How Does Spring Compression Affect Block Motion on an Inclined Plane?

[uchicago2012]

Homework Statement

In the figure the block starts from rest at A, slides down the ramp, compresses the spring 0.75 meters, and goes back. The spring constant is 520 N/m, the block's mass is 12 kg, and the ramp is inclined at 30°. The horizontal part of the sliding is frictionless. If point A is 2 meters above the floor, (a) what is the block's speed at the bottom of the ramp? (b) How much work does friction do while the block descends the ramp? (c) After rebounding, the block starts back up the ramp. What is its speed at the bottom, heading up? (d) How far does it move back up the ramp? (Give a vertical distance.)
See Figure 1

Homework Equations

W_{nonconservative forces} = Change in KE + Change in PE

The Attempt at a Solution

for a.
in the equation W_non = KE₂ - KE₁ + PE₂ - PE₁
where the initial is at the top of the ramp and the final is the point at which the spring is at its maximum compression
is PE₂ = 1/2kx²? I think that should be the only component of PE₂, I just wasn't sure. It gave a reasonable answer once I solved for everything, it just made me a bit nervous to have the only final potential energy of the box be that of the spring.

for c.
now W_non = 0 so
U_i + K_i = U_f + K_f
where the initial is the point at which the spring is at its maximum compression and the final is the point at which the box begins heading up the ramp.
I was confused as to what U_f should be. It's mgy, but I'm not sure what to use for y. The box is beginning to head up the ramp but I'm unclear on whether it actually has an elevation at that point.

 

[jhae2.718]

For the part while the block is sliding down/up the ramp, you need to take friction into account.

Are you given a coefficient of friction?

 

[uchicago2012]

No, but I solved for it. I did in part a and b, but in part c it said the horizontal bit was frictionless.

 

[jhae2.718]

Ok, your work for a) looked like you had neglected friction.

In that case, when the box is just about to head up the ramp, its energy will be entirely in the form of kinetic energy.