Friction and Coefficients of Friction

Frictional force can be expressed as

F_f = μ N        (1)

where

F_f = frictional force (N, lb)

μ = static (μ_s) or kinetic (μ_k) frictional coefficient

N = normal force (N, lb)

For an object pulled or pushed horizontally, the normal force - N - is simply the weight:

N = m g         (2)

where

m = mass of the object (kg, slugs)

g = acceleration of gravity (9.81 m/s², 32 ft/s²)

Frictional Coefficients for some Common Materials and Materials Combinations

Materials and Material Combinations		Static Frictional Coefficient - μs
		Clean and Dry Surfaces	Lubricated and Greasy Surfaces
Aluminum	Aluminum	1.05 - 1.35	0.3
Aluminum-bronze	Steel	0.45
Aluminum	Mild Steel	0.61
Brake material2)	Cast iron	0.4
Brake material2)	Cast iron (wet)	0.2
Brass	Steel	0.35	0.19
Brass	Cast Iron	0.31)
Brick	Wood	0.6
Bronze	Steel		0.16
Bronze	Cast Iron	0.221)
Bronze - sintered	Steel		0.13
Cadmium	Cadmium	0.5	0.05
Cadmium	Chromium	0.41	0.34
Cadmium	Mild Steel	0.461)
Cast Iron	Cast Iron	1.1, 0.151)	0.071)
Cast Iron	Oak	0.491)	0.0751
Cast iron	Mild Steel	0.4, 0.231)	0.21, 0.1331)
Car tire	Asphalt	0.72
Car tire	Grass	0.35
Carbon (hard)	Carbon	0.16	0.12 - 0.14
Carbon	Steel	0.14	0.11 - 0.14
Chromium	Chromium	0.41	0.34
Copper-Lead alloy	Steel	0.22
Copper	Copper	1	0.08
Copper	Cast Iron	1.05, 0.291)
Copper	Mild Steel	0.53, 0.361)	0.181)
Diamond	Diamond	0.1	0.05 - 0.1
Diamond	Metal	0.1 - 0.15	0.1
Glass	Glass	0.9 - 1.0, 0.41)	0.1 - 0.6, 0.09-0.121)
Glass	Metal	0.5 - 0.7	0.2 - 0.3
Glass	Nickel	0.78	0.56
Graphite	Steel	0.1	0.1
Graphite	Graphite (in vacuum)	0.5 - 0.8
Graphite	Graphite	0.1	0.1
Hemp rope	Timber	0.5
Horseshoe	Rubber	0.68
Horseshoe	Concrete	0.58
Ice	Ice	0.02 - 0.09
Ice	Wood	0.05
Ice	Steel	0.03
Iron	Iron	1.0	0.15 - 0.20
Lead	Cast Iron	0.431)
Leather	Oak	0.61, 0521
Leather	Metal	0.4	0.2
Leather	Wood	0.3 - 0.4
Leather	Clean Metal	0.6
Leather fiber	Cast iron	0.31
Leather fiber	Aluminum	0.30
Magnesium	Magnesium	0.6	0.08
Masonry	Brick	0.6 - 0.7
Nickel	Nickel	0.7 - 1.1, 0.531)	0.28, 0.121)
Nickel	Mild Steel	0.641)	0.1781)
Nylon	Nylon	0.15 - 0.25
Oak	Oak (parallel grain)	0.62, 0.481)
Oak	Oak (cross grain)	0.54, 0.321	0.0721
Paper	Cast Iron	0.20
Phosphor-bronze	Steel	0.35
Platinum	Platinum	1.2	0.25
Plexiglas	Plexiglas	0.8	0.8
Plexiglas	Steel	0.4-0.5	0.4 - 0.5
Polystyrene	Polystyrene	0.5	0.5
Polystyrene	Steel	0.3-0.35	0.3 - 0.35
Polythene	Steel	0.2	0.2
Polystyrene	Polystyrene	0.5	0.5
Rubber	Rubber	1.16
Rubber	Cardboard	0.5 - 0.8
Rubber	Dry Asphalt	0.9  (0.5 - 0.8)1)
Rubber	Wet Asphalt	0.25 - 0.751)
Rubber	Dry Concrete	0.6 - 0.851)
Rubber	Wet Concrete	0.45 - 0.751)
Silver	Silver	1.4	0.55
Sapphire	Sapphire	0.2	0.2
Silver	Silver	1.4	0.55
Skin	Metals	0.8 - 1.0
Steel	Steel	0.5 - 0.8	0.16
Straw Fiber	Cast Iron	0.26
Straw Fiber	Aluminum	0.27
Tarred fiber	Cast Iron	0.15
Tarred fiber	Aluminum	0.18
Teflon	Teflon	0.04	0.04, 0.041)
Teflon	Steel	0.05 - 0.2
Tungsten Carbide	Steel	0.4-0.6	0.1 - 0.2
Tungsten Carbide	Tungsten Carbide	0.2 - 0.25	0.12
Tungsten Carbide	Copper	0.35
Tungsten Carbide	Iron	0.8
Tin	Cast Iron	0.321)
Tire, dry	Road, dry	1
Tire, wet	Road, wet	0.2
Wood	Clean Wood	0.25 - 0.5
Wood	Wet Wood	0.2
Wood	Clean Metal	0.2 - 0.6
Wood	Wet Metals	0.2
Wood	Stone	0.2 - 0.4
Wood	Concrete	0.62
Wood	Brick	0.6
Wood - waxed	Wet snow	0.14, 0.11)
Wood - waxed	Dry snow	0.041)
Zinc	Cast Iron	0.85, 0.211)
Zinc	Zinc	0.6	0.04

¹⁾ Kinetic or sliding frictional coefficient - holds only when there is a relative motion between the surfaces; otherwise they are somewhat higher

²⁾ Note! It is commonly thought that the static coefficients of friction are higher than the dynamic or kinetic values. This is a very simplistic statement and quite misleading for brake materials. With many brake materials the dynamic coefficient of friction quoted is an "average" value when the material is subject to a range of sliding speeds, surface pressures and most importantly operating temperatures. If the static situation is considered at the same pressure, but at ambient temperature, then the static coefficient of friction is often significantly LOWER than the average quoted dynamic value. It can be as low as 40 - 50% of the quoted dynamic value.

Kinetic versus Static Frictional Coefficients

Kinetic frictional coefficients are used with relative motion between objects. Static frictional coefficients are used for objects without relative motion. Static coefficients are somewhat higher than kinetic coefficients.

Example - Friction Force

The friction force of a 100 lb wooden crate pushed across a concrete floor with friction coefficient of 0.62 can be calculated as:

F_f = 0.62 (100 lb)

= 62 (lb)

• 1 lb = 0.4536 kg

Example - Car Stopping Distance

A car with mass 2000 kg drives with speed 100 km/h on a wet road with friction coefficient 0.2.

The friction work required to stop the car is equal to the kinetic energy of the car and can be calculated as

E_kinetic = 1/2 m v²      (3)
  = 1/2 (2000 kg) ((100 km/h) (1000 m/km) / (3600 s/h))²

  =  771605 J

The friction work can be expressed as

W_friction = F_f d   (4)

where

d = stopping distance (m)

(4) can be modified to

d = W_friction / μ m g

  = (771605 J) / (0.2 (2000 kg) (9.81 m/s²))

  = 197 m