Frictional forces and their directions

Question

Recently I asked a question regarding frictional forces at math stack-exchange(because its basically part of maths syllabus) and I drew some conclusions.

1. If A and B are in rough contact and are in limiting equilibrium, then there exist two frictional forces. One acting on A and one acting on B.

2. The direction of frictional forces can be determined by working out the direction of motion if the friction was not present.(This is a trick to work out the direction of frictional force which was mentioned on https://physics.stackexchange.com/a/94837/128083)

3. The direction of friction forces on A and B are opposite.

1,2 are trivial and were implied on the math stack exchange link. However, I came up with rule 3 by observing the relative velocities of each object.

Let's say A is a wall and B is a ball that is in rough contact with A. From A perspective, B is falling down hence an upward friction force should be acting on B. From B's perspective, A is moving upwards, hence a downward force should act on A.

However the problem arises in the following question

(By downward and upward, I mean upward tangential and downward tangential at point P)

The ball P is moving downwards. An upwards frictional force acts on P. Edit An upward frictional force at P imply a downward frictional force for the disc at P. Since frictional force is opposite the direction of movement, thus a downward friction imply an upward movement and hence a clockwise moment. However, it seems trivial that the disc should rotate anti-clockwise but using the rule 3, we can conclude that motion is clockwise.

Can somebody please explain the fallacies in the proposed rule 3? If there does exist a fallacy please explain some alternative that I can employ in working out the direction of frictional forces.

The link mentioned above is: https://math.stackexchange.com/q/2303592/335742

Answer 1

Your problem is this sentence:

Since frictional force is opposite the direction of movement...

Frictional forces are opposite the direction of relative movement. It is not the case that it is opposite absolute movement.

Imagine dropping an object onto a conveyor belt moving to the right. Friction from the belt onto the object causes it to move to the right also. So friction and motion are not opposite.

But until it reaches the same speed, the object is moving to the left compared to the belt.

Going back to your disc, if there were zero friction, the disc would not rotate. The ball would simply slide down. In such a case the relative motion between the objects would be the ball moving down compared to the disc. That tells you that the friction between the two would be "up" on the ball and "down" on the disk. That direction of friction is a force and gives you the acceleration. The ball falls more slowly than it would in zero friction, and the disc accelerates counterclockwise.

Answer 2

Assume there is relative motion between the two bodies with one body moving faster than the other one.

The frictional force on the slower body is such as to try to increase its speed so as to reduce the relative motion between the bodies.
The frictional force on the faster body is such to try to decrease its speed so as to reduce the relative motion between the bodies.

So you have the two frictional forces acting in opposite directions both trying to reduce the relative motion between the bodies.

A similar argument can be put forward for two bodies which are moving in opposite directions
In this case the frictional force on one body acts in the same direction as the direction of motion of the other body and the same is true of the other body.
The net result is the frictional forces trying to reduce the relative motion between the two bodies.

In both cases the frictional forces acting on the two bodies are in opposite directions and equal in magnitude - Newton's third law.

Answer 3

You have the mentioned the answer yourself! Since an upward frictional force acts on P, a downward frictional force acts on the disc, at the point of contact of P, and hence it rotates anti-clockwise (Notice, the direction of frictional forces is in opposite direction, not the direction of movement) (Though, to be exact, the frictional forces aren't in the directly upward or downward direction. They are in the direction of the tangent at P)