Question

In the given arrangement, for the system to remain under equilibrium, the ‘\[\theta \]’ should be

A. \[{0^{\text{o}}}\]
B. \[{30^{\text{o}}}\]
C. \[{45^{\text{o}}}\]
D. \[{60^{\text{o}}}\]

Answer 1

Hint:We are asked to find the value of angle \[\theta \] when the system remains under equilibrium condition. By equilibrium it means the net force on the system is zero. First draw a free body diagram for the problem and balance the forces to find the value of \[\theta \].

Complete step by step answer:
Given a system and it is asked to find the value of \[\theta \] when the system is under equilibrium.When a system is in equilibrium position it means the net force on the system is zero or acceleration is zero.Let us draw a free body diagram for the system.

In the diagram above \[T\] represents the tension on the string and \[g\] is the acceleration due to gravity.
Now, balancing the forces for mass \[M\] we get,
\[T = Mg\] (i)
Balancing the forces for mass \[\sqrt 2 M\] we get,
\[T\cos \theta + T\cos \theta = \sqrt 2 Mg\]
\[ \Rightarrow 2T\cos \theta = \sqrt 2 Mg\]
Putting the value of \[T\] in equation (i) we get,
\[2Mg\cos \theta = \sqrt 2 Mg\]
\[ \Rightarrow 2\cos \theta = \sqrt 2 \]
\[ \Rightarrow \cos \theta = \dfrac{{\sqrt 2 }}{2}\]
\[ \Rightarrow \cos \theta = \dfrac{1}{{\sqrt 2 }}\]
\[ \Rightarrow \theta = {\cos ^{ - 1}}\dfrac{1}{{\sqrt 2 }}\]
\[ \therefore \theta = {45^{\text{o}}}\]

Therefore, the value of \[\theta \] when the system remains under equilibrium is \[{45^{\text{o}}}\].

Note:Remember, a system is under equilibrium condition means there is no change in the state of the system. Also, for such types of questions first draw a free body diagram and then proceed for solving the question. A free body diagram is a diagram showing the forces acting on a system with their magnitudes and directions. After drawing a free body diagram, balance the forces to form equations and solve to get the required answer.