Maximum Static Friction

Equivalent forms

f_s \le \mu_s N

An inequality, not an equation — friction is whatever it needs to be, until it can't.

Symbol	Name	SI unit	Dimension	Range
$F_s_max$	Max static frictionoutput Threshold force before slipping	N	MLT^-2	0 – 1000
$mu_s$	Coefficient of static friction Dimensionless surface property		1	0.01 – 1.5
$N$	Normal force Force perpendicular to surface	N	MLT^-2	1 – 2000

Unit systems

Symbol	SI	CGS	Imperial
$F_s_max$	N	dyn	lbf
$mu_s$	—	—	—
$N$	N	dyn	lbf

Where it holds

Dry Amontons–Coulomb regime; not applicable to viscous, lubricated, or extreme micro/nano contacts.

Dimensional analysis

F_s_\max \to [MLT^{-2}]

\mu _s\cdot N \to [1]\cdot [MLT^{-2}] = [MLT^{-2}]

Newtons on both sides.

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Discovery

Charles-Augustin de Coulomb · 1785

Coulomb's friction memoir distinguished static and kinetic regimes and tabulated μ_s for many material pairs — a foundation of tribology.

Try this

How hard can you push a heavy fridge before it finally starts to slide?

A 80 kg fridge sits on linoleum, μ_s = 0.5. Find the push force at which it just begins to move.

Research status: stable

Real-world applications

Why anti-lock brakes work (keep tires in static regime)
Climbing shoe rubber design
Robotic gripper force planning
Earthquake fault stick-slip dynamics

Common misconceptions

Static friction always equals $\mu _s N$ — it can be anywhere from $0\,\mathrm{to} \mu _s N$
$\mu _s = \mu _k$ always — usually $\mu _s$ is 10–50% larger
Heavier objects always have more friction — only because N grows

Experimental verification

Block-on-incline tipping tests: tilt the surface until the block slips.

\tan (\theta

_slip) equals

\mu _s

directly.

Derivation

From real-area-of-contact theory, microscopic asperities form cold welds that must shear before sliding.

The shear force scales with real contact area, which in turn scales with normal load, giving

F_s_\max \propto N

with proportionality

\mu _s

.

Limiting cases

Push < F_s_max⟶ No motionStatic friction matches applied force exactly.

Push

= F_s_\max

⟶ Object on the vergeImpending motion: any small increase causes sliding.

Push > F_s_max⟶ Sliding startsFriction drops to kinetic value

\mu _k N

; net force accelerates object.

What if…

What if I push harder than F_s_max?

The block accelerates; friction drops to $F_k = \mu _k N$ which is smaller.

What

if \mu _s = \mu _k

?

Idealization sometimes used in textbooks — no 'jolt' when motion begins.

What if N goes to zero?

Object floats free; both static and kinetic friction vanish.

1

Fridge on linoleum

Given ·

mu s:: 0.5
N:: 784

Find · F_s_max

Steps

$N = mg = 80 \times 9.8 = 784\,\mathrm{N}$
$F_s_\max = \mu _s N = 0.5 \times 784$
$F_s_\max = 392\,\mathrm{N}$ — any push above this slips the fridge

Result ·

F_s_\max = 0.5 \times 784 = 392\,\mathrm{N}

2

Tilting incline test

Given ·

mu s:: 0.3
N:: 100

Find · F_s_max

Steps

Block tips $at \theta$ such that $\tan (\theta ) = \mu _s = 0.3$
$\theta \approx 16.7^{\circ}$
On flat ground with $N = 100\,\mathrm{N}$ : $F_s_\max = 30\,\mathrm{N}$

Result ·

F_s_\max = 30\,\mathrm{N}

Kinetic Friction Force→Newton's Second Law→