Will a disc or cylinder (rigid body) executing pure rolling on a rough surface stop, neglecting air drag and other heat losses and rolling friction but not static and kinetic friction? If yes, due to which friction it will stop, static or kinetic and how? Assume surface has no rolling friction.
Answer
As Yashas Samaga said, it will not stop on a smooth, but frictional surface. It will stop however on an actual rough surface (as it does in reality – e.g. a steel marble rolling on a rough stone surface will come to a halt quite quickly, although drag / rolling friction is as low as on a smooth glass plate, where the marble would indeed roll very far).
The reason is that a rough surface can in general not be continually tangent to the rolling body. Instead, if the object has rolled over a peak, it will not smoothly traverse the following trough but slightly collide with the next peak. If there's no rolling friction, then the collision will (ideally) be perfectly elastic, i.e. the cylinder will bounce off. When it hits the surface again, the vertical kinetic energy will regenerally not be fully reclaimed to movement in the original direction. In fact, while it has still some velocity in that direction, it will statistically more likely clash with yet another opposing front of the profile, thus losing yet more momentum.
So, I reckon ideally this would eventually lead to a random-walk kind of motion. In reality, this doesn't happen because the collisions are scarcaly sufficiently elastic – actually a good amount or kinetic energy is lost right when the roller hits the next peak.
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