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Consider this situation:

diag

A ball is moving forward and undergoing rotation. Assume that it is not slipping. Eventually, the velocity and rate of rotation of the ball decrease, and it comes to a halt.

But if you observe the direction of friction (when the ball is rotating clockwise), you will see that the friction should have provided a clockwise torque to the ball and the angular velocity of the body should have increased. But this doesn't happen. Why?

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    $\begingroup$ A sphere executing pure rolling never stops. How you concluded that " eventually it stops". $\endgroup$
    – SarGe
    Commented May 26, 2020 at 12:02
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    $\begingroup$ @BobD No, It does not have a constant velocity since rolling friction is slowing it down. $\endgroup$
    – Jdeep
    Commented May 26, 2020 at 12:43
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    $\begingroup$ Wax your floors, put on socks and then slide across them until you hit carpet. Once you hit the carpet, tell me that friction doesn't increase the angular momentum while slowing you down. $\endgroup$
    – Jim
    Commented May 26, 2020 at 12:46
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    $\begingroup$ @NoahJ.Standerson Do you know what "rolling friction" (more properly called rolling resistance) is? It is not the $f$ shown in your diagram. $\endgroup$
    – Bob D
    Commented May 26, 2020 at 12:51
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    $\begingroup$ @BobD What is it then? $\endgroup$
    – Jdeep
    Commented May 26, 2020 at 13:01

6 Answers 6

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First let us clear up some definitions of the terms "reaction force", "normal force", and "frictional force".

Whenever there is a contact between two bodies, there is a reaction force on each body at every point of contact. A reaction force can be split into a normal component (sometimes called the "normal contact force"), and a tangential component (sometimes called the "force due to friction"). The direction of the force due to friction⁠ - the tangential component⁠ - is such that it opposes relative motion due to sliding / slipping.

This means that a perfectly rigid cylinder can roll on a perfectly rigid and rough surface forever since there is no sliding so no friction to provide a torque.

So why do balls we observe normally slow down?

The answer lies in something called "rolling resistance" (sometimes confusingly referred to as "rolling friction", or just "friction"), and entirely explains why a football comes to a stop after rolling it along the ground.

The key is that footballs and dirt are both compressible - they are not rigid bodies. On contact, the weight of the ball deforms both it and the dirt. This means that there are many points of contact between the ball and the dirt. Due to our definition, there are now many normal forces - to be precise, one per point of contact. We will ignore the tangential frictional components for now.

The deformation of the ball and the surface means that the line of action of these normal components is not through the center of the ball (see diagram).

normal forces after

As a result, the ball experiences two torques: a counter-clockwise one from the normal components to the right of the centerline, and a clockwise torque from the normal components to the left.

Since the normal forces are larger on the right side, the counter-clockwise torques are greater, and thus there is a net counter-clockwise torque and the ball slows to a stop.

Notice how we did not even consider any tangential frictional components at all. Just due to the normal components the ball is slowed.

There are a couple of points I have not covered, such as what role the frictional components do actually play and what happens in different types of deformations (e.g. the surface doesn't deform). Also you may be wondering why the normal forces on the right are greater. The answer to all these can be found at: https://lockhaven.edu/~dsimanek/scenario/rolling.htm . This is also where my diagram came from and I credit for these explanations. The exact diagram from your question is also used here and cited as a "naive pictures of friction and a rolling cylinder."

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    $\begingroup$ Nice link. However I don't agree with author view that "{friction same as speed} direction [...] force that would increase the sphere's forward linear velocity" No, you can't just simply translate rolling friction force from contact point into wheel COM, you can't do that. Rolling friction arises in wheel-ground contact point and only gives friction torque - no more, no less. However sliding friction can be transferred to body COM, cause it gives linear movement response, but not torque. $\endgroup$ Commented May 26, 2020 at 14:32
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    $\begingroup$ Nice link. +1. I wonder if the mechanism he proposes is correct. He says the normal forces are asymmetric because of hysteresis without further explanation. I would also take issue with statements like "Textbooks are to blame" for idealizing the situation. In textbooks it presents the essential physics without distracting complications. In real life, it often makes problems no harder to solve than they need to be. $\endgroup$
    – mmesser314
    Commented May 26, 2020 at 15:01
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    $\begingroup$ Hmm, never seen rolling resistance presented in terms of forces before! I've always just thought of it as "energy is lost in the inelastic compression and expansion of the objects", without really thinking about how that'd look if explained in terms of force. $\endgroup$
    – Hearth
    Commented May 27, 2020 at 17:02
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    $\begingroup$ Amusing homophone corner: "what roll the frictional components do actually play". I'd fix it myself, but I don't have the rep to make a small edit, and don't want to make gratuitous edits to bring the count up. $\endgroup$
    – MadHatter
    Commented May 28, 2020 at 6:57
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Rolling resistance, a.k.a., rolling friction, is basically the result of inelastic compression and decompression of the materials at the contacting surfaces (wheel and/or surface upon which it rolls). The inelastic behavior results in friction internal to the materials and a loss of energy dissipated as heat, known as hysteresis.

See the picture of a tire below.

The leading edge of the contact surface materials between the center point of contact and the starting point of contact compress. The resultant reaction force acts backwards on the wheel as shown.

When the materials separate at the rear side they decompress. The reaction force acts forward on the wheel as shown.

The squeezing and un-squeezing of the materials is not purely elastic and therefore internal friction dissipates energy in the form of heat. This loss of available energy for rolling due to heating is called hysteresis. As a result the reaction force acting forwards during un-squeezing is less than the reaction force acting backwards during squeezing eventually bringing the wheel to a stop.

Hope this helps. enter image description here

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Suppose nothing slows the sphere. It rolls forever. In that case, it would roll the same on ice. The friction in your diagram would be $0$.

Suppose it rolls on a rough surface and something (wind resistance? ) slows it. That is, a force on the center of mass acts in the backward direction. Friction of the surface would keep the sphere from slipping. That is, friction would act in the forward direction. Those two forces form the torque that slows the rotation.

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  • $\begingroup$ Assume there is no wind resistance. $\endgroup$
    – Jdeep
    Commented May 26, 2020 at 12:45
  • $\begingroup$ Do you mean to say that the friction force would act in the forward direction in rolling? $\endgroup$
    – Jdeep
    Commented May 26, 2020 at 12:55
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    $\begingroup$ Yes. Another example of that would be to spin up the sphere and then drop it onto the surface. $\endgroup$
    – mmesser314
    Commented May 26, 2020 at 14:34
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The purpose of friction here is to ensure pure rolling of object once it is attained friction self adjusts itself and then acts only to ensure that the lower most point I.e point of contact has zero velocity

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You are writing about static friction (i.e., friction without slipping. This is not the cause of the ball rolling. It makes it possible though.
When the ball rolls it "pushes" itself forward by having a grip (static friction) on the surface.
When the ball is accelerated, say by giving it a push, the static friction allows the ball to get accelerated (you provide the force for the torque to make this happen), unless the push (somewhere near the top) is too big, in which case the ball will accelerate by kinetic friction (like drivers of the cars at the start of the Formula 1 races, where you can see the smoke coming off from the rotating wheels). That's why this kind of acceleration is less effective in giving the ball velocity.
This is the case in the opposite direction too. When you push the breaks in a car and you decelerate by making the wheels stop turning, the distance to come to a full-stop will be larger than if you make the car stop without making the wheels stop turning (but, obviously, with sufficient force). This seems counter-intuitive, but nevertheless.

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Because it would act as a perpetual motion machine, contravening some quite fundamental laws. Also, all the cyclists would be dead, which I observe not to be the case.

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