C. Forces · Forces

Static and Kinetic Friction: Coefficients of Friction

C2.4 C3.1 C2.1About 70 minutes

Learning goals

I can distinguish static friction from kinetic friction.
I can use Ff,k = μk FN and Ff,s ≤ μs FN.
I can calculate coefficients of friction from measurements, for example with a drag sled.
I can explain the factors that do and do not affect friction.

Words to know

static friction (Ff,s)
Friction between surfaces that are not sliding. It adjusts to match the applied force, up to a maximum: Ff,s ≤ μs FN.

kinetic friction (Ff,k)
Friction between surfaces sliding over each other: Ff,k = μk FN. It acts opposite to the sliding.

coefficient of static friction (μs)
The ratio (maximum static friction) ÷ (normal force) for a pair of surfaces. No units.

coefficient of kinetic friction (μk)
The ratio (kinetic friction) ÷ (normal force) for a pair of surfaces. No units. Usually smaller than μs.

drag sled
A block or box pulled across a surface with a force sensor or spring scale to measure friction.

lubricant
A substance (oil, grease, a water layer) that separates two surfaces and reduces friction.

Hard to start, easier to keep going

Try to slide a heavy dresser across a bedroom floor. At first you push and push and nothing happens. Then, suddenly, it lurches into motion, and it is easier to keep it moving than it was to get it started.

Two different kinds of friction are at work. In this lesson you will learn how to measure them and calculate them, which is what engineers do when they design anything from snow tires to hockey skates.

Static friction

When surfaces are not sliding, the friction is static. Static friction is clever: it is exactly as big as it needs to be to stop sliding, up to a limit.

Push a 50 kg box with 20 N: static friction is 20 N.
Push with 60 N: static friction is 60 N.
But it has a maximum: Ff,s,max = μs FN. Push harder than that and the box starts to slide.

So the rule is Ff,s ≤ μs FN. Use the equals sign only when the object is just about to slide.

Kinetic friction

Once surfaces slide, the friction is kinetic:

Ff,k = μk FN

It points opposite to the sliding and stays nearly constant whatever the speed (for everyday speeds). Usually μk < μs, which is why the dresser lurches when it breaks free.

What the coefficients mean

μ (Greek letter mu) depends only on the two surfaces (materials, roughness, wet or dry). It has no units because it is a ratio of two forces. A larger μ means a "grippier" pair of surfaces.

Push harder and harder on a heavy box. At first static friction matches your push exactly, so nothing moves. At the maximum, μs FN, the box breaks free, and friction drops to the smaller kinetic value, which stays about constant.

Surfaces	μs (approx.)	μk (approx.)
Rubber on dry concrete	1.0	0.8
Rubber on wet concrete	0.7	0.5
Steel on steel (dry)	0.74	0.57
Wood on wood	0.25 to 0.5	0.2
Waxed wood on wet snow	0.14	0.1
Ice on ice	0.1	0.03
Teflon on Teflon	0.04	0.04

Typical textbook values. Real values vary a lot with the exact surfaces, temperature, dirt and moisture, so always measure when it matters.

What affects friction?

Does affect it: the kinds of surfaces (μ), and how hard they are pressed together (FN). Doubling the normal force roughly doubles friction.
Mostly does not affect it: the area of contact, and (for kinetic friction) the speed, within everyday ranges. A brick has about the same friction whether it lies flat or on its edge: on its edge it presses harder on a smaller area, which balances out.
Lubricants (oil, wax, a thin layer of water) separate the surfaces and lower μ.
Rolling a wheel or ball is much easier than sliding it. Ball bearings and wheels replace sliding friction with much smaller rolling friction.

Friction is not always the enemy. Without friction you could not walk, drive, hold a pencil or tie your shoes.

Even polished surfaces are bumpy at the microscopic scale. Friction comes from electromagnetic attraction at the tiny contact points and from bumps catching. Pressing harder squashes the bumps together, making more contact, so friction is roughly proportional to the normal force.

Worked example

Example 1: Drag sled

A 2.5 kg drag sled is pulled across a gym floor at a constant speed with a horizontal force of 9.8 N. Find μk.

Constant speed, so Fnet = 0: kinetic friction = applied force = 9.8 N.
Level surface, no other vertical forces: FN = mg = 2.5 × 9.8 = 24.5 N.
μk = Ff,k ÷ FN = 9.8 ÷ 24.5 = 0.40.

μk = 0.40 (no units).

Worked example

Example 2: Moving a dresser

A 40 kg dresser rests on a wooden floor. μs = 0.45 and μk = 0.30. (a) What horizontal force is needed to start it moving? (b) What force keeps it moving at a constant velocity once it is sliding?

FN = mg = 40 × 9.8 = 392 N.
(a) Ff,s,max = μs FN = 0.45 × 392 = 176.4 N. You must push with just over 176 N (about 180 N).
(b) Ff,k = μk FN = 0.30 × 392 = 117.6 N. Constant velocity means Fnet = 0, so push with 118 N (about 120 N).

(a) About 176 N to start it. (b) About 118 N to keep it sliding steadily.

Worked example

Example 3: Static friction is not always μs FN

A 20 kg box sits on a floor with μs = 0.50. You push horizontally with 50 N. What is the friction force?

FN = 20 × 9.8 = 196 N. Maximum static friction = 0.50 × 196 = 98 N.
Your 50 N push is less than 98 N, so the box does not move.
The box is at rest, so Fnet = 0: static friction is 50 N, opposite to your push, not 98 N.

Friction is 50 N. Static friction only matches the push, up to its 98 N maximum.

Just for fun

Teflon has one of the lowest coefficients of friction of any solid, so food slides right off it. That created a puzzle for the people who invented the non-stick pan: how do you get a non-stick coating to stick to the pan?

Hands-on activity

Build a drag sled

You need: a shoebox, a few cans of food (each labelled with its mass), a luggage scale or spring scale (or a long rubber band and a ruler), string, and two or three floor surfaces (wood or tile, carpet, a rubber mat).

Find the mass of the box and cans (kitchen scale or labels). FN = mg.
Tie the string to the box and the scale. Pull horizontally. Read the scale just before the box starts moving (maximum static friction) and while it slides at a steady speed (kinetic friction). Most luggage scales read in kg; multiply by 9.8 to get newtons.
No scale? Use a rubber band: measure how far it stretches. A longer stretch means a bigger force, so you can compare static and kinetic friction, even without newtons.
Add cans to change FN. Make a table of FN, Ff,s and Ff,k.
Calculate μs and μk for each surface. Are they roughly constant as FN changes?
Turn the box on its side (smaller area). Does friction change?

Pull slowly and steadily; don't let the box tip.

Your turn

What is the difference between static and kinetic friction?

A 12 kg box slides across a floor with μk = 0.25. Find the kinetic friction.

It takes 45 N to keep a 15 kg crate sliding at constant speed. Find μk.

A 30 kg box needs 147 N to start it sliding. Find μs.

Why does μ have no units?

A 5.0 kg block rests on a table, μs = 0.60. You push horizontally with 20 N. What is the friction force? Does the block move?

More practice: warm-up

Which is usually larger, μs or μk?

Write the formula for kinetic friction.

Does a larger μ mean more or less friction?

Name one lubricant.

Does doubling the contact area (same box, same mass) double the friction?

More practice: on your own

A 65 kg hockey player slides across the ice on her knees after a goal. μk = 0.10 between her pants and the ice. Find the friction on her.

A 25 kg toboggan with waxed runners is on wet snow (μs = 0.14, μk = 0.10). What force starts it moving, and what keeps it moving at constant speed?

A drag sled with mass 1.8 kg needs 5.3 N to pull at constant speed. A 2.0 kg mass is added. Predict the new force.

Why does a heavy box lurch forward when it finally starts to slide?

Rubber on dry concrete has μs ≈ 1.0; on wet concrete about 0.7. What is the maximum static friction on a 1500 kg car's tires in each case?

Challenge

A 10 kg box rests on the floor (μs = 0.40). You press down on top of it with 50 N while a friend pushes horizontally. What horizontal push will just start it moving?

Explain why friction is roughly independent of contact area, using the idea of microscopic contact points.

A student measures, for wood on wood: FN = 5.0, 10.0, 15.0, 20.0 N and Ff,k = 1.1, 2.0, 3.1, 4.0 N. Find μk from the slope of Ff,k against FN.

Check your understanding

Quiz

A box at rest is pushed with a small force but does not move. The friction on it is:

A 10 kg block slides on a surface with μk = 0.30. The kinetic friction is about:

The units of μs are:

A sled is pulled at constant speed with 20 N. Its normal force is 80 N. μk is:

Which change increases the friction on a sliding box the most?

Why is it easier to keep a heavy box moving than to start it?

For parents and teachers

Big idea: Static and kinetic friction and their coefficients (C2.4); friction as a type of force and its effect on motion (C3.1).

How to help: Help set up the drag-sled activity. Ask your teen to predict before measuring: "What will happen to the force if we add another can? If we turn the box on its side?"

Common difficulties:

Using μs FN for static friction when the object is not about to slide. Static friction equals whatever is needed, up to that maximum.
Using mass instead of normal force (FN in newtons).
Assuming FN = mg when something else pushes down or pulls up on the object.