An old record player of 10 cm radius turns at $10 rad / s$ while mounted on a $3 0^{\circ}$ incline as shown in the figure. A particle of mass $m$ can be placed any where on the rotating record. If the least possible coefficient of friction m that must exist for no slipping to occur is $μ$ , find $23 μ$ .

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ω 2 r = 100 × 0.1 = 10 Limiting condition : ∣ m ω 2 r cos θ ± m g sin θ ∣ = μ ( m g cos θ ∓ m ω 2 r sin θ ) 10 cos 3 0 ∘ + g sin 3 0 ∘ = μ ( g cos 3 0 ∘ − 10 sin 3 0 ∘ ) 5 3 ​ + 5 = μ ( 5 3 ​ − 5 ) μ = 3 ​ − 1 3 ​ + 1 ​ = 2 + 3 ​ 2 3 ​ μ = 2 3 ​ ( 2 + 3 ​ ) = 4 3 ​ + 6 6 ​

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