Dry Friction

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.

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.

Advanced engineering statics simulation

Statics Friction Simulation Suite

Five distinct dry-friction models with explicit assumptions and scenario-specific free-body diagrams.

Simulation

Compare the equilibrium friction demand with the static limit, then use kinetic friction only after sliding begins.

Weight120 N

20–500

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Plane angle30 deg

deg

5–80

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Static friction coefficient0.45

0.05–1.00

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Kinetic friction coefficient0.30

0.02–0.90

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Calculated statesliding

Normal force

103.92 N

Required friction

60.00 N

Capacity 46.77 N

Active friction

31.18 N

Angle of repose

24.23°

F_s\leq\mu_sN,\qquad F_k=\mu_kN

Static friction matches equilibrium demand until its limiting value is reached.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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. Check whether both contact reactions can satisfy force and moment equilibrium.

Advanced engineering statics simulation

Statics Friction Simulation Suite

Five distinct dry-friction models with explicit assumptions and scenario-specific free-body diagrams.

Simulation

Solve a limiting-equilibrium ladder case with upward wall friction fully mobilized, then check the floor-friction capacity.

Weight120 N

20–500

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Ladder angle above floor30 deg

deg

5–80

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Floor friction coefficient0.45

0.05–1.00

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Wall coefficient (fully mobilized)0.30

0.02–0.90

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Calculated statesliding

Floor normal

99.48 N

Wall normal

68.39 N

Wall friction

20.52 N

Assumed at μwNw upward

Moment residual

0.00e+0

F_s\leq\mu_sN,\qquad F_k=\mu_kN

The rough-wall ladder is closed by assuming upward wall friction is fully mobilized; the floor demand is then checked against its capacity.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

Simulation 2 Concept Question

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

Simulation 3 Instructions

Explore a wedge with two rough interfaces. Compare the wedge angle with the combined friction-angle effect.

Advanced engineering statics simulation

Statics Friction Simulation Suite

Five distinct dry-friction models with explicit assumptions and scenario-specific free-body diagrams.

Simulation

Use the standard symmetric impending-motion approximation with two friction angles.

Supported load120 N

20–500

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Wedge angle30 deg

deg

5–80

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Top-interface coefficient0.45

0.05–1.00

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Bottom-interface coefficient0.30

0.02–0.90

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Calculated stateimpending

Required input

347.07 N

Mechanical advantage

0.346

Effective angle

70.93°

Model

symmetric

Impending upward motion

P\approx W\tan(\theta+\phi_1+\phi_2)

This is a clearly labeled symmetric rough-interface approximation, not a universal wedge formula for every contact geometry.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.

Advanced engineering statics simulation

Statics Friction Simulation Suite

Five distinct dry-friction models with explicit assumptions and scenario-specific free-body diagrams.

Simulation

Apply the capstan relation with wrap angle in radians and calculate torque capacity.

Belt friction coefficient0.45

0.05–1.00

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Slack-side tension100 N

10–500

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Wrap angle180 deg

deg

10–360

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Pulley radius0.30 m

0.05–1.00

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Calculated stateimpending

Tight tension

411.12 N

Tension ratio

4.111

Torque capacity

93.34 N·m

Wrap angle

180°

3.142 rad

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

Static friction matches equilibrium demand until its limiting value is reached.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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

Compare the sliding and tipping thresholds. The smaller threshold identifies the first possible motion.

Advanced engineering statics simulation

Statics Friction Simulation Suite

Five distinct dry-friction models with explicit assumptions and scenario-specific free-body diagrams.

Simulation

Compare the horizontal force required for sliding with the force required for first tipping about the base edge.

Weight120 N

20–500

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Static friction coefficient0.45

0.05–1.00

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Base width3.0 m

0.5–5.0

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Horizontal-load height2.0 m

0.2–5.0

Drag for exploration or enter an exact value. Press Enter to apply and Escape to restore.

Calculated statesliding

Sliding threshold

54.00 N

Tipping threshold

90.00 N

First motion

sliding

Governing force

54.00 N

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

Static friction matches equilibrium demand until its limiting value is reached.

Model scope and verification

Scope: Educational rigid-body statics model using the selected geometry, stated idealizations, and displayed SI units.

Acceptance check: Check the governing equilibrium, compatibility, geometry, or limiting-condition statement before accepting the numerical result.

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.

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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