Friction & normal force theory

Macroscopic friction is governed by two empirical laws formulated by Guillaume Amontons (1699) and confirmed by Charles-Augustin de Coulomb (1785): (1) The frictional force is directly proportional to the normal applied load; (2) The frictional force is independent of the apparent macroscopic contact area. Modern surface physics explains this through asperity contact: solid surfaces are rough on a microscopic scale, touching only at microscopic peaks (asperities). The true contact area A_true is proportional to the normal load N (A_true ≈ N / H, where H is material hardness). Because frictional resistance equals interfacial shear strength multiplied by true contact area (F_f = τ A_true = τ N / H), F_f scales linearly with N and remains independent of the apparent macroscopic area.

Static friction F_s resists the initiation of motion and adjusts to balance any applied lateral force up to a limiting maximum: F_s ≤ μ_s N. Once the applied force exceeds the limiting static friction, microscopic cold-welded junctions shear, the surfaces begin to slide, and friction drops to a lower, nearly constant kinetic value F_k = μ_k N. The kinetic coefficient μ_k is almost universally smaller than the static coefficient μ_s because moving asperities do not have sufficient dwell time to form full adhesive bonds with opposing valleys.

When plotting kinetic friction force F_f on the vertical axis against normal load N on the horizontal axis, the data follow F_f = mN + b. The slope m represents the coefficient of kinetic friction μ. In an ideal dry contact system, the theoretical intercept b is zero. An experimental non-zero intercept b indicates an uncorrected horizontal dynamometer tare error, track misalignment, or adhesive surface tack.

Coulomb’s third law states that kinetic friction is independent of sliding velocity over moderate speed ranges. At very low speeds, stick-slip relaxation oscillations can occur; at high speeds, frictional heating alters interfacial viscosity and softens surface asperities, leading to a modest decline in μ. In standard undergraduate experiments, keeping pulling speed uniform at approximately 0.05 m/s ensures the measurement remains squarely in the velocity-independent Coulomb regime.

If the traction force F is applied at an angle θ relative to the horizontal surface, the normal force is no longer simply the gravitational weight W = Mg. Resolving forces gives N = W - F sin(θ) for an upward pull, and the horizontal equilibrium becomes F cos(θ) = μ_k N = μ_k (W - F sin(θ)). Rearranging yields F = μ_k W / (cos(θ) + μ_k sin(θ)). To avoid this trigonometric coupling and measure F_f = μ_k W directly, the pulling line must be kept strictly horizontal (θ = 0).

Rolling friction occurs when a circular cylinder or sphere rolls over a surface without slipping. Rather than shearing interfacial junctions, rolling resistance arises from the viscoelastic hysteresis loss during the cyclic compression and relaxation of the contact materials: F_r = (c_r / R) N, where c_r is the coefficient of rolling resistance and R is the wheel radius. Because no surface shear occurs, rolling friction coefficients (typically 0.001 to 0.01) are orders of magnitude smaller than sliding friction coefficients (0.1 to 1.0).

In a single pulling trial, the relative uncertainty in μ satisfies (u_μ / μ)² = (u_Ff / Ff)² + (u_N / N)². Normal force N is determined from calibrated masses with uncertainty below 0.1%. Friction force F_f measured on a spring balance has fluctuations of ±0.02 N due to microscopic local roughness variations. Multi-point linear regression averages out these local asperity fluctuations across multiple loads, yielding a slope standard error well below 0.5% of μ.