A stone is tied to a string of length l and is moving in vertical circle such that the tension in the string at the highest position of stone is zero. Find the angle θ of string with lowest position when tension in the string is equal to the weight of stone

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$\cos^{- 1} (\frac{1}{3})$

$\sin^{- 1} (\frac{1}{3})$

$\cos^{- 1} (\frac{2}{3})$

$\cos^{- 1} (- \frac{2}{3})$

answer is C.

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Detailed Solution

If tension at the highest point is zero then velocity at the lowest point must be critical i.e. $\sqrt{5 g l}$

now tension at any point if v₀ is the velocity at the lowest point,

$T = \frac{{mv}_{0}^{2}}{l} - 2 mg + 3 mgcosθ = mg 5 mg - 2 mg + 3 mgcosθ = mg solving for θ cosθ = - \frac{2}{3}$

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