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When a wheel rolls without slipping, friction acts backward at the point of contact with the ground. However, the torque due to that friction seems to be in the same direction as the wheel’s rotation.

If that’s the case, why doesn’t the wheel keep accelerating continuously?

I’m trying to understand how the friction force and torque balance during rolling motion. enter image description here

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    $\begingroup$ The friction can act in either direction. You have simply assumed wrongly. $\endgroup$ Commented Oct 15 at 19:20
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    $\begingroup$ This might help - Toppling of a cylinder on a block $\endgroup$ Commented Oct 15 at 19:43
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    $\begingroup$ You assume wrongly. Static friction torque always opposes wheel rotation, so it's force must act in the direction of com movement. $\endgroup$ Commented Oct 16 at 5:20
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    $\begingroup$ Note that a wheel can roll without slipping on a frictionless sheet of ice - it just needs to rotate at the rate that matches its translational speed. Your analysis goes off the rails very early in the question - "When a wheel rolls without slipping, friction acts..." is already incorrect, as a wheel can roll without slipping without friction acting at all. $\endgroup$ Commented Oct 16 at 18:05
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    $\begingroup$ I think you meant to ask why the wheel isn't speeding up. Acceleration, like velocity, is a vector quantity that captures both a rate of change and the direction of that change. If speed of the wheel, as a whole, remains constant, every point on the wheel is constantly changing direction, and thus is constantly accelerating. $\endgroup$ Commented Oct 18 at 15:38

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I think you, to some extent, misunderstand how ordinary friction (i.e. Euler's friction laws with a static and a kinetic friction coefficient) applies to a wheel rolling on the ground. So first I'll explain this, and then I'll explain how actual rolling resistance works (i.e. why wheels rolling on the ground slow down over time)

When a wheel spins on a flat surface, in the absence of energy loss mechanisms like rolling resistance or air resistance, there is no friction between the wheel and the ground. The wheel moves at a constant velocity $v$, and because it's rolling, the top of the wheel moves at a velocity $2v$, and the point in contact with the ground is not moving relative to the ground. So there's no kinetic friction.

Why is there also no static friction? Generally static friction requires some force trying to accelerate the point in contact away from matching the velocity of the ground. For a wheel rolling on a flat surface, there is no such force, and no static friction is needed to keep the wheel in static contact. If the wheel is accelerating due to an external force - like when a wheel rolls downhill, static friction acts exactly how you describe, and causes the wheel's angular velocity to increase as its ordinary velocity increases.

enter image description here

So why do wheels slow down over time? The reason is called "rolling resistance," which is similar to kinetic friction, but not the same. Rolling resistance is a force on the wheel that opposes the velocity of the wheel, given by $F=\mu_{RR}N$, where $N$ is the normal force and $\mu_{RR}$ is the rolling resistance coefficient. Unlike kinetic friction coefficients, $\mu_{RR}$ is typically much less than 1, to show that wheels are more efficient than dragging a rectangle along the ground. The phenomenon of rolling resistance depends on the wheel deforming slightly and having a nonzero-length point of contact.

enter image description here

One example of how rolling resistance can come about is shown above. The point that the wheel first touches the ground receives a larger upward force from the ground than the point that it leaves the ground, resulting in a torque that opposes the rotation of the wheel.

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    $\begingroup$ Good answer. Can you, just for fun, elaborate what happens if a very inelastic wheel (i.e. hardened steel) rolls on a very hard surface? Surely the spring losses you describe are almost negligible then - are they still the only component, or are there other factors at work then as well? $\endgroup$ Commented Oct 17 at 8:32
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    $\begingroup$ Note that being hard does not make something inelastic. $\endgroup$ Commented Oct 17 at 10:26
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    $\begingroup$ @AnoE When an inelastic wheel rolls on an inelastic surface, it doesn't slow down as much. $\endgroup$ Commented Oct 17 at 15:20
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    $\begingroup$ @AnoE As Daniel Wagner says, I wouldn't have called that "very inelastic." Rather just "very hard;" maybe even "not very inelastic." In any case, in the circumstance you describe, the rolling resistance coefficient is simply very small. This is exactly why trains have steel wheels on steel tracks, and why trains are much more fuel efficient than cars. $\endgroup$ Commented Oct 17 at 16:57
  • $\begingroup$ @AXensen ^ I assume you meant "not very elastic", there? $\endgroup$ Commented Oct 17 at 17:32
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When the wheel rotates on a horizontal plane without skidding the friction equals to zero. When skidding is introduced then the force would be forward(if $\omega R>v$ or backward if $\omega R<v$. The overall influence of friction would lower the total energy every time given the skidding is happening.

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    $\begingroup$ In the case of pure roll up or down an incline, friction happens to be a conservative force, so you are wrong that it has to lower the total energy every time. Very pedantic, but it happens. $\endgroup$ Commented Oct 15 at 21:18
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    $\begingroup$ Had to add horizontal plane, my bad. $\endgroup$ Commented Oct 15 at 22:44
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    $\begingroup$ It was not about the horizontal plane; the correction that you added that made it correct is that you now consider only when there is skidding, and whenever there is skidding, it is not pure roll, and of course then you are correct in that sense. However, skidding is way too strong of a condition and makes the result way too obvious; it is really only interesting to deal with the subtleties. $\endgroup$ Commented Oct 15 at 22:53

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