# Is there a contradiction with the frictional force acting on a rolling wheel on a flat surface, if you consider both the net force and the torque? [duplicate]

I have had a physics problem on my mind for a few days and it seems relatively simple at first, but it has been puzzling me, and I was wondering if you could help.

Say a student rolls a tire on a flat level surface. There is no car involved or anything it’s just the tire. As the tire rolls, the tire slows down by friction and eventually comes to a stop. I have drawn an extended free body diagram below in paint.

I have the wheel rolling to the right and gradually slowing down with acceleration to the left. Since it is accelerating to the left, then that means static friction must be pointing to the left. So this is the problem: If static friction is pointing to the left, that means friction is also creating a torque on the wheel that points into the screen. If there is a torque pointing into the screen that torque should speed up the wheel’s rotation! But that doesn’t happen. So the question is: How can we use just Newtonian physics (not energy considerations) to account for the fact that the friction force is doing two seemingly contradictory things at once? It’s slowing the wheel down by pointing to the left, but also speeding the wheel up by applying a torque into the screen. I’ve been struggling with this for a few days. I was thinking maybe I’m missing a force, like air resistance. But that shouldn’t matter. We should be able to perform this same experiment in a vacuum. Somehow, there must be some torque applied out of the screen to cause the wheel to slow in its rotations. But I don’t know how.

• Well, remember that if the surface and wheel were perfectly rigid, there wouldn't be any static friction. The friction is the wheel "climbing uphill" because the roadway deforms to make a slight dip and the wheel squashes a little bit. Commented Jul 5, 2018 at 16:57
• There's a brief & accessible explanation of this paradox in Chapter 10 of Giancoli's Physics for Scientists & Engineers. Commented Jul 5, 2018 at 17:06
• Thank you @MichaelSeifert I found some lecture notes online that explained the phenomenon from his book Commented Jul 5, 2018 at 17:44
• Commented Jul 7, 2018 at 14:35

If there is only single force applied at a single point like in your image, then indeed this will not make the wheel stop both its translation and rotation.

In reality, friction forces act on several points distributed across non-zero contact surface area, not at single points. What happens is that the wheel deforms the ground surface in such a way that pressure is uneven across the contact surface and net torque, being a sum of individual infinitesimal torques due to infinitesimal contact forces, will be such as to slow down the rotation, even while the net force applied at the lowest point of the contact area would not by itself do that.

Why does this happen?

Imagine a wheel dipped in a soft ground surface like mattress. When it moves, the mattress is deformed unevenly. The contact forces acting on the front part are higher than forces acting on the back. Due to deformation and contact friction, these forces acting on the front do not point towards the center of the wheel, but their line of action goes above that center. Such forces both act to slow down the translation and have torque that counteracts the rotation.

This is an example of mathematical fact that net torque acting on a body is not, in general, torque of the net force acting on that body, as if applied at some point. In general, such "reconstruction" of torque with a single force applied at some point is not possible.

• "net torque, being a sum of individual infinitesimal torques due to infinitesimal contact forces, will be such as to slow down the rotation" Could you expand on that? How do these smaller individual torques slow the wheel down? Commented Jul 5, 2018 at 17:05
• @user199884 I've added an explanation. I recommend you try to draw the wheel in the soft deformed ground, with forces on the front being larger than those on the back and pointing above the wheel center. Commented Jul 5, 2018 at 17:13
• Turns out the net normal force is slightly to the right and off-center of directly underneath the center of mass due to the deformation of the wheel. This is what provides the torque out of the screen to slow the wheel down. Wheewww... glad we figured that one out Commented Jul 5, 2018 at 17:39
• What is "net normal force" and how do you know it is to the right? One has to consider all forces due to deformed surface, not just "normal force". Commented Jul 5, 2018 at 18:17
• By "net normal force", I mean the sum of all the individual normal forces applied to the tire by each infinitesimal patch. And when I say it is "slightly to the right" I do not mean that it points to the right. It still points upward, however, it is off center and positioned slightly to the right, instead of being directly under the center of mass. This creates the needed torque to counter the frictional torque and make the tire slow down. Commented Jul 5, 2018 at 19:42