R: Forces and Newton’s laws: Understand and use the F ≤ μR model for friction; coefficient of friction; motion of a body on a rough surface; limiting friction and statics.

The idea

This topic introduces the empirical model of friction, where the frictional force F satisfies F ≤ μR, with μ the coefficient of friction and R the normal reaction. You learn to distinguish static and dynamic situations: when a body is in limiting equilibrium, friction takes its maximum value F = μR; when it moves, friction is usually modelled as F = μR opposing motion; when it is stationary but not at the point of sliding, F < μR. For a body on a rough horizontal surface, R is often equal to weight, but on an incline R = mg cos θ. You apply Newton’s second law along the surface, including friction, to find acceleration or required forces, and use equilibrium conditions for statics problems.

What earns the marks

• State and apply the friction inequality F ≤ μR, recognising that equality holds at limiting equilibrium or when sliding.
• Calculate the normal reaction R correctly for horizontal and inclined rough surfaces, including cases with additional vertical forces.
• Determine the direction of friction as opposing impending or actual motion along the surface.
• Set up and solve equations of motion along a rough surface using F_net = ma, incorporating friction as μR when moving or at limiting equilibrium.
• Solve statics problems on rough surfaces by resolving forces and using equilibrium conditions, often with F = μR at the point of sliding.
• Interpret the coefficient of friction as a dimensionless constant relating maximum friction to normal reaction, and use it to compare roughness.

Common mistakes and how to put them right

• Assuming friction always equals μR.

  Put it right: Use F = μR only when the body is at limiting equilibrium or sliding; otherwise F ≤ μR and must be found from equilibrium or motion equations.

• Using R = mg on an inclined plane.

  Put it right: Resolve perpendicular to the plane to get R = mg cos θ, or include other perpendicular forces.

• Taking friction in the same direction as motion.

  Put it right: Friction always opposes relative motion or impending motion.

• Forgetting that μ is dimensionless and misusing units.

  Put it right: Treat μ as a number with no units; check that F and R are both forces.

Exam tips

• Draw a free-body diagram showing all forces, including friction and normal reaction, and label the direction of motion.
• For inclined plane problems, resolve parallel and perpendicular to the plane to simplify equations.
• If a body is on the point of moving, state F = μR explicitly before solving.
• Check whether the surface is rough or smooth; if smooth, friction is zero.