Torque Balancing and Virtual Work: Solving for Equations in Force Balancing

[cupcake_rainbow]

Homework Statement

Two weights P and Q are suspended from a fixed point 0 by strings OA, OB and are kept apart by a light rod AB. If the strings OA and OB make angles alpha and beta with the rod AB, show that the angle theta which the rod makes with the vertical is given by tan theta = (P+Q)/(Pcot alpha -Qcot beta)

Relevant Equations

torque balancing

So I tried using force balancing.
I have attached files of my solution in my notebook
Torque balancing about O which gives me equation 1
And I used property of triangle, for equation 2
But i can't seem to get the right answer
Also, I was wondering if it could be done by concept of virtual work.

 

[haruspex]

I think you have a couple of errors in eqn (1). α and β seem to have got crossed over, and a sign looks wrong.

 

[cupcake_rainbow]

I think you have a couple of errors in eqn (1). α and β seem to have got crossed over, and a sign looks wrong.

Yup got the answer now, thanks
I am interested in finding out how to write the virtual work equation for this. I thought of increasing angle POQ, say $\phi$ to $\phi +d\phi$ but i couldn't relate $d\alpha$ and $d\beta$ . So if anyone can write the equation, that would be great.
Also I want to know when it's better to use virtual work and when just torque balancing is better.

 

[haruspex]

thought of increasing angle POQ

But POQ is fixed by the geometry. To apply virtual work you wouid consider a small change to theta.

when it's better to use virtual work

Generally speaking, virtual work is useful when there are constraint forces that do no work, such as a bead sliding on wire. Using v.w. you can ignore such forces.