5D Library › Classical and Statistical Mechanics of Human Action · 中文
Chapter 2 Static and Kinetic Friction
Two coefficients of friction
Pushing a box across the floor takes a large force at first; once the box moves, less force keeps it going. The Amontons–Coulomb law of friction describes this with two coefficients. While the box is at rest, friction can take any value from zero up to the maximum static friction, exactly cancelling the push; the maximum is
Once the push exceeds it, the box starts to slide, and friction becomes kinetic friction
where is the normal force. For most dry surfaces, .
Evidence
Why is static friction larger than kinetic friction? The classic work of Bowden and Tabor showed that two surfaces which look flat touch only at tiny bumps (asperities), so the real area of contact is often a very small fraction of the apparent area, and friction comes mainly from shearing the adhesion at these junctions (Bowden and Tabor 1950). Later experiments found that while surfaces rest in contact, the junctions slowly creep, grow and strengthen, so the longer the rest, the larger the force needed to start; once sliding, junctions are constantly renewed and have no time to grow, so friction is lower. This “contact aging” grows roughly with the logarithm of the time at rest and has been measured in rock and many other materials (Dieterich 1972; Rabinowicz 1951; Baumberger and Caroli 2006; Marone 1998).
Stick-slip motion
When , a very common thing happens: stick-slip. Pull a block with a spring whose far end moves at constant speed. The spring stretches, the pull grows, and the block does not move; once the pull exceeds the maximum static friction, the block lurches forward, the spring relaxes, and the block stops and sticks again, over and over. Chalk squealing on a blackboard, a creaking hinge, a violin bow on a string and the periodic slip of earthquake faults are all related to stick-slip (Baumberger and Caroli 2006).
Calculation
Spring–block model: mass 1 kg, spring constant 50 N/m, the spring’s far end moving at 5 cm/s, with a little viscous damping while sliding. Compare above with the two equal.
# Stick-slip: a block on a table, pulled through a spring whose far end moves at constant speed V.
# Coulomb friction: the block stays stuck while the spring force is below mu_s*N; once it slides,
# friction drops to mu_k*N. If mu_s > mu_k the block lurches forward and sticks again.
M, G, K, V = 1.0, 9.81, 50.0, 0.05 # kg, m/s^2, N/m, m/s
C = 1.0 # N s/m, viscous damping of the sliding contact
N = M * G
DT, T_END = 1e-4, 20.0
def run(mu_s, mu_k):
x = v = 0.0
stuck, slips, t = True, 0, 0.0
peak, forces = 0.0, []
while t < T_END:
f_spring = K * (V * t - x)
if stuck:
if f_spring > mu_s * N:
stuck, slips = False, slips + 1
else:
a = (f_spring - mu_k * N - C * v) / M
v_new = v + a * DT
if v_new <= 0.0: # the block comes to rest and sticks again
v, stuck = 0.0, True
else:
v = v_new
x += v * DT
peak = max(peak, f_spring)
if t > T_END / 2:
forces.append(f_spring)
t += DT
swing = max(forces) - min(forces)
return slips, peak, swing, x / T_END
for mu_s, mu_k in ((0.60, 0.40), (0.50, 0.40), (0.40, 0.40)):
slips, peak, swing, mean_v = run(mu_s, mu_k)
print(f"mu_s = {mu_s:.2f}, mu_k = {mu_k:.2f}: {slips:3d} starts from rest in {T_END:.0f} s, peak force {peak:4.2f} N "
f"(kinetic {mu_k*N:4.2f} N), late force swing {swing:4.2f} N, mean speed {1000*mean_v:4.1f} mm/s")
print(f"threshold to start moving: mu_s*N = {0.60*N:.2f} N; force needed to keep it moving: mu_k*N = {0.40*N:.2f} N")
# mu_s = 0.60, mu_k = 0.40: 10 starts from rest in 20 s, peak force 5.92 N (kinetic 3.92 N), late force swing 3.50 N, mean speed 46.9 mm/s
# mu_s = 0.50, mu_k = 0.40: 15 starts from rest in 20 s, peak force 4.97 N (kinetic 3.92 N), late force swing 1.79 N, mean speed 45.7 mm/s
# mu_s = 0.40, mu_k = 0.40: 1 starts from rest in 20 s, peak force 4.29 N (kinetic 3.92 N), late force swing 0.01 N, mean speed 46.0 mm/s
# threshold to start moving: mu_s*N = 5.89 N; force needed to keep it moving: mu_k*N = 3.92 N
In all three cases the block’s average speed is almost the same, about the speed of the spring’s far end, but the motion differs completely. With static friction above kinetic friction, the block started ten times in twenty seconds, each time building the pull up to the maximum static friction before it dropped; with the two coefficients equal, it started once and then slid steadily. Stick-slip needs more than : it also depends on the driving speed, the spring stiffness and the damping, and faster driving or more damping can make it disappear.
The longer at rest, the harder to start?
Calculation
The quantitative form of contact aging (Dieterich’s 1972 rock experiments and the later rate-and-state friction law):
with values of the order typical for rock and similar materials, , , s.
# Static friction grows with the time two surfaces have been at rest in contact ("contact aging"):
# the microscopic contact area creeps and grows roughly with the logarithm of hold time.
# mu_s(t) = mu_0 + b * ln(1 + t / theta) (Dieterich 1972; rate-and-state friction)
# Values of the order measured for rock and many dry materials (Marone 1998): mu_0 = 0.6, b = 0.01, theta = 1 s.
import math
MU0, B, THETA = 0.60, 0.010, 1.0
print("hold time mu_s increase over 1 s hold")
for label, t in (("1 second", 1), ("1 minute", 60), ("1 hour", 3600), ("1 day", 86400),
("10 days", 10 * 86400)):
mu = MU0 + B * math.log(1 + t / THETA)
print(f"{label:16s} {mu:5.3f} +{100 * (mu - (MU0 + B * math.log(2))) / (MU0 + B * math.log(2)):4.1f} %")
print(f"each tenfold longer hold adds about {B * math.log(10):.3f} to mu_s; laboratory tests span seconds to days")
# hold time mu_s increase over 1 s hold
# 1 second 0.607 + 0.0 %
# 1 minute 0.641 + 5.6 %
# 1 hour 0.682 +12.4 %
# 1 day 0.714 +17.6 %
# 10 days 0.737 +21.4 %
# each tenfold longer hold adds about 0.023 to mu_s; laboratory tests span seconds to daysContact aging is a real physical effect, but it grows slowly: each tenfold increase in the time at rest adds only about 0.023 to the static coefficient. From one second to ten days, the increase is about a fifth. As an analogy for “the longer you put it off, the harder it is to start”, the direction is right, but physics itself gives no conclusion such as “a month’s delay makes it ten times harder”.
Hypothesis · not established
Not established. The difficulty of restarting a task that has been left aside resembles contact aging in form: it grows with the logarithm of the time it has been left, not linearly. The testable prediction: among people who keep long-term work logs, the time from sitting down to the first useful action (the “start-up friction” of Chapter 8) rises roughly linearly with the logarithm of the days since the task was last touched. If start-up time is unrelated to the days elapsed, or rises linearly or faster with them, the analogy fails.
Why a wheel needs friction
Friction is often thought of as a force that hinders motion, yet almost all “going forward” on the ground depends on it. As a wheel turns, the tyre’s contact patch is momentarily at rest relative to the road; the tyre pushes the road backwards, and the road pushes the car forwards through static friction. That push has a ceiling:
Whatever the engine demands above the ceiling only spins the wheel (Wong 2008). Walking works the same way: the foot pushes the ground backwards and the ground, through friction, pushes the walker forwards; on ice there is too little friction to accelerate, or even to stay upright. Normal walking needs a friction coefficient of about 0.2 (Redfern et al. 2001).
Calculation
A 1-tonne car with half its weight on the driven wheels, whose engine supplies 1000 N·m of torque at those wheels (asking the contact patch for about 3333 N of push). Compare four surfaces.
# Why a wheel (or a foot) needs friction to move forward.
# The ground can push a vehicle forward only through the friction force at the contact patch,
# and that force is capped: F_drive <= mu * N. Whatever the motor demands above the cap
# goes into spinning the wheel, not into moving the car.
M, G = 1000.0, 9.81 # car mass kg
R = 0.3 # wheel radius m
TORQUE = 1000.0 # N m at the driven wheels
N = M * G * 0.5 # half the weight rests on the driven wheels
demand = TORQUE / R # force the tyres would pass to the road if they gripped
print(f"force the motor asks the tyres to transmit: {demand:.0f} N")
for surface, mu in (("frictionless (ideal)", 0.0), ("wet ice", 0.05), ("packed snow", 0.3),
("dry asphalt", 0.8)):
cap = mu * N
f = min(demand, cap)
state = "grips" if demand <= cap else "wheel spins"
print(f"{surface:21s} mu = {mu:4.2f}: cap {cap:5.0f} N -> forward force {f:5.0f} N, "
f"acceleration {f / M:4.2f} m/s^2, speed after 5 s {5 * f / M:5.1f} m/s ({state})")
print("doubling the motor torque changes nothing on ice: the cap, not the motor, sets the push")
print(f"walking: the largest forward acceleration friction allows is mu*g = {0.2 * G:.1f} m/s^2 at mu = 0.2")
# force the motor asks the tyres to transmit: 3333 N
# frictionless (ideal) mu = 0.00: cap 0 N -> forward force 0 N, acceleration 0.00 m/s^2, speed after 5 s 0.0 m/s (wheel spins)
# wet ice mu = 0.05: cap 245 N -> forward force 245 N, acceleration 0.25 m/s^2, speed after 5 s 1.2 m/s (wheel spins)
# packed snow mu = 0.30: cap 1472 N -> forward force 1472 N, acceleration 1.47 m/s^2, speed after 5 s 7.4 m/s (wheel spins)
# dry asphalt mu = 0.80: cap 3924 N -> forward force 3333 N, acceleration 3.33 m/s^2, speed after 5 s 16.7 m/s (grips)
# doubling the motor torque changes nothing on ice: the cap, not the motor, sets the push
# walking: the largest forward acceleration friction allows is mu*g = 2.0 m/s^2 at mu = 0.2With zero friction, any engine only spins the wheels and the car does not move; on wet ice the push is capped at 245 N, however strong the engine; only on dry asphalt is the ceiling above what the engine asks, so the tyres grip and all of the engine’s force becomes forward force.
One thing to state plainly: grip is not resistance
The idea that first prompted this chapter was: “if the friction coefficient of the environment is zero — perfect comfort, a frictionless easy life — any inner drive can only spin in place, and a moderate amount of outside resistance and friction is exactly what makes forward propulsion possible.” On checking, this needs correcting.
Note
In physics, the static friction between wheel and road is the propulsive force itself: its role is to give the wheel something to push against. Resistance — air drag, rolling resistance, the downhill pull of gravity on a slope — is a different kind of force; it only uses up the propulsive force and never helps forward motion. Adding resistance does not make a car go faster. Moreover, in the Coulomb model a larger friction coefficient always raises the ceiling on propulsion; there is no “moderate” optimum, and dry asphalt always beats wet ice. What physics supports is “moving forward needs something to push against”, not “moving forward needs resistance”, still less “moderate resistance”.
Applied to people, the closer statement is this: someone who has the drive also needs outside structures to push against — a firm deadline, people waiting to see the result, feedback that turns effort into visible progress — or the effort goes out and nothing turns it into forward motion. These structures are not “resistance” but “grip”. They are different from difficulty, adversity or setbacks, which behave more like air drag and hills: they use up effort and do not, in themselves, push anyone forward.
Hypothesis · not established
Not established. Among learners with the same willingness and the same time invested, those with clear outside “grip points” (showing their work to others on a fixed date, specific feedback after each practice) make more measurable progress per hour; adding difficulty on top of this (extra obstacles or penalties) does not raise progress further and may lower it. If randomised studies find no measurable difference in progress from adding grip points, or find that extra difficulty systematically raises progress, the hypothesis is refuted.
Summary
The gap between static and kinetic friction gives “the first step is the hardest” an exact physical counterpart: starting takes more force than continuing, and when the two differ a lot, motion takes the fitful form of stick-slip. Contact aging adds that the longer at rest, the harder to start, though the growth is slow. The wheel shows that moving forward needs something to push against; on checking, this cannot be stretched into “resistance is good for you”. Chapter 3 turns to statistical mechanics: when a system is trapped in one stable state, how long do random disturbances take to carry it to another?
References
Fan, Y. (2026). Classical and Statistical Mechanics of Human Action: Psychological Inertia, Static Friction Thresholds, Kramers Escape Rate in Double-Well Potentials and Habit Phase Transitions, Read Beside East Asian Philosophy of Practice [行动的经典与统计力学] (Version 1.0.0). 5D Library. https://doi.org/10.5281/zenodo.23738774 · Licensed under Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0).