JEE Main · 2022 · 27 Jun 2022 · Shift 2 · Laws of Motion

At the midpoint of the rope, three forces act: tension $T$ in the upper half (at angle $\theta$ to vertical), tension in the lower half (vertical, equal to weight = 100 N), and the horizontal force of 30 N.

For equilibrium at the midpoint:

• Horizontal: $T\sin \theta =30$
• Vertical: $T\cos \theta =100$

Dividing these equations:
$\tan \theta =\frac{30}{100}=0.3=3\times 10^{-1}$

Given $\theta ={\tan }^{-1}(x\times 10^{-1})$, we have $x\times 10^{-1}=0.3$

Therefore, $x=3$.