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