Ramp with block dealing with friction

[Physicsdummy1]

1. Consider a ramp of length L inclined at an angle Ɵ > (the angle of repose) to the horizontal. A block of mass m is placed at the high end and released from rest.
(a) If µ = 0; what is the speed of the block at the bottom of the ramp?
(b) Suppose now, that µ ≠ 0: What is the speed at some distance x down the ramp?
(c) Again µ ≠ 0: What is the speed at the bottom of the ramp?
(d) What value of µ would bring the box to rest at some distance L0 down the ramp?
(e) Suppose the ramp becomes more and more rough the further one travels down along the ramp. In other words, µ = µ(x) = αx; where α is some positive constant and x is a co-ordinate measured along the ramp. What is the speed of the block at the bottom of the ramp?




2. Homework Equations



3.This is as much as I was able to come up with. Not really sure if it is correct? It is in the attachments

 

[Physicsdummy1]

Can anybody help? I need some assistance on this problem so I know what to do on my test tomorrow.

 

[PhanthomJay]

Your answers to a and c appear correct. your answer to e does not appear to be. It is easier i think to use energy methods, where the work done by friction if the friction coefficient is ax is W_f = \int axdx