Static and kinetic friction
Friction has two regimes, and the static one has a maximum rather than a fixed value.
Static vs kinetic
Static friction acts when surfaces do not slide; it adjusts itself to exactly cancel the applied push, up to a maximum of mu_s N. Once sliding starts, kinetic friction takes over with a fixed magnitude mu_k N opposing the sliding direction. Because mu_s exceeds mu_k, it takes more force to start a slide than to keep one going, which is why heavy boxes lurch.
f_s \le \mu_s N, \quad f_k = \mu_k NThe coefficient and N
The coefficient of friction is dimensionless and depends only on the pair of surfaces, not on the size of the forces. The friction formula uses the normal force, not the weight, so on an incline friction is mu times mg cos(theta), weaker than on level ground. On a level surface at rest, N equals mg.
f_k = \mu_k mg\cos\thetaWorked example
A 5.0 kg block slides down a 37 degree incline with mu_k = 0.25, using sin(37) = 0.60 and cos(37) = 0.80. Gravity pulls along the slope with 49(0.60) = 29.4 N. The normal force is 49(0.80) = 39.2 N, so kinetic friction is 0.25 times 39.2 = 9.8 N up the slope. The net 19.6 N along the slope gives a = 19.6/5.0 = 3.92 m/s^2 down the incline.
a = \frac{mg\sin\theta - \mu_k mg\cos\theta}{m} = \frac{29.4 - 9.8}{5.0} = 3.92 \text{ m/s}^2Flashcard
Static friction can be anything up to a maximum
Check your understanding
A 20 kg crate sits on a level floor with mu_s = 0.40 and mu_k = 0.30. What horizontal force keeps it sliding at constant speed?