A homogeneous rod AB of mass m and length ℓ resting with its lower end against a vertical wall and kept in inclined position by a string CD as shown in the figure. Given that AD = 4 AB . Angles form

Question

A homogeneous rod AB of mass m and length ℓ resting with its lower end against a vertical wall and kept in inclined position by a string CD as shown in the figure. Given that AD = 4 AB ​ . Angles formed by the string and the rod with the wall are α and β respectively. Coefficient of friction between end A of the rod and vertical wall is μ . Choose the correct options.

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Equilibrium: m g = f + T cos α , N = T sin α , and torques about D : m g + f = N cot β . Eliminating: f = N ( cot β − cot α ) . α = β ⇒ f = 0 - (A). β > α ⇒ f < 0 , i.e. friction acts downward - (B). β < α ⇒ f upward - (C). β = 5 3 ∘ , α = 3 7 ∘ : μ ≥ cot 5 3 ∘ − cot 3 7 ∘ < 0 , so any μ works - (D) wrong. (A), (B), (C)

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