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A simple pendulum has a bob of mass m and swings with an angular amplitude $ϕ$ . The tension in the thread is T. At a certain time, the string makes an angle $θ$ with the vertical $(θ \leq ϕ)$

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$T=mg cos θ, for all values of θ$

$T=mg cos θ, only for θ= ϕ$

${T=mg, for θ=cos}^{-1} [\frac{1}{3} (2cos ϕ +1)]$

T will be larger for smaller values of $θ$

answer is B, C, D.

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Detailed Solution

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

$o r \frac{m v^{2}}{ℓ} = 2 m g (\cos θ - \cos ϕ)$

also $T = m g (3 \cos θ - 2 \cos ϕ)$

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