Friction

Friction is the part of the contact force that acts along the surfaces. It opposes relative motion between them, or the motion that is just about to start.

Push a heavy cupboard gently and it does not move: static friction pushes back exactly as hard as you push. Push harder and static friction grows to match, but only up to a limit:

• Static friction: f_s ≤ μ_s N. It is only 'up to' this value. Do not write f_s = μ_s N unless the body is just about to slide.
• Limiting friction: the largest static friction, μ_s N, reached just before sliding starts.
• Kinetic friction: f_k = μ_k N, once the body slides. μ_k is usually a little less than μ_s, which is why it is harder to start a box moving than to keep it moving.

Example: a 10 kg box on a floor with μ_s = 0.4 and μ_k = 0.3 (g = 10 m s⁻², so N = 100 N). Push with 25 N: friction is 25 N and nothing moves. The box starts to slide only at 40 N. Once sliding, friction is 30 N, so a 40 N push gives a = (40 – 30)/10 = 1 m s⁻².

Friction depends on the normal force N, not directly on the weight. N equals mg only on level ground with nothing pressing down or lifting. Friction does not depend on the area of contact or on the sliding speed; it depends on the two surfaces, through μ.

On a slope at angle θ, the weight splits into mg sin θ down the slope and mg cos θ into it. The block is just about to slide when mg sin θ = μ_s mg cos θ, that is when tan θ = μ_s. The mass cancels, so heavy and light blocks start to slide at the same angle. This angle is called the angle of repose.

Rolling friction is much smaller than sliding friction: that is why wheels, ball bearings and rollers make work easier, and why oil and smooth surfaces help.

Friction is not only a nuisance. Without it we could not walk, cars could not start or stop, and a box on the floor of a speeding-up train would slide back. Static friction can carry the box with the train up to an acceleration of μ_s g.