A uniform rod of length L rests against a smooth roller as shown in Fig. Find the friction coefficient between the ground and the lower end if the minimum and that the rod makes with the horizontal is θ θ .
Text Solution
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Balance horizontal force μ μ N 2 = N 1 sin θ θ
Balance vertical forces N 1 cos θ θ + N 2 = mg
or N 1 
= mg
or N 1 = 
Apply ∑ ∑ τ τ = 0 about point A.
N 1 sin θ θ h + N 1 cos θ θ h cot θ θ = mg 1/2 cos θ θ
hN 1 [sin 2 θ θ + cos 2 θ θ ]= 
cos θ θ sin θ θ
2 μ μ h = μ μ l cos 2 θ θ sin θ θ + l cos θ θ sin 2 θ θ
μ μ (2h – l cos 2 θ θ sin θ θ ) = l cos θ θ sin 2 θ θ
or μ μ = 
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