Push gently on the side of a box resting on the floor. It does not move. There is something worth confirming here: if it is being pushed and still does not move, then a force balancing the push must be coming from somewhere.
That force is friction. Watch the arrows in the figure. Increase the push and the friction grows by exactly as much. Ease off and it shrinks by exactly as much. Push not at all and no friction appears either.
This is where friction differs from gravity or the normal force. Gravity is fixed at . Static friction, on the other hand, has no size of its own. Push hard and it is large, push softly and it is small; it comes out to whatever is needed at the time.
So while the box stays put, the friction is . The letter has not appeared yet. The coefficient is used only at the limit, which comes in the next scene. Writing the friction on a stationary box as straight away is an extremely common mistake.
Coming out to match the push does not mean coming out without end. Keep increasing the push and at some point the friction stops growing.
The marker standing in the figure is that limit. The friction grows this far and no farther. This value is the maximum static friction, written , where is the coefficient of static friction and is the force pressing on the surface, the normal force.
The instant the push passes that marker, there is nothing left to balance it. The forces no longer sum to zero, so the box begins to move. The condition for slipping is .
Turned around, this says that as long as the friction remains . is not the value of the friction but the ceiling the friction can reach. Do not run the two together.
Once the box slips, the friction changes to a different value. Friction on something already moving is called kinetic friction, and it is , with the coefficient of kinetic friction.
Two markers stand in the figure. The outer one is the limit of static friction, the inner one is kinetic friction. The friction arrow grows out to the outer marker, then shrinks back to the inner one the moment the box slips, because .
The hand doing the pushing has not changed. The friction alone has dropped, so a net force appears all at once, and the box accelerates from there. Pushing a heavy cabinet and feeling it turn light the moment it starts to move is this same drop.
One more point: kinetic friction does not depend on speed. Sliding slowly or sliding fast, it stays . Variable while the object is at rest, constant once it slips. That exhausts the behaviour of friction.
Draw everything so far on a single graph, with the push across and the friction that comes out of it up the side.
The left half is the line , at a slope of exactly 45 degrees. That line is what it means to say the friction gives back what you give it. The two axes are drawn to the same scale, which is why the angle comes out at 45 degrees.
On reaching the line breaks off and jumps downward. That jump is the friction falling at the instant of slipping. It is bridged with a dashed line because the friction does not pass through the values in between.
To the right of the jump the line is horizontal. However hard the box is pushed, the kinetic friction stays at . This one broken line draws the whole character of friction as a force.