JEE MAIN - Physics (2023 - 6th April Evening Shift - No. 26)

A simple pendulum with length $$100 \mathrm{~cm}$$ and bob of mass $$250 \mathrm{~g}$$ is executing S.H.M. of amplitude $$10 \mathrm{~cm}$$. The maximum tension in the string is found to be $$\frac{x}{40} \mathrm{~N}$$. The value of $$x$$ is ___________.
Answer
101

Explanation

Given the amplitude $$A$$ and the length $$l$$ of the pendulum, we can find the maximum angular displacement $$\theta_0$$:

$$ \sin \theta_0 = \frac{A}{l} = \frac{10}{100} = \frac{1}{10} $$

By conservation of energy, the following equation holds:

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

The maximum tension occurs at the mean position (i.e., when the pendulum is vertical). At this point, we have:

$$ \begin{aligned} & T - mg = \frac{m v^2}{l} \ & \Rightarrow T = mg + \frac{m v^2}{l} \end{aligned} $$

Substituting the conservation of energy equation, we get:

$$ \begin{aligned} & T = mg + 2 m g(1 - \cos \theta) \\\\ & = mg\left[1 + 2\left(1 - \sqrt{1 - \sin^2 \theta}\right)\right] \\\\ & = mg\left[3 - 2 \sqrt{1 - \frac{1}{100}}\right] \\\\ & = \frac{250}{1000} \times 10\left[3 - 2\left(1 - \frac{1}{200}\right)\right] = \frac{101}{40} \\\\ & \therefore x = 101 \end{aligned} $$

So, the value of $$x$$ is 101.

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