A thin ring of volume mass density ρ , cross-sectional area A , radius R and Young's modulus of elasticity E , is rotated about an axis passing through its centre with angular velocity ω . If inc

Question

A thin ring of volume mass density ρ , cross-sectional area A , radius R and Young's modulus of elasticity E , is rotated about an axis passing through its centre with angular velocity ω . If increase in circumference (assumed small) is a E 18 π R 3 ρ ω 2 ​ , find a .

TestHub · Accepted Answer

Take a small element subtending d θ : 2 T sin 2 d θ ​ = d m ω 2 R with d m = ρ A R d θ . T d θ = ρ A R 2 ω 2 d θ ⇒ T = ρ A R 2 ω 2 Stress = A T ​ = ρ R 2 ω 2 strain = C Δ C ​ = E ρ R 2 ω 2 ​ Δ C = 2 π R ⋅ E ρ R 2 ω 2 ​ = E 2 π R 3 ρ ω 2 ​ = 9 E 18 π R 3 ρ ω 2 ​ So a = 9 . Answer: 9.

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