finding distance of block that is attached to a spring

Question

A block of mass m = 2.5 kg is attached to a spring with spring constant k = 530 N/m. It is initially at rest on an inclined plane that is at an angle of θ = 27° with respect to the horizontal, and the coefficient of kinetic friction between the block and the plane is μk = 0.14. In the initial position, where the spring is compressed by a distance of d = 0.11 m, the mass is at its lowest position and the spring is compressed the maximum amount. Take the initial gravitational energy of the block as zero.

If the spring pushes the block up the incline, what distance, L in meters, will the block travel before coming to rest? The spring remains attached to both the block and the fixed wall throughout its motion.

Answer 1

I am interpreting the phrase "If the spring pushes the block up the incline"

as implying no oscillation -- the block travels only up the incline before stopping.

In that case the initial spring energy gets converted to the final potential gravitational

energy plus the work performed the force of friction over the unknown distance L.

The initial spring energy is kd²/2.

The final potential energy of the block is mgLsin(θ).

The work performed by the force of friction is Lmgµ_kcos(θ).

Thus

kd²/2 = mgLsin(θ) + Lmgµ_kcos(θ)

L = kd²/(2mg(sin(θ) + µ_kcos(θ)))

Substituting actual numbers

L = 530 N/m x 0.11² m² / (2 x 2.5 kg x 9.8 m/s² x (sin(27°) + 0.14 x cos(27°))) = 0.226 m