Displacement of a Pendulum to find Work done by Force

[shinify]

I'm having trouble with this problem. I did: F_Tcosθ-mg= 0, solving for F_T, getting F_T= mg/cosθ. Then, along the x-axis, F-F_Tsinθ=0, solving for F, getting F= mgtanθ. Not sure how to go about it from here. What is the direction of the displacement?

 

[haruspex]

The force is always horizontal, so for the work done you want the horizontal displacement. But.. why not just use conservation of energy?

 

[shinify]

What do you mean?

 

[haruspex]

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.

 

[Nickie]

First of all, great job on solving for the necessary forces in the pendulum system! To find the work done by the force, we need to consider the direction of the displacement. In this case, the displacement is along the path of the pendulum's swing, which is perpendicular to the force of gravity. This means that the displacement is perpendicular to the direction of the force, and therefore, no work is done by the force.

To further explain, work is defined as the product of the force applied and the displacement in the direction of the force. In this case, the force of gravity is acting in the vertical direction, while the displacement is in the horizontal direction. This means that there is no component of the force acting in the direction of the displacement, resulting in no work done.

In summary, your calculations are correct, but the direction of the displacement is perpendicular to the force of gravity, resulting in no work done by the force. I hope this helps clarify your understanding of the problem. Keep up the good work!