A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (?) of thread deflection in the extreme position will be

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Q.
A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (θ) of thread deflection in the extreme position will be ______. [2024]

1 $\tan^{- 1} (\sqrt{2})$

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

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

4 $2 \tan^{- 1} (\frac{1}{\sqrt{5}})$

Ans.
(2)

Loss in kinetic energy = Gain in potential energy

$\Rightarrow \frac{1}{2} m v^{2} = m g ℓ (1 - \cos θ)$

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

Acceleration at lowest point $= \frac{v^{2}}{ℓ}$

Acceleration at extreme point $= g \sin θ$

$∴$ $\frac{v^{2}}{ℓ} = g \sin θ$

$∴$ $\sin θ = 2 (1 - \cos θ)$

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

Solution

Q.
A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (θ) of thread deflection in the extreme position will be ______. [2024]

1 $\tan^{- 1} (\sqrt{2})$

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

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

4 $2 \tan^{- 1} (\frac{1}{\sqrt{5}})$

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Solution

Q. A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (θ) of thread deflection in the extreme position will be ______. [2024]

1 tan-1(2)

2 2tan-1(12)

3 tan-1(12)

4 2tan-1(15)

Ans. (2) Loss in kinetic energy = Gain in potential energy ⇒12mv2=mgℓ(1-cosθ) ⇒v2ℓ=2g(1-cosθ) Acceleration at lowest point =v2ℓ Acceleration at extreme point =gsinθ ∴ v2ℓ=gsinθ ∴ sinθ=2(1-cosθ) ⇒tanθ2=12⇒θ=2tan-1(12)

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Q.
A ball suspended by a thread swings in a vertical plane so that its magnitude of acceleration in the extreme position and lowest position are equal. The angle (θ) of thread deflection in the extreme position will be ______. [2024]

1 $\tan^{- 1} (\sqrt{2})$

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

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

4 $2 \tan^{- 1} (\frac{1}{\sqrt{5}})$