A ball suspended by a thread, swings in a vertical plane so that its acceleration in the extreme position and lowest position are equal in magnitude. Angle θ of thread deflection in the extreme positi

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A ball suspended by a thread, swings in a vertical plane so that its acceleration in the extreme position and lowest position are equal in magnitude. Angle θ of thread deflection in the extreme position will be:

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The image shows a diagram of a pendulum.

$a_{A} = g sin θ$ (only tangential)

$a_{B} = \frac{v ^{2}}{l}$ (only radial)

$K . E . + P . E . = \frac{1}{2} m 0^{2} + m g l (1 - cos θ) = \frac{1}{2} m v^{2}$

$v^{2} = 2 g l (1 - cos θ)$ ...........(i)

$∴ a_{B} = \frac{v ^{2}}{l} = 2 g (1 - cos θ)$

Since, $a_{A} = a_{B}$

$∴ g sin θ = 2 g (1 - cos θ) ⟹ 2 sin \frac{θ}{2} cos \frac{θ}{2} = 2 \times 2 sin^{2} \frac{θ}{2}$

$⟹ tan \frac{θ}{2} = \frac{1}{2} ⟹ θ = 2 tan^{- 1} (\frac{1}{2})$

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