# Ideally speaking, will a rolling disk with no slipping come to a stop because of friction from the ground?

Consider a rotating disk on a horizontal plane with static friction. The contact point of the disk with the plane has null instantaneous velocity. Assuming the center of the disk has fixed $v_0$ velocity at a time $t_0$, is it right to conclude that the disk will stop in a time $t_1 > t_0$ by effect of the momentum of friction force? (By the Euler second law we get a negative angular acceleration)

• ? t1>t0 doesn't place much of a limit. Protons will decay in that timeframe. By definition, only rolling friction, not static, applies, since the disk never moves across the surface. – Carl Witthoft Dec 6 '13 at 13:14
• It is not necessary for the contact point to be also the instant center of rotation. Is that a condition in your question for sure? – ja72 Dec 6 '13 at 15:05
• If there is friction the disk will stop at some point $t_1>t_0$ regrardless if initial conditions. – ja72 Dec 6 '13 at 15:06
• @ja72 The disk is moving but not sliding; hence by definition of rolling vs static friction the static friction is not involved. Unless you are assuming a combination of rolling and sliding, in which case things get ugly. – Carl Witthoft Dec 6 '13 at 15:18
• I suggest to change the title to "Will a rolling disk come to a stop with friction?" This would prevent the question from being closed. – ja72 Dec 6 '13 at 15:22

The contact point of the disk with the plane has null instantaneous velocity

This implies that there is no slippage, and as such there are no non-conservative forces doing work on the disk. Assuming the disk is perfectly rigid and is not being subjected to any linear or angular accelerations, the disk will continue to roll forever, and will not come to a rest in a finite amount of time as you are suggesting.

Consider a rotating disk on a horizontal plane with static friction

Note that there is no static friction experienced by a freely rolling disk on a flat plane. If you state that the disk is experiencing static friction, then there must be some sort of force or torque being applied to it.

• Many thanks to you all for helping. I'd like to be more precise: the case is that of "pure rolling". – user35352 Dec 8 '13 at 15:55
• Many thanks to you all for helping. I'd like to be more precise: the case is that of "pure rolling". As is well known, in pure rolling static friction is fundamental, even if its work is null because the contact point has 0 instantaneous velocity. I'm totally sure about all these things. My doubt come just from considering Euler second law, which implies that there is some negative angular acceleration, so the disk should stop in a finite time. What do you think about E. second law? – user35352 Dec 8 '13 at 16:03
• @user35352 There is no static friction on a freely rolling disk. – Asad Saeeduddin Dec 8 '13 at 19:04
• Thank you Asad, this completely answer the question. Thanks a lot! – user35352 Dec 16 '13 at 23:20
• @Secret If a disk spinning at an angular velocity of $\omega$ rad/s comes into contact with a plane relative to which it is travelling at a velocity of $\omega \cdot R$ (in a direction given by the cross product of the disk's angular velocity and the plane's normal), there will be no slip. This is a purely kinematic consequence, and there are no angular accelerations or decelerations involved. The plane could be frictionless or frictionful; it wouldn't matter. The reason I chose a frictionless plane to illustrate is because it is inherently incapable of exerting any torque on the disk. – Asad Saeeduddin Jul 21 '15 at 14:11

A rolling disk will eventually come to a stop due to rolling friction. While an ideal disk may not be compressible, and so would avoid the bulk of rolling friction, there must be a contribution from surface adhesion. The molecules of the disk bond with the molecules of the surface when pressed together. As the disk moves forward, it must constantly spend energy to break these bonds and (over a long time perhaps) gradually it will slow to a stop.

To convince yourself of this, consider an ideal disk with duct tape around the rolling surface, sticky side out. It is incompressible, we can assume there is no slippage, and there is a null instantaneous velocity at the contact point. However, there is no doubt in anyone's mind that this disk will not continue rolling forever.

Most materials don't stick like duct tape, but the principles remain. In a finite amount of time, the adhesive forces between the disk and the flat surface will bring the disk to a stop.

A rolling disk will come to a stop eventually because any incidental friction will decellerate the center of mass. Ideally with a flat surface, and constant motion there should be no change as there will be not friction required to keep the disk rolling.

In real life though, for sure a rolling disk will stop eventually.