Dry Friction - Theory & Concepts

Dry Friction

$Learning Objectives$

Distinguish required static friction from limiting static friction.
Predict the direction of friction from actual or impending relative motion.
Analyze inclined blocks, ladders, wedges, and belt systems.
Compare sliding and tipping thresholds for a rigid body.
Identify configurations in which equilibrium is impossible.

Friction forces on an inclined contact

The normal force is perpendicular to the surface, friction acts tangent to the contact and opposes impending relative motion, and weight remains vertical.

$Dry Friction$

Dry friction is the tangential contact reaction that opposes actual or impending relative motion between unlubricated solid surfaces.

Static and Kinetic Friction

Static friction adjusts to the equilibrium demand until its limiting value is reached; kinetic friction applies after sliding begins.

F_s \leq \mu_s N \qquad F_{s,\mathrm{max}}=\mu_s N \qquad F_k=\mu_k N

Variables

Symbol	Description	Unit
$F_s$	Actual static-friction force	N
$F_{s,\mathrm{max}}$	Limiting static-friction force at impending motion	N
$F_k$	Kinetic-friction force during sliding	N
$\mu_s$	Coefficient of static friction	unitless
$\mu_k$	Coefficient of kinetic friction	unitless
$N$	Normal contact reaction	N

$Do Not Assume Limiting Friction Prematurely$

For a body that remains at rest, solve the equilibrium equations for the friction demand first. Use $F_s=\mu_sN$ only when motion is impending or when testing a possible impending-motion mode.

$Friction Direction$

Friction opposes relative motion at each contact interface. Reversing the assumed impending motion reverses the friction direction, and an incorrect direction can make an otherwise correct equilibrium calculation invalid.

$Worked Example Summary$

A block of weight $100\ \text{N}$ rests on a $20^\circ$ incline with $\mu_s=0.50$ . The required friction is $100\sin20^\circ=34.2\ \text{N}$ , while the limiting value is $0.50(100\cos20^\circ)=47.0\ \text{N}$ . Because the demand is below the limit, the block remains in static equilibrium and the actual friction force is $34.2\ \text{N}$ , not $47.0\ \text{N}$ .

$Simulation 1 Instructions$

Adjust the incline and friction coefficients. Observe when the actual static friction reaches its limiting value and changes to kinetic friction.

Simulation

120 N

30 deg

0.45

0.30

sliding — kinetic friction active

Downslope demand

60.000 N

Normal reaction

103.923 N

Static capacity μsN

46.765 N

Active friction

31.177 N

Critical angle

24.228°

F_s\le\mu_sN,\quad F_k=\mu_kN\;\text{ only after sliding}

$Simulation 1 Concept Question$

Why does the actual static-friction force equal the downslope demand before impending motion?

$Simulation 2 Instructions$

Change the ladder angle and the friction coefficients at the floor and wall. The rough-floor/rough-wall ladder has one static reaction degree of freedom, so inspect the admissible reaction range rather than assuming both contacts are simultaneously at $F=\mu N$ .

Simulation

120 N

30 deg

0.45

0.30

no admissible static reaction set

Static equilibrium

not admissible

Required Nw minimum

68.388 N

Allowed Nw maximum

36.477 N

|F_f|\le\mu_fN_f,\quad |F_w|\le\mu_wN_w

$Simulation 2 Concept Question$

Which contact interface reaches its friction limit first as the ladder becomes flatter?

$Simulation 3 Instructions$

Explore impending rightward wedge insertion with two rough interfaces. The friction arrows oppose that assumed relative motion, and the limiting model combines the wedge angle with the two interface friction angles.

Simulation

120 N

30 deg

0.45

0.30

impending-motion friction-angle model

Required input

347.068 N

Mechanical advantage

0.346

Top friction angle

24.228°

Bottom friction angle

16.699°

Effective angle

70.927°

P=W\tan(\alpha+\phi_1+\phi_2),\quad \phi_i=\tan^{-1}\mu_i

$Simulation 3 Concept Question$

Why can a small increase in either interface friction produce a large increase in required input force?

Belt Friction

Limiting belt tensions for impending slip over a rough cylindrical surface.

\frac{T_2}{T_1}=e^{\mu\beta}

Variables

Symbol	Description	Unit
$T_2$	Tight-side tension	N
$T_1$	Slack-side tension	N
$\mu$	Belt-to-drum friction coefficient	unitless
$\beta$	Wrap angle measured in radians	rad

$Simulation 4 Instructions$

Change the coefficient of friction, wrap angle, slack-side tension, and pulley scale to inspect tension ratio and torque capacity.

Simulation

100 N

0.45

180 deg

0.30 m

limiting capstan tension relation

Tight-side tension

411.121 N

Tension ratio T2/T1

4.111

Wrap β

3.1416 rad

Torque capacity

93.336 N·m

\frac{T_2}{T_1}=e^{\mu\beta},\quad \beta=\theta_{\mathrm{deg}}\pi/180

$Simulation 4 Concept Question$

Why does the tension ratio grow exponentially rather than linearly with wrap angle?

Sliding and Tipping Thresholds

Threshold comparison for a rectangular block under a horizontal force applied at height h.

P_{\mathrm{slide}}=\mu_sW \qquad P_{\mathrm{tip}}=\frac{Wb}{2h}

Variables

Symbol	Description	Unit
$P_{\mathrm{slide}}$	Horizontal force that produces impending sliding	N
$P_{\mathrm{tip}}$	Horizontal force that produces impending tipping	N
$W$	Block weight	N
$b$	Base width	m
$h$	Load application height	m

$Simulation 5 Instructions$

Change the applied horizontal force as well as the block width, load height, and static-friction coefficient. Compare the current demand with the separate sliding and tipping thresholds; the smaller threshold identifies the first possible motion.

Simulation

120 N

3.0 m

2.0 m

60 N

0.45

sliding first

Applied force

60 N

Sliding threshold

54.000 N

Tipping threshold

90.000 N

First-motion mode

sliding

Base reaction xR

+1.000 m

Contact state

partial-contact

P_{\mathrm{slide}}=\mu_sW,\quad P_{\mathrm{tip}}=\frac{WB}{2h}

$Simulation 5 Concept Question$

How can increasing friction make tipping govern even though the applied force is unchanged?

$General Friction Analysis Procedure$

STEP-BY-STEP

Draw a separate free-body diagram for every body.
Predict actual or impending relative motion at every contact.
Direct each friction force opposite that relative motion.
Solve equilibrium for the required contact reactions.
Compare every static-friction demand with $\mu_sN$ .
If a demand exceeds its capacity, revise the assumed state to sliding or declare equilibrium impossible.

Friction-State Decision Workflow

Solve the required static-friction demand first, enforce unilateral contact, compare competing limit states, and distinguish static, impending, and sliding behavior.

Draw a separate FBD for each body and contact → Predict possible relative or impending motion at each contact; Predict possible relative or impending motion at each contact → Solve equilibrium for required static friction and normal reactions; Solve equilibrium for required static friction and normal reactions → All assumed contact normals satisfy N ≥ 0?; All assumed contact normals satisfy N ≥ 0? — No → Contact loss or uplift: revise the contact model; All assumed contact normals satisfy N ≥ 0? — Yes → If load varies, compute candidate sliding, tipping, and uplift thresholds; Contact loss or uplift: revise the contact model → Contact state identified; Does tipping, uplift, or another limit state occur before sliding? — Yes → Use the earlier governing non-sliding limit state; Does tipping, uplift, or another limit state occur before sliding? — No → |Frequired| < μs N?; Use the earlier governing non-sliding limit state → Contact state identified; |Frequired| < μs N? — Yes → Static contact admissible: Fs = Frequired; |Frequired| < μs N? — No → |Frequired| = μs N within tolerance?; Static contact admissible: Fs = Frequired → Contact state identified; |Frequired| = μs N within tolerance? — Yes → Impending sliding: Fs = μs N; |Frequired| = μs N within tolerance? — No → No static equilibrium; if sliding occurs use Fk = μk N and dynamics as needed; Impending sliding: Fs = μs N → Contact state identified; No static equilibrium; if sliding occurs use Fk = μk N and dynamics as needed → Contact state identified; If load varies, compute candidate sliding, tipping, and uplift thresholds → Does tipping, uplift, or another limit state occur before sliding?

Draw a separate FBD for each body and contact: terminator
Predict possible relative or impending motion at each contact: process
Solve equilibrium for required static friction and normal reactions: process
All assumed contact normals satisfy N ≥ 0?: decision
Contact loss or uplift: revise the contact model: process
If load varies, compute candidate sliding, tipping, and uplift thresholds: process
Does tipping, uplift, or another limit state occur before sliding?: decision
Use the earlier governing non-sliding limit state: process
|Frequired| < μs N?: decision
Static contact admissible: Fs = Frequired: process
|Frequired| = μs N within tolerance?: decision
Impending sliding: Fs = μs N: process
No static equilibrium; if sliding occurs use Fk = μk N and dynamics as needed: process
Contact state identified: terminator

Key Takeaways

Static friction is a bounded reaction, not automatically $\mu_sN$ .
Limiting static friction applies only at impending motion.
Multiple-contact problems require a consistent motion assumption at every interface.
Belt friction depends on wrap angle in radians.
Sliding and tipping are different limiting states, and the lower threshold governs first motion.

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