# Friction in Slipping and Rolling

I'm solving a rotational motion problem that involves slipping and rolling. The problem's basic setup has a sphere rotated and then placed onto the ground, at which point it begins to acquire translational velocity. A solution in my textbook assumes the following:

"Once the ball starts rolling without slipping, there is no more frictional sliding force, and so the velocity will remain constant."

Why does friction stop exerting a force? Am I mistaken that friction exerts a force whenever an object and its surface might be distanced from each other? • Why does friction stop exerting a force?

It doesn't! Static friction still works. The wheel is not slipping; it is rolling. At the contact point the wheel and surface stay together because of static friction. This point of the wheel doesn't slide over the surface. So no kinetic friction, but certainly static friction. If not, then how would you start your car? Your tires need grip (static friction) on the surface, and must not slip.

The quote you give does though mention constant velocity. That is the same as saying no acceleration and of course also no angular acceleration. When that is the case, which it will be after some time, all torques must sum up to zero. So, if friction is the only force that causes a torque, then friction must be zero. Friction is only making the wheel start speeding up it's rotation - it makes the wheel start to turn, when your car speeds up. But when the rotation is constant, there is no more friction - just like when the block is sliding with constant speed on the ice, or when the space shuttle is drifting at constant speed, there is no friction that brakes it.

• Am I mistaken that friction exerts a force whenever an object and its surface might be distanced from each other?

Yes, you are mistaken. If I lift a bag of potatoes from the floor, there is no friction.

But I know what you mean: what if you move the object in parallel along the surface. Still, ideally you can have a no-friction icy surface, but apart from that I guess not.

• Comments are not for extended discussion; this conversation has been moved to chat. Mar 28, 2017 at 14:29

Friction will not stop. Friction is responsible for the wheel's horizontal motion. However, the velocity will become (and remain) constant.

It's called static friction. What you don't have anymore, is kinetic friction.

The rigid body, or ball in the picture described above has an applied force on it. The frictional force will continuously increase until it reaches a maximum static frictional force.

The static frictional force is defined by

$$f_{s,max} = \mu_s F_N$$

At that point, the object begins to slide and static friction no longer applies. Instead, between the object and the surface that it is sliding on, there is a more or less constant kinetic frictional force which can be defined by: $$f_k = \mu_k F_N$$

Am I mistaken that friction exerts a force whenever an object and its surface might be distanced from each other?

It depends on what you mean by "distanced". If the object is still in contact with a surface, it will experience friction. However, if you pull the object and lift it away from said surface, it will of course, no longer be under the influence of friction. Instead you are holding the object in a fluid (e.g. air) and depending on the applied force, can experience another non-conservative force such as air resistance. The prevailing answers seem to all suggest that friction does not stop. If that were the case, since there are no more horizontal forces, there would be some kind of horizontal acceleration due to $$F_{net}=ma$$, but that's not the case.

The answer is that friction no longer exerts a force when the ball is rolling without slipping. Let's try to unpack that statement, by considering examples where friction does and does not exert a force:

1. A sack of potatoes sits on the ground, with no horizontal forces acting on it.

In this case, friction does not act, since no relative motion.

1. A sack of potatoes is dragged across the ground.

In this case, there would be friction, since there is relative motion between the ground and the sack of potatoes.

1. A horizontal force attempts to drag a sack of potatoes, but it does not budge.

In this case, there is friction, but in the form of static friction, which prevents what would have been relative motion had the friction not been present.

In this case, the rolling ball is most similar to case 1, because at every moment in time, the bottom of the ball has no relative motion with the ground, and there are no other forces that would perturb it. This is why, in an ideal scenario with no other sources of friction (i.e. rolling friction), a ball that is rolling without slipping would keep rolling indefinitely.

• This is the correct answer. Static friction vanishes if the ball is rolling without slipping. We can prove this by contradiction. Let us assume that the static friction persists even as the ball is rolling without slipping, then the center of mass must accelerate in a direction opposite to the linear velocity of the center of mass of the ball and the ball must have an angular acceleration in the direction opposite to the angular speed, thus causing it to alter it's linear and angular velocities. Since this does not happen, we conclude that static friction vanishes in rolling without slipping. Apr 20, 2021 at 6:26