its centre in its own plane with constant angular speed ω. The tension in the rope is:

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G1 Physics · Accepted Answer

Take a small rope element subtending angle dθ at the centre. Its length is R dθ and mass dm = λR dθ. The required inward centripetal force is dm·ω²R = λω²R² dθ. If the tension in the rope is T, the inward resultant of the tensions at the two ends of the small element is 2T sin(dθ/2) ≈ T dθ. Hence T dθ = λω²R² dθ, giving T = λω²R².