Friction: coefficients and calculations

Static and kinetic friction, the coefficient of friction μ and the model f = μ·N, inclines and the energy friction turns into heat. With an experiment to measure μ two ways.

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Friction opposes relative motion between surfaces in contact. While the surfaces do not slide, static friction matches the applied force up to a maximum: f_s ≤ μ_s·N. Once sliding, kinetic friction is f_k = μ_k·N, where N is the normal force and μ is the dimensionless coefficient of friction.[1]

Typical values (static / kinetic): rubber on dry concrete 1.0 / 0.7; wood on wood 0.5 / 0.3; dry steel on steel 0.6 / 0.3 but oiled 0.05 / 0.03; ice on ice 0.1 / 0.03; waxed wood on wet snow 0.14 / 0.1. This is an approximate empirical model, so the values are given to only one or two significant figures.[1]

μ_s is usually larger than μ_k. Friction is nearly independent of the apparent contact area, because only the high spots of the rough surfaces actually touch; adhesion between surface molecules also contributes.[1][2]

Example: a 100 kg crate on a level floor has weight and normal force 980 N. With μ_s = 0.45 you must push with more than 0.45 × 980 ≈ 440 N to start it; with μ_k = 0.30 it then takes about 290 N to keep it moving at constant speed.[1]

On a slope of angle θ, an object slides down at constant velocity when the friction balances the component of its weight along the slope, which gives μ_k = tan θ. Because μ_s > μ_k, a resting object does not start to slide until the angle is larger.[1]

Friction does negative work, W = −f·d, and converts kinetic energy into thermal energy: a car stopped by friction on level ground loses its kinetic energy as heat. Rolling greatly reduces losses, since sliding coefficients are typically 100 to 1,000 times larger than rolling ones.[5][2]

Try it: Measuring the coefficient of friction two ways

⏱ 50 minutes

You need

Wooden block with a hook (about 200–500 g)
Force meter (0–5 N or 0–10 N)
Balance
A board about 1 m long that can be tilted
Protractor (or metre rule to measure height and length)
Slotted masses to load the block

Safety

Keep the board low and clamp or hold it so it cannot slip; stand to the side so the block cannot slide onto feet. Don't overload the force meter.

What to do

Method 1: measure the mass of the block and calculate N = m·g. Pull it along the level board at a slow, steady speed and read the force meter: this is f_k. Calculate μ_k = f_k / N.
Repeat with two or three different loads on the block and plot f_k against N. The gradient is μ_k.
Method 2: put the block on the board and slowly raise one end. Each time, give the block a small nudge. Find the angle θ at which it just keeps sliding at a steady speed after the nudge.
Calculate μ_k = tan θ (or height ÷ horizontal length).
Repeat each method at least three times. Compare the two values of μ_k and estimate the uncertainty in each.
Optional: also record the force (method 1) or angle (method 2) at which the block first starts to move without a nudge, and compare with μ_k.

What happens, and why?

At constant velocity the net force is zero, so the pulling force equals kinetic friction (method 1), and on the slope the friction balances the weight component along the slope, giving μ_k = tan θ (method 2). The f–N graph should be a straight line through the origin if f = μ·N holds. The starting force or angle is larger because static friction is usually larger than kinetic friction. Values scatter because the model is only approximate and depends on the state of the surfaces.[1]

Play: Static, kinetic or calculate

Is it static friction (no sliding yet) or kinetic friction (sliding)? Calculation cards: work out f = μ·N with g ≈ 9.8 m/s² and pick the closest value.

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Other ages: Ages 6–7 · Ages 10–11