Understand and use the F ≤ μR model for friction; coefficient of friction;…

The idea

Friction opposes relative motion or tendency to move. The model F ≤ μR relates friction F to normal reaction R and coefficient μ. When a body is in limiting equilibrium, F = μR; otherwise F < μR. For a body on a rough surface, resolve perpendicular to the surface to find R, then apply F = μR at limiting equilibrium or F ≤ μR when stationary. In statics, friction balances the component of weight or applied force along the surface. For motion, use Newton’s second law along the surface with F = μR if sliding. Example: a block of mass 5 kg on a rough horizontal plane with μ = 0.3 has R = 5g N, so limiting friction is 0.3 × 5g = 1.5g N. If the applied horizontal force is 10 N and g = 9.8 m s⁻², then 1.5g = 14.7 N, so the block remains stationary and friction is 10 N.

What earns the marks

• State that friction acts parallel to the contact surface and opposes relative motion or the tendency to move.
• Define the coefficient of friction μ as the constant in the relationship F = μR at limiting equilibrium, and note that μ is dimensionless.
• Resolve forces perpendicular to the rough surface to determine the normal reaction R, including cases where an applied force has a component perpendicular to the surface.
• Apply F = μR when the body is in limiting equilibrium or sliding, and apply F ≤ μR when the body is stationary but not at the point of sliding.
• For statics on a rough inclined plane, resolve parallel and perpendicular to the plane and use F = μR at the point of impending motion.
• For motion on a rough surface, use Newton’s second law along the direction of motion with friction F = μR, and include acceleration or deceleration as appropriate.

Common mistakes and how to put them right

• Assuming friction always equals μR.

  Put it right: Use F = μR only at limiting equilibrium or when sliding; otherwise friction is whatever value is needed to maintain equilibrium, up to μR.

• Using R = mg without resolving perpendicular to the surface.

  Put it right: Resolve perpendicular to the surface; on an inclined plane R = mg cos θ for a body with no other perpendicular forces.

• Treating μ as having units.

  Put it right: μ is dimensionless because it is the ratio of two forces.

• Forgetting that friction can act up or down an inclined plane depending on the direction of impending motion.

  Put it right: Determine the direction of relative motion or tendency to move before assigning the direction of friction.

Exam tips

• Draw a clear force diagram showing weight, normal reaction, friction and any applied forces, with friction opposing the direction of motion or impending motion.
• State whether the body is in limiting equilibrium, stationary but not limiting, or moving before choosing F = μR or F ≤ μR.
• Show the resolution perpendicular to the surface to find R, then substitute into F = μR if appropriate; keep g as a symbol until the final numerical evaluation unless a numerical value is required.
• Check the direction of friction on an inclined plane by considering which way the body would slide if friction were absent.