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

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The discussion focuses on the challenges of using force balancing and torque balancing to solve equations related to a mechanical system. Initial attempts revealed errors in the equations, particularly with the variables α and β. After corrections, the user successfully found the answer but sought guidance on formulating the virtual work equation. It was noted that virtual work is advantageous when dealing with constraint forces that do no work, while torque balancing is more straightforward for direct force applications. The conversation emphasizes the importance of understanding when to apply each method based on the system's constraints.
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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.
1.jpeg
2.jpeg
 
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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.
 
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haruspex said:
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.
 
cupcake_rainbow said:
thought of increasing angle POQ
But POQ is fixed by the geometry. To apply virtual work you wouid consider a small change to theta.
cupcake_rainbow said:
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.
 
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