Work with tension and angles problem

[bcd201115]

Homework Statement

A child of weight w sits on a swing of length l. A variable horizontal force P that starts at zero and gradually increases is used to pull the child very slowly (so the kinetic energy is negligible small) until the swing makes an angle θ with the vertical. Calculate the work done the force P

Homework Equations

Can anyone help with this? I am completely lost as to how to even start.

The Attempt at a Solution

 

[Chestermiller]

Try drawing a free body diagram showing the 3 forces acting on the child (and attached swing) when the angle is θ (vertical gravity force, tension in cord, horizontal force P). At any point in this process, the child (and attached swing) are in equilibrium. Therefore the net horizontal and vertical components of force on the child are always zero. The vertical force balance gives you the tension in the cord. The horizontal force balance then gives you the horizontal force P as a function of the child's weight and the angle θ. This should get you started in determining the work.

 

[Chestermiller]

Incidentally, the force exerted by the cord is always perpendicular to the motion of the swing, so it does no work. Therefore, from an energy balance perspective, the work done by the horizontal force P is equal to the change in potential energy of the child. You can calculate this directly, or you can use the information in my previous reply to integrate the force P over the arc distance.

 

[fisselt]

I have a similar problem that I'm trying to understand.
So would it be correct to say that the work=Δy=l(1-cosθ)? Is there no work done calculated in the x direction? I'm a little lost.

 

[fisselt]

Actually, it should be the integral of the change in y then.

That would be w= L∫(1-cosθ) dθ

Closer to the correct answer now?