Spring and Inclined Plane

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: Ui + Ki = Uf + Kf.

Because there is both gravitational Ug and spring potential energy Us, energy conservation becomes:   Ugi + Usi + Ki = Ugf + Usf + Kf

Ki =0 and Kf = 0 because the block starts and ends at rest.
Ugi = 0 and Usf = 0 by construction.

Energy conservation becomes:   Usi = Ugf → ½ kX2 = mgh

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