We want to analyze the mass-spring system discussed in Problem 20 for the case in which the weight W is dropped onto the platform attached to the center spring. If the weight is dropped from a height h above the platform, we can find the maximum spring compression x by equating the weight’s gravitational potential energy W(h + x) with the potential energy stored in the springs. Thus

 

which can be solved for x as

With

which gives the following quadratic equation to solve for x:

a. Create a function le that computes the maximum compression x due to the falling weight. The function’s input parameters are k₁, k₂, d, W, and h. Test your function for the following two cases, using the values k₁ = 10⁴ N/m; k₂ = 1.5 × 10⁴ N/m; and d = 0.1m.

W = 100N h = 0.5m

W = 2000N  h = 0.5m

b. Use your function le to generate a plot of x versus h for 0 ≤ h ≤ 2 m. Use W = 100 N and the preceding values for k₁, k₂, and d.

W (h + x) = k if x