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Free Body Diagrams

In the figure...

Question

In the figure pulley $p_{1}$ is fixed and $p_{2}$
is movable. If $W_{1} = W_{2} = 100 N,$ what is the angle (in degrees) $A P_{2} P_{1}$ when the system is in equilibrium?

The pulleys are massless and frictionless

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Solution

Considering $P_{2}$ and $W_{2}$ to be system and drawing its FBD

Let the tension in the string $A P_{2}$ and $P_{2} P_{1}$ be T.
Accordingly tension in the entire string is T.

Considering the force on weight $W_{1}$ , we get
$T = W_{1}$
Further, let $∠ A P_{2} P_{1} = 2 θ$
Resolving tensions in horizontal and vertical directions and considering the forces on weight $W_{2},$ we get
$2 T c o s θ = W_{2}$
or $2 W_{1} c o s θ = W_{2} o r c o s θ = \frac{1}{2}$
or $θ = 60^{\circ}$
So $∠ A P_{2} P_{1} = 2 θ = 120^{\circ}$

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