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 :
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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 :

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G1 Physics · Accepted Answer

Loss in kinetic energy = Gain in potential energy ½mv² = mgℓ(1 - cosθ) ⇒ v²/ℓ = 2g(1 - cosθ) Acceleration at lowest point = v²/ℓ Acceleration at extreme point = g sinθ Given that both accelerations are equal: v²/ℓ = g sinθ ⇒ sinθ = 2(1 - cosθ) Using sinθ = 2 sin(θ/2) cos(θ/2) and 1 - cosθ = 2 sin²(θ/2), ⇒ tan(θ/2) = 1/2 Therefore, θ = 2 tan⁻¹(1/2)