Unit 3 · Topic 05

Static and kinetic friction

Friction is a tangential contact force. Introductory friction models are empirical and approximate: they capture useful regularities without modelling all microscopic details of real surfaces.

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Concept 01

Direction of friction

Essential The minimum you should retain

Friction opposes relative sliding between surfaces or the tendency to slide that would occur without friction.

UnderstandInterpret and connect

It does not always point opposite the object's motion relative to the ground. When walking without slipping, for example, static friction from the ground on the foot can point forward.

DeepenFormulation and conditions

Determine the direction by first imagining the relative motion at the contact if friction vanished. Friction acts to oppose that relative motion or tendency.

ExploreConnections for further study

For wheels, belts, or stacked bodies, friction direction can be unintuitive. Therefore it should not be assigned from a memorized rule such as «always opposite the centre-of-mass velocity.»

Concept 02

Static friction

Essential The minimum you should retain

Static friction acts when the surfaces do not slide relative to one another. Its magnitude adjusts to what is needed to prevent sliding, up to a maximum.

UnderstandInterpret and connect

The model is 0 ≤ f_s ≤ μ_s N. Equality f_s = μ_sN applies only at impending slip, not in every static situation.

DeepenFormulation and conditions

To test whether rest is possible, first calculate the friction required by equilibrium. Then check whether |f_s,required| ≤ μ_sN. If not, the assumed static regime is inconsistent.

ExploreConnections for further study

The coefficient μ_s summarizes properties of a surface pair under specified conditions. It is not a fundamental constant and may vary with contamination, temperature, surface preparation, and other factors.

Worked example

Static friction is not always μ_sN

A 10 kg block rests on a horizontal floor with μ_s = 0.50 while a 20 N horizontal force is applied.

Given
  • m = 10 kg
  • μ_s = 0.50
  • F_app = 20 N
  • g = 9.8 m/s²
Target

Determine whether it remains at rest and find the actual static friction.

  1. Normal force

    With no vertical acceleration, N = mg = 98 N.

  2. Available maximum

    f_s,max = μ_sN = 49 N.

  3. Required value

    Equilibrium requires only 20 N of friction opposite the push.

  4. Check

    Because 20 ≤ 49, the block remains at rest and f_s = 20 N.

Conclusion

The actual static friction is 20 N, not 49 N. 49 N is only the available maximum.

Mathematical relation

Static-friction range

|fs|≤μsN |f_s|\leq\mu_sN
Represents

The range of values static friction can adopt before sliding.

Physical interpretation

Actual friction adjusts up to a maximum; equality holds only at the threshold.

DeepenVariables, conditions, and checks

Variables

f_s
required static friction; usual unit: N
μ_s
static-friction coefficient; usual unit: dimensionless
N
normal force; usual unit: N

Conditions of application

  • There is no relative sliding.
  • The dry Coulomb-friction model is used.

Dimensional check

Both sides have unit N.

Errors it helps prevent

  • Always setting f_s=μ_sN.

Static friction adjusts up to f_s,max

Required static friction versus applied force, ending at the static maximum.
  • f_s=F_app up to f_s,max

The f_s=F_app branch ends at the threshold. Beyond it, the graph no longer represents a static state.

Concept 03

Kinetic friction

Essential The minimum you should retain

When surfaces slide, the introductory model uses kinetic friction with approximate magnitude fk=μkN f_k=\mu_kN , opposite the relative sliding velocity.

UnderstandInterpret and connect

Kinetic friction is not chosen to balance other forces. Within the model its magnitude follows from μ_k and N; the resulting net force then determines acceleration.

DeepenFormulation and conditions

For many dry surface pairs, μ_k is approximately smaller than μ_s, but this is not a fundamental law. The simple model also neglects dependence on speed, temperature, and microscopic contact conditions.

ExploreConnections for further study

When Coulomb friction is inadequate, a more specific force law is needed. Recognizing a model's limit is better than forcing μ_kN onto every situation.

Mathematical relation

Kinetic-friction model

fk=μkN f_k=\mu_kN
Represents

The approximate friction magnitude while surfaces slide.

Physical interpretation

Its direction opposes relative sliding velocity.

DeepenVariables, conditions, and checks

Variables

f_k
kinetic-friction magnitude; usual unit: N
μ_k
kinetic coefficient; usual unit: dimensionless
N
normal force; usual unit: N

Conditions of application

  • The surfaces are sliding.
  • The introductory dry-friction approximation is used.

Dimensional check

N on both sides.

Errors it helps prevent

  • Opposing ground velocity.
  • Adjusting it like static friction.

Example transition between regimes

Two branches compare static friction up to the threshold and an approximate kinetic level after sliding.
  • required static
  • illustrative kinetic

The static line follows the required force to its maximum; after sliding, the illustrative kinetic level is lower. This comparison does not claim a universal μ_k<μ_s law.

Concept 04

Rolling resistance and model limits

Essential The minimum you should retain

Rolling without slipping does not mean that friction is absent. Static friction may act at the contact, and real systems can also lose energy through deformation.

UnderstandInterpret and connect

Rolling resistance is not the same as kinetic sliding friction. A rolling tyre may have nearly static instantaneous contact while still dissipating energy through deformation of tyre and road.

DeepenFormulation and conditions

This unit does not introduce a universal quantitative law for rolling resistance. The distinction is conceptual so that μ_kN is not used when there is no sliding.

ExploreConnections for further study

Friction models are phenomenological. Choosing among static friction, kinetic friction, rolling resistance, or fluid drag depends on the dominant physical mechanism and required accuracy.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Choosing friction opposite the object's ground velocity.

Analyse relative motion or its tendency between the surfaces in contact.

Always setting f_s=μ_sN.

First find the required static friction; μ_sN is only the maximum.

Adjusting f_k as though it were static friction.

In the kinetic model its magnitude is approximated by μ_kN.

Assigning units to μ.

In this model it is a ratio of forces and is dimensionless.