Crates on a Ramp with Friction and a spring at bottom. Finding the spring constant.

Brainsplosion

You are designing a delivery ramp for crates containing exercise equipment. The crates weighing [itex]F_1[/itex] will move at a speed of v at the top of a ramp that slopes downward at an angle [itex]\phi[/itex]. The ramp exerts a kinetic friction force of [itex]F_2[/itex] on each crate, and the maximum static friction force also has this value. Each crate will compress a spring at the bottom of the ramp and will come to rest after traveling a total distance of L along the ramp. Once stopped, a crate must not rebound back up the ramp.

Calculate the force constant of the spring that will be needed in order to meet the design criteria.



This is what I've tried so far:
initial energy: [itex](Lsin\phi)F_1+1/2*mv^2[/itex]
final energy: [itex]1/2*kx^2[/itex]
lost energy: [itex]F_2L[/itex]

initial= final + lost
[itex](Lsin\phi)F_1+1/2*mv^2 = 1/2*kx^2 + F_2L[/itex] (we shall call this equation 1)

From the free body diagram of the crate resting on the spring at the bottom of the ramp:
[itex]kx=F_1sin\phi+F_2[/itex] (we shall call this equation 2)

I'm not quite sure all of the above is correct, but if so.. I don't think there's supposed to be the variable of m in there. Should I substitute [itex]F_1/g[/itex] ?
Then, am I supposed to solve for x in equation 2 then plug that into equation 1?

LowlyPion

Looks like a plan.

F1/g looks workable.

2 equations, 2 unknowns ...

Brainsplosion

Edit: I found what I was doing wrong. Thanks.