Dry friction and impending motion
Friction is the force that resists sliding between surfaces. Engineers rely on it to hold things in place and fight it to make things move efficiently.
Origins of the laws
Guillaume Amontons stated basic laws of friction in 1699, and Charles-Augustin de Coulomb refined them in 1781. Coulomb's model of dry friction says friction is proportional to the normal force and roughly independent of contact area and, for sliding, of speed. Leonardo da Vinci had noted similar ideas around 1500 in notebooks published much later.
Static friction
While surfaces do not slide, static friction adjusts to whatever is needed for equilibrium, up to a maximum: \(F \le \mu_s N\). At impending motion, the point just before slipping, friction reaches its maximum \(F = \mu_s N\).
Kinetic friction
Once sliding begins, kinetic friction is approximately \(F = \mu_k N\), usually somewhat less than the maximum static friction. That is why it takes more force to start a heavy crate moving than to keep it moving.
Typical coefficients
Coefficients depend on the materials and surface conditions: roughly 0.6 to 0.7 for wood on wood, 0.7 to 1.0 for rubber on dry concrete, about 0.15 for lubricated steel, and around 0.05 for PTFE (Teflon) on steel. Measured values vary, so designs use conservative estimates.
Three kinds of problem
In some problems, impending motion is given, so set F = mu_s N. In others, you must check: assume equilibrium, find the friction force needed, and compare it with mu_s N; if it exceeds the limit, the body slides. In a third type, motion is possible in several ways, and each must be tested.
The angle of friction
The friction angle \(\phi_s\) satisfies \(\tan\phi_s = \mu_s\). A block on an incline is on the verge of sliding when the incline angle equals \(\phi_s\). With \(\mu_s = 0.5\), that is \(\tan^{-1} 0.5 \approx 26.6°\); this is the basis of the 'angle of repose' of piled sand and gravel.
Is it sliding?
A 50 kg crate rests on a floor with mu_s = 0.4, and a worker pushes horizontally with 150 N. The maximum static friction is 0.4 x 490.5, about 196 N. Since the push is less, the crate stays put, and the actual friction force is exactly 150 N, not 196 N.
Worked example
A 200 kg crate is to be pushed up a 15-degree ramp with mu_s = 0.3, the push parallel to the ramp. At impending motion up the slope, the push P must overcome the weight component and maximum friction: P = W sin 15 + 0.3 W cos 15 = 1,962 x (0.259 + 0.290), about 1,077 N. Without friction it would be only about 508 N.
Key takeaways
Amontons (1699) and Coulomb (1781) described dry friction: proportional to the normal force and roughly independent of contact area. Static friction adjusts up to mu_s N, reached at impending motion; kinetic friction is about mu_k N and usually smaller. Coefficients range from about 0.05 for PTFE to around 1 for rubber on concrete. Check whether bodies slide by comparing required friction with mu_s N. A block on an incline slips when tan theta exceeds mu_s, the friction angle.
Flashcard
Resists sliding between surfaces
Check your understanding
A 50 kg crate (mu_s = 0.4) is pushed horizontally with 150 N. What happens?