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A L shaped, uniform rod has its two arms of length l and 2l. It is placed on a horizontal table and a string is tied at the bend. The string is pulled horizontally so that the rod slides with constant speed. Find the angle $θ$ that the longer side makes with the string. Assume that the rod exerts uniform pressure at all points on the table.

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$π - \tan^{- 1} \frac{1}{4}$

$π - \tan^{- 1} \frac{1}{2}$

$π - \cos^{- 1} \frac{1}{4}$

$\sin^{- 1} \frac{1}{4}$

answer is A.

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Detailed Solution

Let friction on arm BC be f. Then friction on arm AB is 2f.
$∴$ F = 3f for uniform motion.

Net torque about B must be zero

$∴ 2 flsin α = f \cdot \frac{l}{2} \sin (90 - α) \Rightarrow \tan α = \frac{1}{4} \Rightarrow α = \tan^{- 1} (\frac{1}{4}) ∴ θ = π - \tan^{- 1} (\frac{1}{4})$

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