# Smooth rolling motion - conservation of energy?

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This isn't about a specific physics problem, but rather a question:
Given I have a ball or cylinder rolling smoothly along some path, is it generally true that mechanical energy is conserved?
I.e. if ##E_mech = K+U = K_{trans} + K_{rot} + U##, then ##\Delta E_mech = 0##?

I have been able to formulate a proof for a cylinder rolling down an inclined plane, with a change in height ##\Delta h##. I've been able to show that, at the bottom, ##K_{rot}+K_{trans} = mgh##.

But I just wanted to check that this is generally true along any path (e.g. curved paths), given that the rolling is always smooth? And also, are there any caveats here where this assertion doesn't work?

Thanks!

But I just wanted to check that this is generally true along any path (e.g. curved paths), given that the rolling is always smooth? And also, are there any caveats here where this assertion doesn't work?

The total amount of mechanical energy, in a closed system in the absence of dissipative forces (e.g. friction, air resistance), remains constant.
so, if you have conditons of perfect rolling (without slipping) the energy should be conserved.

Hmm. I was under the impression that the static friction present was a friction force and hence the there are non-conservative forces at work.
However, now that you mention it, I realize that static friction, by definition, can do no work. Hence there are no non-conservative forces doing work per se (although there are non-conservative forces present!)

Is that why mechanical energy is conserved?

Grinkle
Gold Member
What is the relative motion between the smoothly rolling object and the surface along which it is rolling at the point / line of contact?

The answer to that plus an equation for energy dissipated by static friction between two surfaces should provide you the insight you are looking for.