A simple pendulum with length 100 cm and bob of mass 250 g is executing S.H.M. of amplitude 10 cm. The maximum tension in the string is found to be The value of is ________ . [2023]

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Solution

Q.
A simple pendulum with length 100 cm and bob of mass 250 g is executing S.H.M. of amplitude 10 cm. The maximum tension in the string is found to be $\frac{x}{40} N .$ The value of $x$ is ________ . [2023]

Ans.
(99)

$\sin θ_{c} = \frac{A}{l} = \frac{10}{100} = \frac{1}{10}$

$From conservation of energy:$

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

$Maximum tension occurs at mean position.$

$∴$ $T - m g = \frac{m v^{2}}{l} \Rightarrow T = m g + \frac{m v^{2}}{l}$

$∴$ $T = m g + 2 m g (1 - \cos θ)$

$= m g [1 + 2 (1 - \sqrt{1 - \sin^{2} θ})] = m g [3 - 2 \sqrt{1 - \frac{1}{100}}]$

$= \frac{250}{1000} \times 9.8 [3 - 2 (1 - \frac{1}{200})] = \frac{99}{40}$

$∴$ $x = 99$

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