A thin uniform elastic ring of radius R and density ρ rotates about its centre in its own plane. The material has breaking stress σ(breaking). Neglect the small elastic change in radius while calculating the centrifugal force. What is the maximum safe angular speed?
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Question

A thin uniform elastic ring of radius R and density ρ rotates about its centre in its own plane. The material has breaking stress σ(breaking). Neglect the small elastic change in radius while calculating the centrifugal force. What is the maximum safe angular speed?

Options

G1 Physics · Accepted Answer

For a rotating elastic ring, the tensile stress is σ = ρω²R². At the limiting safe condition, σ = σ(breaking). Therefore, σ(breaking) = ρω(max)²R². Hence, ω(max)² = σ(breaking) / [ρR²]. Thus, ω(max) = [1/R]√[σ(breaking)/ρ]. Correct option: A.