Static friction

Static friction is the odd one out among AP forces: it has no formula of its own. The AP Physics 1 CED says it adopts the value and direction required to prevent an object from slipping or sliding on a surface. Push a crate with 20 N and it stays put, static friction is 20 N backward. Push with 50 N and it still stays put, static friction is 50 N backward.

That obliging behavior runs out somewhere. The CED states that there exists a maximum value for which static friction will prevent an object from slipping on a given surface, and gives the derived equation $F_{f,s,\text{max}} = \mu_s F_N$. The equation sheet prints the relationship as $|\vec{F}_f| \leq |\mu \vec{F}_N|$, an inequality rather than an equation, which is the sheet telling you the same thing.

The condition is no relative sliding, not being at rest. A box riding on the bed of an accelerating truck is moving quickly over the road, but the box and the bed are not sliding on each other, so the friction between them is static, and it is the forward force accelerating the box.

Two traps: writing $F_f = \mu_s F_N$ when nothing in the problem says the object is on the verge of slipping, and fixing the direction of static friction before working out which way the object would otherwise slide.

Deciding which of the two frictions applies is worked through in static vs kinetic friction.