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

Assume that all the energy initially stored in the spring is transferred to the block.

Should we do this using forces or using energy? It's tricky to analyze a spring problem using forces, because the spring force is not constant. Energy is much easier to deal with.

Use the master energy equation:

U_i + K_i + W_nc = U_f + K_f

The initial position is where the block starts. The final position is where the blocks stops (it may well start to slide back down, but we don't care about that).

There is no friction so W_nc = 0.

One thing to remember is that there are two kinds of potential energy here, gravitational potential energy U_g and spring (elastic) potential energy U_s. Expanding the master energy equation a little gives:

U_gi + U_si + K_i = U_gf + U_sf + K_f

Take the zero level for gravitational potential energy to be the block's starting point. The final position is where the blocks stops (it may well start to slide back down, but we don't care about that).

All sorts of things are zero. K_i and K_f are zero because the block starts and ends at rest.
U_gi = 0, and we're assuming U_sf = 0. The energy equation becomes:

U_si = U_gf

½ kX² = mgh

From geometry, h = dsin(θ)

½ kX² = mgdsin(θ)

Solving for d gives:

d

=

kX2 2mg sin(θ)

For a particular case where k = 50 N/m, X = 0.10 m, m = 0.080 kg, and θ = 30 degrees, d works out to:

d = 0.64 m