A block of mass m is held at rest on a frictionless incline. The block compresses a spring by a length X from equilibrium. The spring constant is k. When the block is released, it travels a distance d up the slope. What is d?

Let's analyze this problem by D0EL.

Step 1: Define/draw diagram and coordinate system.
Step 2: Choose a consistent zero.
Step 3: Energy conservation
Step 4: Losses.

Step 2: The gravitational potential energy zero is the block's starting point.

Step 3: U_i + K_i = U_f + K_f.

Because there is both gravitational U_g and spring potential energy U_s, energy conservation becomes:   U_gi + U_si + K_i = U_gf + U_sf + K_f

K_i =0 and K_f = 0 because the block starts and ends at rest.
U_gi = 0 and U_sf = 0 by construction.

Energy conservation becomes:   U_si = U_gf → ½ kX² = mgh

Using h = dsin θ, we solve for d and obtain:  d = kX²/(2mg sin θ).