A-Level Maths: Forces on Slopes and Friction
Inclined planes are where A level mechanics brings together resolving forces, the friction model and Newton's second law. This material trains those skills through the kind of structured problems that appear in the mechanics section of every A level Mathematics specification, based on the DfE subject content for forces, Newton's laws and friction.
You start by resolving weight into components parallel and perpendicular to a slope and finding the normal reaction, including when an extra force pulls at an angle to the plane or acts horizontally. The friction model is used carefully: you learn to tell apart a body in ordinary equilibrium, where friction is only as large as it needs to be, from one in limiting equilibrium or sliding, where . Typical questions ask for the coefficient of friction when a block is on the point of slipping, the least force needed to stop a parcel sliding down a ramp, or the horizontal force that holds a particle on a smooth slope.
Dynamics questions ask for the acceleration of a body sliding down a rough slope or pulled up it, and for the distance a body projected up a slope travels before stopping, followed by the key follow-up: will it stay at rest or slide back? You compare the component of weight down the slope with the maximum friction, which reduces to comparing with . Connected-particle problems place one particle on a rough slope and another hanging over a smooth pulley, so you write an equation of motion for each particle, find the acceleration and the tension, and check whether the system moves at all.
All values are given in each question, with and answers to 3 significant figures, and the standard modelling assumptions (particle, light inextensible string, smooth pulley) are stated or discussed.
The material offers three formats. The quiz has 12 worked multiple-choice problems with explanations that show each resolving step and the common mistakes behind the wrong options. The flashcards summarise the resolving rules, the friction model and the modelling assumptions. The written work gives 8 longer problems to solve by hand, from checking whether a block slides to a full connected-particle analysis.
- Resolve weight and applied forces parallel and perpendicular to an inclined plane
- Use the friction model F ≤ μR and recognise limiting equilibrium
- Find the coefficient of friction, least holding force or horizontal holding force for a body on a slope
- Apply Newton's second law to bodies moving up or down rough slopes
- Decide whether a body at rest on a rough slope stays at rest or slides back
- Solve connected-particle problems with one particle on a slope and one hanging over a pulley
- State and use standard modelling assumptions
Practice material written by Zestly, based on the DfE A level Mathematics subject content (mechanics: forces and Newton's laws, resolving forces on inclined planes, the F ≤ μR model of friction, connected particles).
Sample question
A box of mass 4 kg rests in equilibrium on a rough plane inclined at $25^\circ$ to the horizontal. It is not on the point of slipping. What is the magnitude of the frictional force acting on the box? (Use $g = 9.8 \text{ m s}^{-2}$.)
- $35.5$ N
- $16.6$ N
- $39.2$ N
- It cannot be found without $\mu$
See the answer
$16.6$ N
The box is in equilibrium, so the forces parallel to the slope balance: $F = mg \sin 25^\circ = 4 \times 9.8 \times 0.4226 = 16.6$ N (3 s.f.). The coefficient of friction is not needed, because friction only takes the value $\mu R$ when the box is in limiting equilibrium or moving; here $F$ is whatever is needed to prevent motion. $35.5$ N is $mg \cos 25^\circ$, the normal reaction.