Friction, limiting equilibrium and Newton's third law

Push gently on a heavy box and it doesn't move. Something is pushing back exactly as hard as you push. That is friction: a force from the floor, along the surface, that opposes sliding.

The key fact about friction is that it adjusts itself. Imagine a box whose friction can reach at most 24 N:

So friction is only as big as it needs to be to stop sliding, up to a maximum. When an object is at rest but would start to slide if pushed any harder, friction is at its maximum and the object is in limiting equilibrium. Exam questions say this as "about to slip", "on the point of sliding" or "in limiting equilibrium".

What decides the maximum? Two things:

• How hard the surfaces are pressed together. That is the normal reaction $R$. A heavier box presses harder and grips more.
• How grippy the surfaces are. Rubber on concrete grips far better than ice on steel. This is measured by the coefficient of friction $\mu$. It has no units, and it is usually between 0 and 1.

Multiply them to get the maximum:

$F_{\max} = \mu R$

So the rule is:

• Whenever an object is at rest, $F \leqslant \mu R$. Work out $F$ from equilibrium (it is whatever is needed), not from $\mu R$.
• Only when it is about to slip (or already sliding), $F = \mu R$.

Rearranging gives the definition of the coefficient of friction: in limiting equilibrium, $\mu = \dfrac{F}{R}$.