When we usually draw friction, it is opposite the direction of motion. However, in rotational motion problem when a cylinder is rolling on the floor, it is in the same direction as the motion. I did some math, and figured out that indeed, if you make it point to the left then it will have a negative sign. While I am convinced of its validity, I do not understand why this is intuitively true, and why rolling friction is then opposite to sliding friction.

EDIT: Upon request, here is a diagram. The friction at the bottom should be negative and point the other way, but I don't know why. Here is what the friction should be like...

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    $\begingroup$ If the friction on rolling objects and sliding objects really were in opposite directions, wouldn't rolling objects speed up instead of slowing down? That's not what we generally see. $\endgroup$ – Peter Shor Oct 31 '15 at 20:24
  • $\begingroup$ Not following this question. Maybe it would be helpful to draw a picture illustrating what the problem is. $\endgroup$ – user93237 Oct 31 '15 at 20:25
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    $\begingroup$ Er ... the direction of the rolling friction depends on many things. The direction of rolling friction on your car tires as you brake is against the motion. The direction of the rolling friction on your drive tires as you pull away from a stop is forward. Details matter. $\endgroup$ – dmckee --- ex-moderator kitten Oct 31 '15 at 20:28
  • $\begingroup$ @SamuelWeir good point. I have added an image. $\endgroup$ – John Targaryen Oct 31 '15 at 22:12
  • $\begingroup$ @dmckee OK, so assume that this case is when there is force applied to the top that is larger than the frictional force, so it is accelerating in a positive direction. $\endgroup$ – John Targaryen Oct 31 '15 at 22:14

Firstly, a more general advice(something that was told to my class by my professor):
The "opposite to the motion" direction of friction is not the best way to see it. In fact, nothing in physics should be viewed as being an absolute rule except from the very basic foundations of physics, which are its laws. One case in which friction is not opposite of motion(well, the one component of friction) is the case with a motorcycle making circles in a cylinder. In order for the motorcycle to be able to stay at a constant height, friction has to be applied upwards. So, here the direction of motion is tangent to the circle that the motorcycles does but one component of friction is perpendicular to that direction. So, use your intuition to find out where does the friction point.

Now, for your particular example, you can think of it like the friction that "helps" us walk. In order for you to take a step forward, you need to push the Earth to the back. This can be achieved via the friction between your shoe and Earth. The same applies for your example. In order for the cylinder to roll, it needs to push the Earth backwards, and so the Earth needs to push it forward. So, that forward force is the rolling friction. The sliding friction is the same case as if you had a box that was moving forwards and the friction was pushing backwards(here, it opposes the motion and indeed it is opposite of the direction of motion, where direction of motion is the direction of the sliding velocity).

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  • $\begingroup$ " In order for the motorcycle to be able to stay at a constant height, friction has to be applied upwards." Not so. Friction is not actually needed, although it always helps. A marble on a frictionless conic surface only needs to move fast enough to stay on circular trajectory. Also, friction will NOT act upwards on a motorcycle unless it is falling downslope. $\endgroup$ – udrv Nov 1 '15 at 5:02
  • $\begingroup$ Yes, I agree with you of course. The motorcycle MIGHT not need the friction. But friction COULD be applied upwards to help keep it at constant height. And that is the important thing here and not if it happens all the time or if its neseccary. The important for my point is that you can also have that friction. As for your last point, it need not be falling downwards in order for the friction to be acting upwards. That's why it is called static friction. If it was falling downwards, then the friction would be called kinetic. $\endgroup$ – TheQuantumMan Nov 1 '15 at 9:45
  • $\begingroup$ I changed the cone to a cylinder. $\endgroup$ – TheQuantumMan Nov 1 '15 at 10:38
  • $\begingroup$ I think the case with a cylinder and static friction has the cylinder rotating and the object stationary wrt the surface of the cylinder. When the object is moving or rolling wrt to the cylinder surface, friction is by definition kinetic or rolling, respectively. $\endgroup$ – udrv Nov 1 '15 at 21:03
  • $\begingroup$ I used the cylinder-motorcycle example to show that we should not be thinking of friction as something that always acts opposite of direction of motion. To not standardize friction, if I may say. My answer to the question is after the advice. $\endgroup$ – TheQuantumMan Nov 1 '15 at 21:07

Frictional forces between two surfaces try and prevent relative movement between the two surfaces (static friction) or reduce the relative movement between the two surface (kinetic friction).
This does not necessarily mean that friction opposes motion.

If a ball of radius $r$ is rolling on horizontal surface without slipping $v = r\, \omega$ where $v$ is its translational velocity $v$ and $\omega$ is its angular velocity.
In such a state the frictional force due to the surface over which it is rolling is zero.

Now consider a ball projected along a horizontal surface with a velocity $v$ and angular speed $\omega$ zero ie it is not rotating.

enter image description here

If there are no frictional forces between the ball and the surface then the ball will continue to move a velocity $v$ and not rotating.

Now suppose that there is a frictional force $F_{\rm bot}$.

The no slipping condition $v=r\omega$ is obviously not satisfied and at the point of contact there is relative movement between the bottom of the ball and the surface.
At the point of contact the frictional force due to the surface acting on the ball affects the ball in two ways to try and get the ball to the no-slipping condition.

  • The frictional force reduces the translational velocity of the ball as $F_{\rm bot}$ is acting in the opposite direction to the translational velocity.

  • The frictional force applies a torque about the centre of mass of the ball $C$ such that the angular velocity of the ball is increased.

So $v$ is decreasing and $\omega$ is increasing until you get to the no-slipping condition $v=e \omega$.
Si the kinetic frictional force has acting in a direction such as to reduce the relative movement at the point of contact between the ball and the surface.

Now you should be able to state and explain the direction of the frictional force in this case where the ball is rotating and initially has zero translational velocity.

enter image description here

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Well, though it's true that direction of friction is not always in the opposite direction of the motion, above example isn't really accurate.

Rolling friction is actually a misnomer, actual term is Rolling Resistance and direction of Rolling Resistance is diffused as it's origin is different than 'Friction'. Rolling resistance arises due to deformation in contact surfaces and thereby loss of kinetic energy in addition to sliding friction in different directions tangent to the deformed surfaces.

Thus, there is one component of rolling resistance that is in the opposite direction of the motion.

If Rolling friction is assumed to be in the direction of motion as explained above in the figure, the ball will continue to accelerate since there will be no linear stopping force.

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