Friction

Friction resists sliding between surfaces. Learn static and kinetic friction, the coefficient of friction, angle of friction, inclined planes, belt and rope friction, and where engineers want more or less friction.

Without friction you couldn't walk, drive, hold a tool or keep a bolt tight. With too much, machines waste energy and wear out. Engineers constantly design for the right amount: high in brakes and clutches, low in bearings and slides.

The Coulomb friction model — friction force proportional to the normal force — is simple and remarkably useful for everyday design calculations.

An analogy

Microscopically, surfaces are like two brushes pressed together: their bristles (asperities) interlock. Press harder (more normal force) and they interlock more, so it's harder to slide.

Static vs kinetic

Up to a limit, static friction matches whatever force tries to slide the object, so it stays put. Once the applied force exceeds μ_s·N, sliding starts and friction drops to the kinetic value μ_k·N. That's why it's harder to start pushing a heavy box than to keep it moving.

Typical coefficients

Steel on steel (dry) ≈ 0.6–0.8; lubricated ≈ 0.05–0.15; rubber on dry asphalt ≈ 0.7–1.0; wet ≈ 0.4–0.6; PTFE on steel ≈ 0.04. Values vary with surface finish, temperature and contamination, so designs use conservative ranges.

Inclined planes

On a slope of angle θ, weight splits into mg sin θ down the slope and mg cos θ into it. The block slides if tan θ > μ_s. Screws and wedges are inclined planes wrapped or stacked, which is why they can be self-locking.

Belt and rope friction

Friction around a curved surface grows exponentially with wrap angle: T₁/T₂ = e^{μθ}. A few turns of rope on a bollard let a person hold a ship. Belt drives use this to transmit torque; slipping occurs when the ratio exceeds e^{μθ}.

Beyond Coulomb

Real friction depends on sliding speed (Stribeck curve: boundary, mixed and hydrodynamic lubrication), temperature and time (stick-slip causes squeals and chatter). Rolling resistance, tyre slip curves and tribology — the science of friction, wear and lubrication — refine the simple model.

Formulas

• Limiting static friction: F_f ≤ μ_s · N
• Kinetic friction: F_f = μ_k · N (μ_k < μ_s)
• Angle of repose: tan θ = μ_s
• Belt / capstan equation: T₁ / T₂ = e^{μθ} (θ in radians)

Worked example: Force to start a crate moving

Given: m = 50 kg; μ_s = 0.4, μ_k = 0.3

Find: Force to start sliding and to keep it sliding.

• N = 50 × 9.81 = 490.5 N
• Start: 0.4 × 490.5 = 196 N
• Keep sliding: 0.3 × 490.5 = 147 N

Answer: 196 N to start, 147 N to keep moving

Worked example: Block on an incline

Given: m = 20 kg on 30° slope; μ = 0.25

Find: Does it slide? Acceleration?

• Down-slope: 20 × 9.81 × sin 30° = 98.1 N
• Max friction: 0.25 × 20 × 9.81 × cos 30° = 42.5 N
• 98.1 > 42.5 → slides; a = (98.1 − 42.5)/20 = 2.78 m/s²

Answer: Slides at 2.78 m/s²

Common mistakes

• Wrong: Assuming friction always equals μN. — Right: Static friction is only as large as needed to prevent motion, up to the limit μ_s·N.
• Wrong: Using degrees in e^{μθ}. — Right: The wrap angle must be in radians.

Applications

• Brakes & clutches: Engineered friction materials (μ ≈ 0.3–0.5) convert kinetic energy to heat or transmit torque.
• Bolted joints: Friction between clamped plates carries shear load in slip-critical connections.
• Bearings & lubrication: Rolling-element and hydrodynamic bearings cut friction to μ ≈ 0.001–0.005.

Key takeaways

• F ≤ μ_s N (static); F = μ_k N (sliding).
• Slope slides if tan θ > μ_s.
• Rope/belt: T₁/T₂ = e^{μθ} with θ in radians.
• Design for high friction where needed and low where wasteful.

Going further

• Self-locking screws: A screw is self-locking when its lead angle is less than the friction angle; jacks rely on this, while ball screws (low friction) need brakes.
• Tyre friction: Tyres generate force through slip; the Pacejka 'magic formula' models force vs slip ratio and slip angle for vehicle simulation.
• Friction coefficients in design: Standards (e.g. EN 1993-1-8 for slip-critical bolts) specify μ by surface treatment class; design uses lower-bound values.