Friction: static and kinetic

Friction as a contact force

Friction is a force between surfaces that resists relative sliding or the tendency to slide. Although microscopic friction is complex, classical mechanics often uses simple models that work well in many everyday situations.

There are two main types in introductory mechanics: static friction and kinetic friction. Static friction acts when surfaces are not sliding relative to each other. Kinetic friction acts when they are sliding.

Static friction adjusts

Static friction is flexible. If you push gently on a heavy box and it does not move, static friction matches your push in the opposite direction. If you push harder and the box still does not move, static friction grows.

But static friction has a maximum value:

$f_s leq mu_s N$

Here $mu_s$ is the coefficient of static friction and $N$ is the normal force. The maximum possible static friction is

$f_{s,max}=mu_s N$

The equality holds only when the object is just about to slip.

Kinetic friction

Once surfaces slide, kinetic friction acts opposite the direction of relative motion. Its magnitude is modeled as

$f_k=mu_k N$

where $mu_k$ is the coefficient of kinetic friction. Usually $mu_k<mu_s$, meaning it often takes more force to start sliding than to keep sliding.

Kinetic friction is commonly treated as approximately independent of speed and contact area, though this is an idealized model.

Direction of friction

Friction opposes relative motion or impending relative motion, not necessarily the motion of the object relative to the ground. This distinction matters.

When you walk, your foot tends to push backward on the ground. Static friction from the ground pushes your foot forward. In this case, friction helps you move forward.

For a car accelerating without skidding, static friction on the tires points forward. During braking, static friction may point backward, slowing the car.

Block on a rough horizontal surface

Suppose a block is pulled horizontally by force $F$. If it remains at rest, then

$f_s=F$

as long as

$Fleq mu_s N$

If $F$ exceeds the maximum static friction, the block starts sliding. Then the friction force becomes kinetic:

$f_k=mu_k N$

The horizontal equation is

$F-f_k=ma$

if the block accelerates in the direction of the pull.

Block on a rough incline

On an incline, weight has a component down the plane:

$mgsin heta$

The normal force is

$N=mgcos heta$

If the block is at rest, static friction acts as needed to prevent sliding, up to its maximum. If the block is just about to slide down, friction points up the plane and

$mgsin heta = mu_s mgcos heta$

This gives

$an heta=mu_s$

for the critical angle in this ideal model.

Energy and friction

Kinetic friction does negative work on sliding objects:

$W_f=-f_k d$

for motion over distance $d$ opposite the friction force. This transforms mechanical energy into thermal energy. Mechanical energy is not conserved when kinetic friction is present, though total energy is still conserved if thermal energy is included.

The big idea

Static friction prevents slipping and adjusts up to a maximum. Kinetic friction acts during sliding and is modeled by $f_k=mu_kN$. Friction's direction depends on relative motion or impending motion. Although friction is microscopically complex, simple friction models are powerful tools for analyzing surfaces, ramps, vehicles, walking, and energy loss.