Question

An object of mass m is traveling on a horizontal surface. There is a coefficient of kinetic friction μ 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.

Part A 

Find k, the spring constant.
Express kin terms of μ, m, g, and v.

Answer 1

Part A Answer

This problem must be solved in several steps. First, figure out the potential energy of the spring, based on the starting velocity, as well as the change in energy from friction:

1/2mv2 = μmgx + 1/2kx^2

When the spring recoils backwards, there will be further energy loss to friction:

1/2kx^2 = μmgx

So we can say that:

1/2mv^2 = μmgx + 1/2kx^2

And substitute in for (1/2kx^2):

1/2mv^2 = μmgx + μmgx
1/2mv^2 = 2 * μmgx
1/4mv^2 = μmgx
1/4v^2 = μgx
x = (1/4v^2) / (μg)

Now we can substitute x back in to the formula 1/2kx^2 = μmgx. It’s simpler to isolate k before substituting, so we’ll do that:

1/2kx^2 = μmgx
1/2kx = μmg
k = 2μmg/x
k = 2μmg / ((1/4v^2) / (μg))
k = 2μmg * ((μg) / (1/4v^2))
k = 2 * m * (μg)^2 / (1/4v^2)
k = 8m * (μg/v)^2

8m * (μg/v)^2