Displacement of a Pendulum to find Work done by Force

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SUMMARY

The discussion focuses on calculating the work done by a force on a pendulum during its displacement. The user derived the tension force (FT) as FT = mg/cosθ and the horizontal force (F) as F = mgtanθ. The conversation emphasizes the importance of considering the direction of displacement, specifically the horizontal component, and suggests using integration to find work done, expressed as ∫F.dx. The discussion also highlights the relevance of energy conservation, noting that work can be converted into potential energy (PE), kinetic energy (KE), or lost to friction.

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I'm having trouble with this problem. I did: FTcosθ-mg= 0, solving for FT, getting FT= mg/cosθ. Then, along the x-axis, F-FTsinθ=0, solving for F, getting F= mgtanθ. Not sure how to go about it from here. What is the direction of the displacement?
 
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The force is always horizontal, so for the work done you want the horizontal displacement. But.. why not just use conservation of energy?
 
What do you mean?
 
The force has done work on the system. Where has that work gone? There are three possibilities: PE, KE and frictional losses. Which apply here?
If you want to do it by integration, ∫F.ds, since the force acts horizontally that becomes ∫F.dx, where F is the scalar magnitude of F and dx is the horizontal component of ds. You have F as a function of θ, so next you need dx in terms of θ and dθ.
 

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