Skip to content

The coefficient of friction

Forces and motion · level 6, lesson 2 of 6

A question to start

On Mars, a crate presses on the floor with only 0.38 of its Earth weight. Is it easier to slide? By how much?

The idea

Friction depends on two things: how grippy the two surfaces are, and how hard they press together. The pressing force is the normal force \(N\), at right angles to the surfaces. Experiments show that, roughly:

\[F_\text{static} \le \mu_s N, \qquad F_\text{kinetic} = \mu_k N.\]

\(\mu\) (the Greek letter mu) is the coefficient of friction, a number with no units. It depends on the pair of surfaces, not on how big the contact area is.

Surfaces \(\mu_s\) static \(\mu_k\) kinetic
Rubber on dry concrete about 1.0 about 0.8
Wood on wood 0.25 to 0.5 about 0.2
Steel on steel, dry about 0.7 about 0.5
Ice on ice about 0.1 about 0.03

(Typical textbook values; real ones vary with dirt, wetness and wear.)

On flat ground, \(N = mg\). So on Mars, where \(g\) is smaller, \(N\) is smaller and friction is smaller in the same proportion:

A 100 kg crate, \(\mu_s = 0.5\), on a flat floor Mars Earth
Normal force \(N = mg\) 371 N 981 N
Push needed to start it sliding, \(\mu_s N\) 186 N 491 N
Acceleration with a 300 N push, \(\mu_k = 0.4\) 1.52 m/s² sliding needs over 491 N: it stays put
EarthMarsNN

That is good news for moving cargo on Mars, but bad news for grip. A vehicle's tyres can push it forward, brake it or turn it with only about 0.38 of the force they could on Earth.

Worked examples

1. The crate on Mars with a 300 N push. It slides, since \(300 > 186\). Kinetic friction \(= 0.4 \times 371 = 148\) N. Net force \(300 - 148 = 152\) N. \(a = 152 \div 100 = 1.52\) m/s².

2. Braking a truck on Mars. A 2,000 kg truck with \(\mu_s = 0.8\) between tyres and rock: the most braking force is \(0.8 \times 2000 \times 3.71 = 5{,}900\) N, giving at most \(5900 \div 2000 = 3.0\) m/s². On Earth, 7.8 m/s².

Try it

  1. A 50 kg box sits on a floor with \(\mu_s = 0.6\). What push starts it sliding on Earth? On Mars?
  2. Why does friction on Mars come out smaller for the same object and surfaces?
  3. A crate slides with \(\mu_k = 0.3\). Find the kinetic friction on a 200 kg crate on Mars.
  4. What is the greatest deceleration a car with \(\mu_s = 0.9\) can brake at, on Mars?
Answers
  1. Earth \(0.6 \times 50 \times 9.81 = 294\) N; Mars \(0.6 \times 50 \times 3.71 = 111\) N.
  2. The normal force \(mg\) is smaller, and friction is \(\mu\) times the normal force.
  3. \(0.3 \times 200 \times 3.71 = 223\) N.
  4. \(a = \mu_s g = 0.9 \times 3.71 = 3.3\) m/s². The mass cancels.

Common mistake

"A wider tyre has more friction because more of it touches." In the simple model, friction does not depend on area: the same normal force spread over more area presses less at each point. Wide tyres help for other reasons, such as not sinking into soft ground.

Where it leads