An 8 lb box slides down slope (kinetic coefficient of friction, μ_k = 0.1) and compresses a spring. How far is the spring compress before the box is pushed back up the slope?

FBD and Spring Compression

The problem involves both energy and forces. The friction force needs to be determined before energy methods can be applied.

Find the normal and friction forces,
     ΣF_y = 0
     N = 8 cos55 = 4.589 lb

     f = μN = 0.3 (4.589) = 1.377 lb

Energy balance
     T₁ +  U_1-2 = T₂
     0 + (mgh - fd) = 0.5 k s²
     8 (8 + s) sin55 - 1.377 (8 + s) = 0.5 (100) s²
     52.43 + 6.553s - 11.01 - 1.377 s = 50 s²
     50s² - 5.176s - 41.42 = 0

Solving the quadratic equation
     

  s = 0.05176 +/- 0.9116     (use positive answer)

s = 0.9634 ft