The Coefficient of Friction: A-Level Maths Revision

Introduction to Friction

• Definition: Friction is "the resistance an object encounters in moving over another".
• Comparative Example: Easier to drag an object over glass than sandpaper due to different levels of frictional resistance.
• Assumptions in Problems:
  • "Smooth" surface: assumed to exert no frictional force.
  • "Rough" surface: offers some frictional resistance.

Limiting Equilibrium

• Concept: When pushing an object (e.g., a book), initially, the frictional force equals the applied force.
• Behavior of Frictional Force:
  • Increases with the applied force until a maximum value is reached.
  • At maximum value, friction is "limiting".
  • A stationary object under maximum friction is in "limiting equilibrium".
  • If pushed slightly harder, the object begins to move.
• Summary:
  • Frictional force increases to a maximum, beyond which the object either moves or is on the verge of moving.

Coefficient of Friction

• Definition: Number representing friction between two surfaces.
• Symbol: Typically denoted as ( \mu ).
• Maximum Frictional Force Formula:
  • ( F = \mu R )
  • ( \mu ): Coefficient of friction.
  • ( R ): Normal reaction force.
• Direction: Acts parallel to surfaces in contact, opposing motion.

Example Problem

• Given: Particle of mass 5 kg on a rough incline (angle: 30 degrees).
• Objective: Find the coefficient of friction between the particle and the plane.
• Equations:
  • Resolving up the plane: ( F - 5g \sin 30 = 0 )
  • Resolving perpendicular to the plane: ( R = 5g \cos 30 )
• Result in Limiting Equilibrium:
  • ( F = \mu R )
  • ( 5g \sin 30 = \mu 5g \cos 30 )
  • ( \mu = \frac{\sin 30}{\cos 30} = 0.577 ) (3 significant figures)