What are the rules for static and kinetic friction?

[terryds]

A bottle is placed on a sheet of paper at the edge of a table. The sheet is pulled slowly at first but very quickly when it has reached the very end of the edge. The bottle remains on the table.
Why does this happen?

I think what happens in the initial condition (when it's still pulled slowly) is that there is static frictional force between bottle and the paper which makes the bottle goes the same way as the paper.
But, when the paper is pulled very quickly, the frictional force becomes kinetic frictional force. Since the coefficient of kinetic friction is much less than the static one, the acceleration of the bottle becomes very very small so the bottle doesn't fall off the table.

But, after googling (https://www.quora.com/What-is-the-principle-behind-the-table-cloth-trick), I found out that it's due to the inertial resistance. What is actually inertial resistance? Is it a force or a vector? What's the formula? I really don't get it.

 

[kuruman]

The reference that you provide is confusing in my opinion. I understand mass and acceleration but not inertial resistance. Remember, static friction has an upper limit that it cannot exceed. When you start pulling on the paper slowly, your action accelerates the paper with some acceleration a regardless of how hard you pull. When you pull gently, the acceleration is small so that an object of mass m placed on the paper is accelerated by static friction and moves with it without sliding with the same acceleration as the paper. This is because ma is less than the upper limit of the force of static friction. When you pull rapidly, ma exceeds the maximum force of static friction almost immediately and the object slides relative to the paper.

 

[terryds]

The reference that you provide is confusing in my opinion. I understand mass and acceleration but not inertial resistance. Remember, static friction has an upper limit that it cannot exceed. When you start pulling on the paper slowly, your action accelerates the paper with some acceleration a regardless of how hard you pull. When you pull gently, the acceleration is small so that an object of mass m placed on the paper is accelerated by static friction and moves with it without sliding with the same acceleration as the paper. This is because ma is less than the upper limit of the force of static friction. When you pull rapidly, ma exceeds the maximum force of static friction almost immediately and the object slides relative to the paper.

Another question :

1. Does the friction between the bottle and the paper has a reaction conjugate? I mean, if there is action (in this case: friction), is there any reaction??
For example, the static friction between bottle and paper has the same direction as the direction I pull the paper, so there is reaction force which exerts on the paper caused by the ball, right? (as a reaction of static frictional force)

2. Does the kinetic friction has the same direction as the direction I pull the paper?? I doubt it since the bottle remains on the table when yanked quickly. So, does the direction of friction force change (become the opposite) as it goes from static to kinetic in this case? Or, does the direction remain the same, but the acceleration is just so slow?

 

[kuruman]

Good questions.

1. According to Newton's Third Law, every action has a reaction. The paper pushes the bottle forward and the bottle exerts a backward force on the paper. You accelerate the paper not the bottle, you are not even touching the bottle. It is the forward force of static friction exerted by the paper that accelerates the bottle. Of course, the paper cannot accelerate the bottle all by itself, that's why you might think that it is the force from your hand that does it. Your hand accelerates the paper not the bottle. Now, your hand that pulls the paper feels (through the paper) the backward reaction force that the bottle exerts on the paper. It's that backward reaction force that your reference calls "inertial resistance", an unfortunate name.

2. Don't forget that when you pull the paper forward very rapidly, the bottle is at rest relative to the table but slides relative to the paper. Kinetic friction on an object is always opposite to the velocity of the object. Since the paper slides forward, the kinetic friction on it will be backward. Therefore the kinetic friction on the bottle will be forward. It's the same situation as if you held the bottle with your hand and slid the paper under it.

Here are two simple rules for static and kinetic friction:

Static friction adjusts itself in magnitude and direction to provide the observed acceleration, but only up to a certain limit that it cannot exceed.
Kinetic friction only exists between two surfaces that are rubbing as they move past each other and points in the opposite direction of the velocity of one surface relative to the other.