QJEEM23Ph358e: JEE Main 2023 Physics - Class 11 JEEM23Ph358e Solution: A simple pendulum with length and bob of mass is executing…

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.