With friction, does the path followed from A to B affect energy transfer? Consider I have a roller coaster that can travel across one of three paths, all of which are half circles of the same size.


*

*The half circle is an arch that starts vertically up, and ends vertically down.

*The half circle is U-shaped, starting vertically down then ending vertically up. 

*The half circle is horizontal the entire way, being on a parallel plane to the ground.
The train starts at the same velocity on each of these paths.  If we ignore friction, then all should have the same ending velocity.  However, when we consider friction, I'm not sure which path has the fastest ending velocity.
 A: In this system:
Around half circle #1, gravity is a component of centripetal force.  The normal force between coaster and path is due to centrifugal force.
Around half circle #2, gravity is also a component of centripetal force, via the normal force between coaster and path.  The normal force is a combination of gravity and centrifugal force.
Around half circle #3, gravity is a component of centripetal force to the extent that the normal force between the coaster and the path is due to gravity, and assuming the path is banked the normal force is also due to centrifugal force.
I think we need to assume that the initial velocity is sufficient to hold the coaster to the path around half circle #1, but not so great that the coaster flies off the path around half-circle #3.  In order to find such a speed, half-circle #3 probably would need to be banked.
The net force in all three cases is toward the center of the half circle.  It seems to me the question is:  What components of the net force are due to friction?  The half circle with the least net force due to friction would have the fastest ending velocity.
A: If friction is constant, then the path followed doesn't matter.  This is true because the only force involved is gravity, which is a conservative field.  And that means (or is the definition of :-) ) that the energy expended to go from point A to point B is path-independent.  
Now, if you're interested in like the required initial velocity, you'll see that your different paths will lead to different terminal velocities.  This leads to a discussion of the "Principle of Least Action."   
