On a frictionless level plane, will a cylinder roll forever?

Question: If a rigid solid metal sphere (or cylinder) is rolling without slipping on a perfectly smooth, rigid and frictionless horizontal surface would it ever stop rolling?

This often-heard question exposes pervasive misconceptions about friction that can arise from inadequate textbook treatments of the subject. Typically textbooks introduce friction before they take up rotation. Their exposition goes something like this:

Bodies that touch at surfaces or points experience reaction forces at all points of contact. A reaction force at a point on a surface can be resolved into two components, a normal force (perpendicular to the surface), and a tangential force (tangent to the surface at the point of contact). The latter is called the force due to friction. The force due to friction acts on a body in a direction to oppose its motion.
This leaves much to be desired, for it uses careless and misleading language. We'll tighten it up a bit, later.

Do rolling bodies stop because of friction?

Now look at that question again. The rolling body and the plane surface are described as rigid. Students often interpret "rolling without slipping" to be synonymous with "frictionless". There is of course also that normal force due to the plane acting upward, and in this idealized analysis, that force passes through the center of the rolling body. It has zero lever arm and therefore zero torque about that center and cannot change the sphere's angular velocity. It cannot accelerate the body, for it has no component in either direction the body could move.

So the student concludes that there's no force that can change the horizontal motion of the body, and no torque to change its rotation, so therefore if it's rolling, it will roll forever without deceleration. Some even cite as an example that perpetual motion can occur in nature. We recognize that this is a highly idealized problem.

Now lets get a bit more realistic. Suppose there is some friction at the surface. Following a habit used in other problems, the student resolves the force at the point of contact into a normal component, N, and a tangential component, f (the force due to friction), drawing the arrow representing the force due to friction as opposing the forward motion of the rolling object, i.e., opposite to its velocity. For a level surface, N = -mg. This looks good, for it suggests that this friction force, being the only horizontal force acting, will slow the body and eventually bring it to a stop. This satisfies student intuition.

Friction opposite to velocity.Friction in direction of velocity.
Naive pictures of friction and a rolling sphere.

But wait, that friction force has a torque about the center of the sphere. The torque is t = fR where f is the friction force and R is the radius of the sphere. This torque has a direction such as to cause a positive angular acceleration of the sphere, i.e., causing the sphere to increase its angular speed! This would necessarily increase its forward speed if it rolls without slipping. Something is clearly wrong with this picture.

Perhaps the student says, "I must have the friction in the wrong direction. Perhaps it is in the forward direction. This direction would give a counterclockwise torque to slow the sphere's angular velocity, but it is a force that would increase the sphere's forward linear velocity. It's still a contradiction. A paradox! Friction isn't the answer.

So, it appears that even if there is friction, it cannot decelerate the rolling body. Either our analysis was deficient, or all bodies should roll forever!

A lot of things are wrong here. The textbook description of friction was seriously deficient. The question postulates perfectly rigid solids, which can't exist in nature. Our accumulated experience of physics comes from observing real solids, which can deform due to their elastic properties. That's what gives rise to reaction forces at surfaces—deformation near the contact surfaces.

Here's a better explanation of friction.

Bodies that touch at surfaces or points experience reaction forces at all points of contact. A reaction force at a point can be resolved into two components, a normal component of force (perpendicular to the surface at the point of contact), and a tangential component of force (tangent to the surrface at the point of contact). The latter is called the force due to friction. The force due to friction acts at the surface in a direction to oppose slipping or sliding at the surface.
That last sentence is important. The force due to friction doesn't necessarily act to oppose the motion of the body but acts to oppose slipping or sliding along the contact surface. A body can roll without slipping. Likewise, torques due to friction don't necessarily oppose the rotation of a body (consider a wheel or pulley being driven by a belt).

In our problem, we may wonder whether any force acts on a rolling body to oppose its motion. Students become accustomed to thinking of rolling without slipping as being a "no friction" situation. Yet our intuition tells us that this rolling body must continually slow and finally stop. We've seen it happen with marbles rolling on a smooth floor. So what force and/or torque slows them to a stop?

Are there such things as perfectly rigid bodies?

Here the textbooks are to blame again, for using the term "rigid body" without bothering to explain the consequences of perfect rigidity and the reason why perfectly rigid bodies aren't found in nature. Here's an example that shows that bodies cannot be perfectly rigid.

When bodies collide, the reason they transfer momentum and energy is because of elastic compressions during the small (but finite) duration of impact. A "soft" collision (as between two rubber balls) has a longer impact time than a "hard" one (steel balls or marbles). If bodies were perfectly rigid they could not deform on contact, for every rigid body collision would involve an infinite force acting for an infinitesimally small time. That's a bit hard to conceive in the world of large objects, isn't it?

Here's another way to look at it. To avoid complications due to rolling friction, consider a collision between two balls suspended by strings. Pull one ball aside and let it go so it strikes an identical ball that is initially stationary. The moving ball must slow to a stop. The stationary one must speed up from rest. Both processes take a bit of time. During this time, the balls must either be compressing and decompressing at the point of contact, or they must somehow "overlap" and co-penetrate. The latter scenario isn't the way solids behave. That's really what "rigid" means—that rigid bodies can't occupy the same space at the same time. Gases can. Some liquids can mix and comingle. [Even the atoms of solids can migrate, in effect causing the solid surfaces to interpentetrate when the solids are in intimate contact over long periods of time. But that isn't our concern here.]

When rigid bodies come into contact, parts of the body at and near the contact surface must "get out of the way". Both bodies deform somewhat.

Rolling resistance.

In the case of a rolling body, surface deformations give rise to tangential reaction force components acting at the surface. They also give rise to force components which, though normal to the deformed surface where the bodies contact, are not necessarily normal to the undeformed flat surface well away from the contact. These deformations, though often small, are still larger than the microscopic processes responsible for friction phenomena. Their effect on a body rolling without slipping is called "rolling resistance" in the engineering literature. Introductory physics books often lump this together with resistance, or even call it "rolling friction", a misleading term. Some even lump the two together and call it "friction". In some simple, or idealized, problems one can get away with that carelessness. Not in this case.

Because of the shape deformation, the force due to friction can have components that affect the motion of a rolling body, but the normal components at the deformed surface do also, and these are generally of larger effect. So lets concentrate our attention on them.

Ball rolling to the right, with surface deformation.
The deformation is greatly exaggerated.
Normal force components across the deformed region are not uniform in size.
They are greater on the forward side, producing a counterclockwise net torque.

As we analyze a problem such as this we should show the deformation explicitly in our diagrams. Several cases are possible:

  1. The rolling body deforms, the surface it rolls on doesn't.
  2. The rolling body doesn't deform, the surface it rolls on does.
  3. The body and the surface it rolls on both deform.
Several things can happen due to the surface deformation:

  1. The contact area at the surface enlarges.
  2. The contact area at the surface may be asymmetric.
  3. The force loading across the surface of contact is generally not uniform across the contact area. If the ball is rolling, the normal force components are greater on the forward side (relative to the forward motion).
Why should the forces on the forward side be greater? If the ball were not rolling we would expect a symmetric deformation unbiased in either direction. The dynamics of the initiation of motion would need to be considered, but we need not get bogged down in the details. Once motion is established, the front side of the ball is compressing itself and the surface below, while at the back side, the surfaces are both relaxing back to their undeformed condition. Typically, elastic materials display hysteresis, that is, the compression and relaxation forces are unequal functions of the amount of deformation.

One generalization can be made, whatever the deformation. While tangential contact components of force (due to friction) always act in a direction to oppose slipping or sliding at the surfaces, the force components normal to the surface always act to oppose the acceleration of the rolling body.

Consider deformations of type (1) and (3) together, but without (2). If the loading is greater in the forward direction of a body's rolling motion, the net force vector at the surface will pass in front of the rolling body's axis of rotation. This gives a torque about the axis in the correct sense to slow the body's rotation. Notice that we haven't even mentioned friction to reach this conclusion!

Consider deformations of type (2). Imagine a round wheel more rigid than the surface, so it deforms very little but makes a nearly round indentation in the more resiliant surface. All of the forces normal to the contact surface pass nearly through the wheel's rotation axis, and produce negligible torque on the wheel. Now, if we are to get anywhere (literally) we must also have good old fashioned friction, otherwise the wheel moving forward would slimply slide or slip at the forward rise of the deformation, and just spin around in that indentation. At the front of the indentation, friction opposes that slippage, acting forward and upward on the wheel. Friction provides a torque in the correct sense to decelerate the wheel, slowing its rotation. There is a small forward horizontal component of friction (due to the very small angle between the friction force's direction and the horizontal, but that is small compared to the horizontal components of the normal forces, in the opposite (backward) direction.

Consider the case where the wheel deforms against a very rigid surface which does not deform much. Then all of the normal forces are nearly vertical, but those acting at the forward surface (in the direction of forward motion) are greater, contributing to a torque that slows the rotation.

So we see that rolling dynamics in the real world of non-rigid elastic materials is a complex interplay of contact forces due to deformation, and consideration of friction alone can lead to contradictory and unrealistic conclusions.

So where do the energy and momentum go?

We seem to have discovered the forces that can slow a rolling body's motion. But if the body slows, it loses kinetic energy of rotation, loses forward momentum, and loses angular momentum. We need to account for where these went, since they are conserved quantities in closed systems.

The rolling body alone is not a closed system. Our closed system is the rolling body and the surface it rolls on.

Bodies in contact exchange momentum due to the action-reaction force pairs at their point or surface of contact. The "lost" momentum of the rolling body is imparted to the surface on which it rolls. Both linear and angular momentum are conserved. Since the surface is generally much more massive than the rolling body, this doesn't cause noticable displacement of it. But an interesting experiment could be arranged with a lightweight horizontal surface free to move (say on rollers, an air-suspension table or a pendulum) and its gain of momentum could be observed and measured.

The mechanical energy loss of the rolling body goes to the surface also through the work done by the contact action-reaction forces, resulting in heating of the surface and the rolling body. Two mechanisms are present. (1) Thermal energy may increase at the horizontal surface and in the ball due to compression and relaxation of materials that aren't perfectly elastic. (2) For materials that exhibit hysteresis and other non-linear stress-strain effects, these may cause small changes in the kinetic energy of the horizontal surface. This isn't usually noticed as a visual displacement, for that surface is likely very much more massive than the rolling object.

Let's idealize some more.

Some skeptic will likely suggest that we consider the hypothetical case of a really rigid body, a non-deformable body, even though no such bodies are known. Surely in that case there should be nothing to slow the rolling body.

Sorry, nature always has another "gotcha" to prevent perpetual motion.

When charges accelerate, they emit energy in the form of electromagnetic radition. Bodies moving in circular paths are accelerating. So all the charges within the atoms of our rolling body will gradually lose energy by radiation and the body's rotation will slow continually.

Some folks hold up the atom as an example that demonstrates there is perpetual motion in nature, because the electrons in atoms orbit around the nucleus forever if the atom is undisturbed.

The old (early 20th century) picture of the atom imagined it to consist of a positively charged nucleus with electrons orbiting around it. We no longer think of atoms this way for several good reasons. Heisenberg's uncertainty principle prevents us from simultaneously measuring both the position and velocity of an electron within such a small scale object as an atom, so we really can't know simultaneously where an electron would be there and what its velocity might be. We can't even know whether the electrons within an atom maintain some kind of integrity as an "object" or whether their mass and charge are distributed through the entire volume of the atoms outer shell.

So we don't think of the electrons in an atom moving in orbits. The atom isn't an example of perpetual motion. It is an example of a body that can, so far as we know, maintain its energy indefinitely if not disturbed by interacting with something else. It's rather hard to confirm this, for "indefinitely" is a mighty long time, and atoms are naturally subject to continuing disturbances and interactions. All we can say is that no one has yet observed a stable atom undergo decay without being disturbed.

Another example is the superconductivity experiments in which a body is cooled to near-absolute zero, and electric current in it can persist indefinitely. The electrons in the material are in a quantum state something like a larger scale analogy of the state of electrons within an atom. We shouldn't forget that this is not a self-sustaining system, for considerable energy must be continually supplied to it to maintain the very low temperature required.

Some cite the earth orbiting the sun as an example of perpetual motion. Not quite. The earth's orbital kinetic energy slowly decreases over time due to energy-dissipative tidal deformations. We can actually measure this. This won't cause the earth and other planets to crash into the sun, in fact, it causes them to move farther away from the sun.

So, in fact there are no known examples of perpetual motion in nature.


Re: Answers left as exercises for the reader. Send your answers to Donald Simanek at the address shown to the right. The earliest good one(s) which arrive may be posted here, with credit to author. I will post (at my discretion) answers which are in simple to explain, clear, correct, perceptive, and which stimulate thinking and further discussion. Posted answers, whether written by me or by others, do not always represent the final word on a given puzzle. On several occasions perceptive readers have noticed things we missed, or suggested simpler ways to explain something. So don't hesitate to skeptically rethink given "answers".


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