The problem describes a system of masses connected by strings over pulleys. The masses are $m$, $\sqrt{2}m$, and $m$. The system is in equilibrium. The question asks to find the value of the angle $\theta$.

1. Problem Description

The problem describes a system of masses connected by strings over pulleys. The masses are $m$, $\sqrt{2}m$, and $m$. The system is in equilibrium. The question asks to find the value of the angle $\theta$.

2. Solution Steps

Since the system is in equilibrium, the tension in the strings is equal to the weight of the masses $m$. Let $T$ be the tension in the strings. Therefore, $T = mg$. The tension in the middle string holding $\sqrt{2}m$ is equal to $\sqrt{2}mg$. At the point where the three strings meet, the vertical component of the tension from the two strings with mass $m$ must equal the tension in the middle string.

The vertical component of the tension in each string is $T\sin\theta$. Therefore:

$2T\sin\theta = \sqrt{2}mg$

Since $T = mg$,

$2mg\sin\theta = \sqrt{2}mg$

$2\sin\theta = \sqrt{2}$

$\sin\theta = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}$

$\theta = \sin^{-1}(\frac{1}{\sqrt{2}}) = 45^\circ$

3. Final Answer

(4) 45°