Static & Kinetic Friction

A surface pushes on whatever touches it with a single contact force. Resolve that force into a component perpendicular to the surface, the normal force N, and a component along it, friction f. Why should the parallel part track N? Even polished surfaces are rough on the micrometre scale, so they touch only where the tallest asperities meet, and those junctions flatten until they carry the load. The true contact area is then about N/H, where H, the indentation hardness of the softer surface, is of order 10⁹ Pa for steel. A 10 kg steel block presses down with 98 N, giving a true contact area near 1 × 10⁻⁷ m², roughly a hundred-thousandth of a 10 cm × 10 cm face. Sliding means shearing those junctions, so friction is about their shear strength times that area. Double the load and the true area doubles; stand the block on a smaller face and it does not change. That is Amontons' rule, friction proportional to load and independent of apparent area, recovered from the microscopic picture.

While two surfaces grip, no formula gives static friction. Like the normal force, it is whatever Newton's second law needs to keep the surfaces from slipping, so you find it by assuming no slip and solving. Its only rule is the inequality fₛ ≤ μₛN. A 30 kg filing cabinet on carpet with μₛ = 0.50 has N = 294 N and a ceiling of 147 N. Push it with 60 N and it does not move, so a = 0 and friction is exactly 60 N; push with 120 N and friction is 120 N. Friction reaches μₛN only at the instant slipping begins. That gives a three-step routine for every friction problem: assume the contact holds, so both surfaces share one acceleration; solve for the friction force this requires; compare its size with μₛN. If it fits, the answer stands. If it does not, the assumption was false, the surfaces slide, and you solve again with kinetic friction.

Friction on a body points against the slipping of its surface over the other surface: the actual slipping for kinetic friction, and for static friction the slipping that would happen if the contact were frictionless. That is often along the motion. When a pickup truck speeds up, a frictionless bed would slide forward beneath a crate and leave it behind, so static friction on the crate points forward and supplies all of its acceleration. A car's drive tyres push backward on the road, and the road's friction pushes the car forward. For kinetic friction, work out the velocity relative to the other surface. A suitcase dropped at rest onto a baggage belt moving at 0.60 m s⁻¹ slides backward relative to the belt, so kinetic friction points forward and speeds it up at μₖg. With μₖ = 0.40 that is 3.92 m s⁻²: the speeds match after 0.15 s, the suitcase has slipped 4.6 cm along the belt, and static friction takes over. Every friction force has a third-law partner on the other surface, pointing the opposite way.

When the surfaces slip, the inequality is replaced by an equation: friction has magnitude μₖN whatever the applied force, and in Coulomb's model it does not depend on speed or apparent area. For most dry pairs μₖ is smaller than μₛ, largely because junctions that have had time to settle under load are stronger than those formed and broken on the move. Drag a 6.0 kg sandbag with a spring scale whose reading rises slowly, with μₛ = 0.55 and μₖ = 0.40. The reading climbs to μₛN = 32.4 N, and the instant the bag breaks free friction falls to μₖN = 23.5 N. Hold the pull at 32.4 N and the unbalanced 8.8 N accelerates the bag at 1.47 m s⁻²; to drag it at steady speed you must ease off to 23.5 N. That drop from static to kinetic drives stick–slip, the alternation of gripping and lurching behind a squeaking hinge, chalk juddering on a board and the sustained note of a violin bow.

Both friction laws scale with N, so anything that changes the normal force changes friction, and N is often not mg. On an incline N = mg cos θ while gravity pulls down the slope with mg sin θ, so a block stays put only if mg sin θ ≤ μₛmg cos θ, that is tan θ ≤ μₛ. Mass and g cancel: a block that first slips at 25° reveals μₛ = tan 25° = 0.47. A rope pulled at angle φ above the horizontal carries part of the weight, so N = mg − F sin φ, and a crate starts moving when F cos φ reaches μₛ(mg − F sin φ). That gives F(φ) = μₛmg / (cos φ + μₛ sin φ), and setting dF/dφ = 0 leaves −sin φ + μₛ cos φ = 0, so tan φ = μₛ. For a 20 kg crate with μₛ = 0.60, a horizontal pull must reach 118 N, but at the best angle, 31°, 101 N is enough. Pull more steeply and the horizontal component you lose outweighs the friction you save. The best rope angle equals the angle of repose of the same pair of surfaces.

A table listing μₛ = 0.6 for “steel on steel” is reporting particular experiments. A coefficient belongs to a pair of surfaces in a particular condition, and it shifts with polish, oxide films, grease from fingers, humidity and temperature; measurements on nominally identical pairs routinely differ by 20% or more. Values of 1 or more are allowed: rubber on dry concrete can reach it, and clean metals in vacuum can cold-weld. Static friction also creeps up with time at rest, so a part left standing overnight can need a harder push to start. The linear law itself fails in three familiar regimes. Soft rubber drapes over surface roughness, so tyre grip depends on contact area, speed and temperature, and μ falls as the load rises. In a lubricated contact a fluid film carries the load, and the drag grows with sliding speed. And at very high loads, once the true contact area approaches the whole footprint, friction stops growing in proportion to N. Rolling resistance is a different mechanism again and needs its own model.