Question

The pulleys and strings shown in the figure are smooth and a negligible mass. For the system to remain in equilibrium, Find the angle $\theta $.

Answer 1

Hint: The given system consists of weights attached to string through pulleys, Therefore figuring out all the types of forces acting in the system ( resolved forces) at the string at each points where weights are attached and then applying the condition for equilibrium conditions will give the value of the required angle for the given system.

Complete step by step answer:
Figure out which forces affect the block then choose a suitable coordinate system and rewrite the force equation to scalar form then decide and find the magnitude of the acceleration. According to the
question for block $A$ is $T - mg = m\left( 0 \right)$
For block $B$ is $T - mg = m\left( 0 \right)$
Since the both $A$ and $B$ blocks are similar and tension in the strings are mg. It has zero acceleration, as the system is in equilibrium.
So, $2T\cos \theta = \sqrt 2 mg$
$ \Rightarrow 2mg\cos \theta = \sqrt 2 mg$
$\Rightarrow \cos \theta = \dfrac{1}{{\sqrt 2 }}$
$\Rightarrow \theta = 45^\circ $

Note:
TENSION- Two or more physical objects that are in contact, exert forces on each other. Based on the objects in contact we give these contact forces different names. If the objects in contact are a string, rope, cable or a spring, the force is known to be tension.
Tension is a force which acts along the length of a medium, in other words a force carried by any flexible medium, such as a rope or cable.
Tension is defined as an action-reaction pair of forces acting at each end of the elements in contact. Let us consider a rope, the force of tension is felt by each and every section of the rope in both the directions, apart from the endpoints. The tension experienced by the endpoints is due to tension on one side and the force from the weight attached. The value of Tension varies throughout the string in some special cases.