Math 2.0

Mechanics of particles and systems Definition

Friction

Builds on Classical mechanics

Definition

Let a particle slide on a fixed surface of class C1C^1 in ℝ3\R^3, in a Newtonian system. Let N≥0N \ge 0 be the size of the normal force that the surface exerts on the particle. Let vv be the velocity of the particle, which is tangent to the surface. Let 0≤μk≤μs0 \le \mu_k \le \mu_s be constants, the coefficients of kinetic and static friction. A Coulomb friction force is a force ff tangent to the surface such that:

  1. if v≠0v \ne 0, then f=−μkN v/∣v∣f = -\mu_k N\, v / \abs{v};
  2. if v=0v = 0, then ∣f∣≤μsN\abs{f} \le \mu_s N.

Remarks

In case 2 the force is not given by a formula: it takes the value needed to keep the particle at rest, as long as that value satisfies the bound. Example: a block of mass mm at rest on a plane tilted by the angle θ\theta has N=mgcos⁡θN = m g \cos\theta and needs the friction mgsin⁡θm g \sin\theta along the plane. It stays at rest exactly when mgsin⁡θ≤μsmgcos⁡θm g \sin\theta \le \mu_s m g \cos\theta, that is, tan⁡θ≤μs\tan\theta \le \mu_s. Non-example: the force f=−μkNvf = -\mu_k N v for a sliding particle fails clause 1, since its size grows with the speed; such linear drag is viscous friction, not Coulomb friction. The model is an empirical law, also called the Amontons–Coulomb law.

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Differential equation, concept, no evidence levelDifferential equationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptInterval, concept, no evidence levelIntervalconceptOrdinary differential equation, concept, no evidence levelOrdinary differential…conceptOrdinary differential equation, definition, no evidence levelOrdinary differential…definitionLipschitz condition, definition, no evidence levelLipschitz conditiondefinitionSolution of an ordinary differential equation, definition, no evidence levelSolution of an ordina…definitionInitial value problem, definition, no evidence levelInitial value problemdefinitionPicard–Lindelöf hypotheses, definition, no evidence levelPicard–Lindelöf hypot…definitionPicard operator, definition, no evidence levelPicard operatordefinitionIntegral equation form of an initial value problem, statement, no evidence levelIntegral equation for…statementAn iterate of the Picard operator is a contraction, statement, no evidence levelAn iterate of the Pic…statementPicard–Lindelöf theorem, statement, no evidence levelPicard–Lindelöf theor…statementClassical mechanics, definition, no evidence levelClassical mechanicsdefinitionFriction, definition, no evidence levelFrictiondefinition
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  1. Differential equation concept
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  5. Ordinary differential equation concept
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  7. Lipschitz condition definition
  8. Solution of an ordinary differential equation definition
  9. Initial value problem definition
  10. Picard–Lindelöf hypotheses definition
  11. Picard operator definition
  12. Integral equation form of an initial value problem statement
  13. An iterate of the Picard operator is a contraction statement
  14. Picard–Lindelöf theorem statement
  15. Classical mechanics definition
  16. Friction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Friction
About this page
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MSC70F40, 70A05
DefinesFriction
UsesClassical mechanics
Changed3 November 2026

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