Energy & Springs: Find Max Compression & Rebound Distance

[Niles]

Homework Statement

A 2.0 kg package is released on a 53.1◦
incline, 4.0 m from a long spring with force
constant 120 N/m that is attached at the
bottom of the incline. The coefficient of
friction between the package and the incline
are µs=0.40 and µk=0.20. The mass of the
spring is negligible.
a. Show that the maximum
compression of the spring is 1.06m.
b. The package rebounds back up the
incline. Show that the package
comes to rest at a distance 1.32 m
below its initial position.

The Attempt at a Solution

I want to use that K_1 + U_1 + W_other = K_2 + U_2

W_other in this case is the work of friction and the work of the spring?

 

[Niles]

- which are both negative, because they work opposite of the direction of motin?

 

[Astronuc]

Sliding down the incline, before the spring contact, the gravitational potential energy transform into kinetic energy as the mass accelerations and frictional energy which is dissipated (lost). Then the spring is being compressed, the mass is doing work on the spring (compressing it) and the spring is storing the energy (which is equal to the KE - energy lost to friction). Then the spring stops at some deflection and recoils the mass.

Without friction, the springs stored energy would become the KE of the mass. But with friction some energy is lost.

 

[Niles]

So at the point where the spring and block hit each other (I know the velocity at this point), I can use:

K_1 - W_fric - W_spring = U_2

to find the distance the spring is compressed?

 

[Niles]

I mean

K_1 + W_fric + W_spring = U_2

but W_fric and W_spring are negative.