# Why does a rolling sphere stop?

I've just started studying Rotational Mechanics and one thing I'm confused about is that if friction is equal to zero when a body is rolling purely.. then why does a rolling body e.g. sphere.. stop rolling in reality

• I'm not sure what it is you are asking. In reality, rolling friction is non-zero in general. Are you asking for the microscopic mechanisms for rolling friction? The Wikipedia article on rolling resistance has a lot of material on the topic.
– Pirx
Nov 23, 2016 at 17:34
• "if friction is equal to zero"...clearly, but that isn't true. Why do you think it is? Nov 23, 2016 at 17:35
• Air resistance? Nov 23, 2016 at 17:49
• Nov 23, 2016 at 19:29
• @Intellex In an "ideal" world the sphere will roll for ever but in the real world even with no air resistance rolling friction would play a part. Nov 21, 2019 at 13:24

This is a great question that, I think, naturally follows any explanation of rolling objects. Hopefully this webpage will help.

http://www.lhup.edu/~dsimanek/scenario/rolling.htm

I will explain below, beginning with a brief summary of friction.

## Friction

Friction is defined as a force that opposes slipping or sliding between two surfaces. If you are dragging a bag of money:

You will notice it is hard. Why? There's just paper there, and you're pulling perpendicular to gravity so there shouldn't be any force you are fighting. Well, we know from physical experience that there is a force called friction:

On Earth's surface, gravity does its best to push objects downwards. And when objects that aren't perfectly smooth get pushed together all of their little bumps and holes will get pushed together as well and you will have something like the picture above. To move the green object you will have to go around all those contact points. How hard this will be depends on the roughness of the object (the coefficient of friction, µ) and how much the ayssmetrical surfaces are pressed together, $mg$.

The reality is that there are no perfectly smooth surfaces, and even if there were you would still experience some friction from intermolecular forces because the objects in contact are both made of atoms.

The key thing to keep in mind is that friction opposes sliding between two surfaces, and that there is no realistic contact where this isn't present.

## Perfect Rolling Objects

Now, lets apply this definition to a rolling object. A perfect sphere will only make contact at one point. And we have said there is always friction between two contact surfaces. So to move with friction we make the natural assumption that a sphere must be slipping or sliding.

However, if we take a closer look at that single contact point we see immediately that this is not the case. For when the sphere moves forward at all, even just a little bit, that contact point rotates off the ground and there is a new contact point. A rolling object never has a surface in contact with the ground for a more than a split second. So, if there is not sliding (no two surfaces are continually fighting each other) then we conclude that there must not be friction.

There seems to be some contradiction here. Two surfaces are moving relative to each other but are not experiencing the traditional definiton of friction.

To explain this we will take one more hypothetical before getting to the reality of this situation.

Imagine a pencil balancing on its tip. If you push at the center of mass, about halway up the pencil, the pencil will fall forward, essientially rotating about its tip.

Now imagine this were repeated in space. Not even deep space, but the upper atmosphere of planet earth. Gravity is still present and pulling the same way on the pencil. The only difference is that there is no surface for the pencil to balance on. If you repeated the experiment and pushed on the center of mass the pencil would just move forward. It would not rotate. Things do not rotate unless there is a torque.

So where is the torque that causes the pencil to rotate forward on earth? It comes from the contact point with the surface, from friction. When you push on the center of mass there is an equal resistance force at the tip from the contact. There is now a torque and the pencil rotates around the fixed point, the tip. Once it begins to rotate gravity then helps with additional torque downwards at the center of mass.

So we now have an example when friction causes an object to move without sliding taking place between the surfaces.

The reality of the perfect sphere is that it is made up of an infinite amount of pencil tips. When force is applied to the sphere the single contact point resists motion and the rest of the sphere begins to move around that point. But as soon as motion begins there is a new contact point, and torque from gravity and forward motion are causing the sphere to rotate about that new point. The sphere is continual falling over itself. Hopefully this picture helps to illustrate:

The reality is that rolling is caused by friction, though in a unique way. Rolling would not happen without those instantaneous contact points resisting motion.

If a rolling object had zero friction it would not roll at all. It would just translate. But this is never the case because there is always friction. In the case of a rolling object, friction causes the object to be continually tipping over.

So we have identified a common misconception about the nature of rolling, but your question still remains, could an object roll forever.

## Perpetual Motion

We have been looking at a perfect sphere with one contact point. There isn't any obvious reason why a rolling body should every stop in this case. But, as we have discussed, smooth rigid objects are an idealization. In reality both the body and the surface are slightly deformed by one another's presence.

Whether the object deforms or the surface does, there is now more than one contact point. The normal force of these contact points are not all aligned with the center of the sphere, as was the case with a single contact point. If an object is rolling forward it is pushing against these new contact points. Since these reaction forces are off center there is now a torque opposite to the sphere's direction of motion. Over time this will bring the sphere to a stop.

Because all objects are made of atoms, there is not a case where this deformation doesn't occur. No object in this scenario will roll forever. Even without deformation there are quantum mechanical reasons why the objects would slow, as Simanek explains in his article.

• Does the body always get deformed? Isn't it possible to have a body which is perfectly rigid? Do correct me if my thinking is incorrect Nov 16, 2017 at 11:45
• @AbhigyanC It always would. If you read the last paragraph, "because all objects are made of atoms, there is not a case where this deformation doesn't occur." So your question can be rephrased, "are there ever atoms' outer electron shells which touch other atoms' outer electron shells and do not influence each other electromagnetically. But such closeness and influence is the definition of "touching," so the body always gets deformed. If a Ph.D or professor could comment here, that would be appreciated Jun 11, 2018 at 22:53
• @ BoddTaxter Hello. Thanks for your explanation. I want to ask a doubt . You have stated in second last paragraph that there are normal forces on the sphere which are off the center and they give rise to a torque opposite to the sphere's direction of motion. I cannot understand how they will give rise to a torque in the opposite direction to sphere's motion. Will this net torque affect the rotation of the sphere or its translation. If either of them is affected , why doesn't the frictional force act in the required direction to result in pure rolling again . Thank you very much. May 5, 2019 at 14:25
• same question here ,why would an off-centre deformation give rise to a torque in the opposite direction? Oct 1, 2020 at 4:08

I've just started studying Rotational Mechanics and one thing I'm confused about is that if friction is equal to zero when a body is rolling purely.. then why does a rolling body e.g. sphere.. stop rolling in reality

Both these statements cannot be true at the same time. If the friction really is zero, then the sphere rolls forever and if the sphere stops then the friction was not really zero.

Any rolling object in a gravitational field, deforms the surface it rolls on and creates an indentation. The rolling object has to continually climb out of its own indentation and so is effectively trying to roll uphill all the time. Think of a car when it gets stuck in soft mud and the drive wheel digs a deep hole for itself. It very difficult to get the car moving again until the wheel is lifted out the hole. Generally any rolling object is not infinitely rigid so the rolling object itself is continuously being deformed and this burns up energy too. Then there is air friction to take into account. It is almost impossible to have zero friction unless the rolling object is in space and there is no gravity or road. It is also a myth that friction is required for rolling motion. An object can continue to rotate and translate at the same time in space due to angular and linear momentum and no friction is required.