2.7 Kinetic and Static Friction

When one surface slides across another, kinetic friction opposes the sliding, pointing opposite to the motion of one surface relative to the other. Its size is:

$$F_{f,k} = \mu_kF_N$$

The coefficient of kinetic friction, $\mu_k$, is a unitless number that depends on the two materials (rubber on concrete is high, ice on steel is low). For problems in this course it doesn't depend on the speed or on the area of contact. It's multiplied by the normal force, not by the weight, which matters on inclines and with angled pulls.

When surfaces are in contact but not sliding relative to each other, static friction does whatever is needed to keep them from slipping, up to a limit:

$$F_{f,s} \le \mu_sF_N$$

Push a heavy box gently and static friction matches your push exactly, so the box stays put. Push harder and static friction grows to match, until it hits its maximum $\mu_sF_N$. Push beyond that and the box breaks free and starts to slide; then kinetic friction takes over.

Usually $\mu_s > \mu_k$, which is why it takes more force to get something sliding than to keep it sliding.

Static friction can point in the direction of motion. When you walk, your shoe pushes backward on the ground, and static friction on your shoe points forward. That forward friction is what accelerates you (and a car).

Even smooth-looking surfaces are rough at the microscopic scale. Where tiny bumps touch, the atoms of the two surfaces attract each other electrically, and the bumps catch on each other. Friction is the combined effect of all those contact points.

Kinetic friction always turns some kinetic energy into thermal energy, which is why rubbing your hands warms them. Static friction, with no sliding, doesn't (3.4).

Friction can also act between two moving objects. If a box rides on the bed of an accelerating truck without slipping, static friction from the truck bed is the force that accelerates the box. If the truck accelerates too hard, the needed friction exceeds $\mu_sF_N$ and the box slides backward relative to the truck.

• Find the normal force first, from the forces perpendicular to the surface. Don't assume it's mg.
• If the object is sliding, use $F_f = \mu_kF_N$ opposite the sliding.
• If it might not slide, find the friction needed to keep it still and compare with $\mu_sF_N$. If the needed amount is less than or equal to the maximum, it stays put and friction equals the needed amount.
• On an incline with no other forces, a block just starts to slip when $\tan\theta = \mu_s$. Sliding down, its acceleration is $g(\sin\theta - \mu_k\cos\theta)$.