**A Mass-Spring System with Recoil and Friction**

Part A = F = mg/2

Part B = W_{d} / W_{p} = 1.00

**Solution Below:**

Part A

Express kin terms of μ, m, g, and v.

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