# Conservation of Energy with Rotation Caused by Friction

A book that I am using to study poses the following problem:

Suppose you wish to use a spring with a force constant $k$ to launch a spherical ball or radius $r$ and mass $m$ up a ramp of inclination $\theta$ and hypotenuse $L$. Suppose also that the ball stops just at the top of the ramp so that it travels a distance $L$, and that friction is sufficient for the ball to roll without slipping immediately after launch. What distance should the spring be compressed?

It is clear to me that in the absence of friction that one simply relates the potential energy stored in the spring to the gravitational energy the ball obtains at the top of the ramp. My book states that even when the ball rolls without slipping as a result of friction the approach is still the same.

I find this last statement confusing since friction is a non-conservative force, so some energy should be lost, and the ball will not rise to the same height. Is this reasoning incorrect? Am I misinterpreting "rolling without slipping"?

• What they mean is: consider that the ball has tiny, frictionless teeth, like some sort of gears, which makes sure that it always rolls without gliding. It is very unfortunate side effect of high school physics teaching that friction, which, as you say, is a lossy process, is being used to represent certain types of holonomic constraints: en.wikipedia.org/wiki/Holonomic_constraints. – CuriousOne Jul 19 '16 at 16:03
• @CuriousOne, Thinking about the problem in terms of holonomic constraints makes more sense to me, thank you. This is a GRE prep question, so it makes sense why the problem simply states there is friction. – Zachary Castillo Jul 19 '16 at 17:50
• Possible duplicate : Energy dissipation due to frictional force in rotational motion: physics.stackexchange.com/q/206118 – sammy gerbil Jul 20 '16 at 16:43