How Does Friction Affect the Final Speed of a Particle on a Curved Path?

[Picrill]

Homework Statement

A particle of mass m is moving along a path composed by two two quarters of circumference of radius R (see attached picture). The spring acting on the particle has a known constant equal to k and an equilibrium length equal to R (the radius again). Due to friction, on the particle acts a constant force F. Find the speed of the particle when it gets on the table.

Homework Equations

Final speed of the particle?

The Attempt at a Solution

I would start by finding the total initial energy of the system, namely the potential energy stored in the spring plus the potential energy of the particle. The spring is compressed by mg/k so its energy is 1/2 (mg/k)^2. Therefore the mass energy is mg(R - mg/k). Summing this with some algebra I get:

E = 1/2 * (mg/k) * (2Rk - mg)

Then I have no idea how to use this. The kinetic energy theorem does not work, because there is friction, right? Anything?

Thanks!

 

[Picrill]

I feel retarded but I don't see where the edit button is, so I'm just going to write something here. I wanted to make clear that F in the problem is the friction force experienced by the particle. And also, my calculations above are wrong, the spring is stretched by R at the initial position (see picture).

 

[Redbelly98]

I can't see your figure yet, as it is pending approval by the forum administrators, so my following comment may or may not apply.

You can use energy "conservation" if the friction force F acts over a known distance D. The final energy will not be equal to the initial energy, but instead will be reduced by a certain amount.

That "certain amount" is the work done by the friction (and there is a simple formula relating work to force and distance).