A mass spring system w/ recoil and Friction ?

[gills]

Homework Statement

An object of mass, m, is traveling on a horizontal surface. There is a coefficient of kinetic friction, mu , between the object and the surface. The object has speed, v, when it reaches x=0 and encounters a spring. The object compresses the spring, stops, and then recoils and travels in the opposite direction. When the object reaches x=0 on its return trip, it stops.

Find k, the spring constant.
Express k in terms of mu , m, g, and v.

Homework Equations

Work-energy theorem

The Attempt at a Solution

I know that the work-energy theorem must be used in this problem. What is throwing me off is the friction i think because of the x and because it's a two stage problem? Not quite sure. Any help would be much appreciated.

 

[Astronuc]

The mass m initially passes x=0 with velocity v, from which one determines the kinetic energy. If the system were frictionless the spring would compress then rebound and the mass would pass with velocity v.

But with friction, it returns to x = 0 with v = 0, and therefore no kinetic energy. The spring only stores energy. Where did the kinetic energy go?

Remember work is force over distance. When the spring compresses some length L, what is the work/energy dissipated by the friction? What is then stored in the spring at deflection L? What happens on the recoil?

 

[gills]

What I'm working with right now:

--Net energy after the whole process (i.e. after the compression and recoil of spring):

E_f - E_i = W_nc

= 0 - (1/2)mv^2 = -mu*mg*2x (2x because it moves over the x distance twice-compression and recoil)

x = v^2/(4mu*g) -----> spring compression

--Net energy up until compressed spring:

E_f - E_i = W_nc

=(1/2)kx^2 - (1/2)mv^2 = -mu*mg*x

-->k = [m(v^2 - mu*g*x)]/x^2

Then to get rid of x variable I subbed in the x I solved in first eq. and i ended up with:

k = (12m*(mu)^2*g^2)/v^2

That's the best I've got so far. According to the answers in the back of my book, I'm only off by a multiplicative factor. I can't seem to find where that is.

 

[gills]

got the answer. I was making a stupid algebraic mistake.

Answer is = [8*(mu)^2*g^2*m]/v^2