About this chapter
Forces and friction brings rough surfaces and slopes into the dynamics of Chapter 4. The key step is resolving perpendicular to the surface to find the normal reaction, because friction depends on it, and a force pulling at an angle changes it.
This chapter covers resolving a force, inclined planes, the normal reaction, friction, equations of motion on slopes, exact trig from a given tangent, deciding whether a particle moves, pulleys on slopes, and what happens when a force is removed. The last question follows a particle projected up a rough slope, then sliding back down.
The ten questions
- Q1 (6 marks): a box pulled by a rope at 30°: the normal reaction in terms of P, then P itself.
- Q2 (6 marks): a horizontal force pushing a particle up a smooth 20° slope: the force and the normal reaction.
- Q3 (6 marks): a stone sliding to rest on ice: its deceleration, the coefficient of friction, and why mass does not matter.
- Q4 (6 marks): a trolley pushed at 25° below the horizontal at constant speed, then pushed horizontally.
- Q5 (6 marks): show a particle slides on a rough slope, find its acceleration, and the least friction to hold it.
- Q6 (12 marks): a parcel pulled up a rough 25° slope by a force at 20° to it, and its later motion.
- Q7 (11 marks): a particle on a rough slope connected over a pulley to a hanging particle.
- Q8 (11 marks): a particle on a rough table pulled by a hanging particle, which later hits the floor.
- Q9 (10 marks): a car towing a trailer up a slope: the acceleration, the towbar tension, and later motion.
- Q10 (12 marks): a particle projected up a rough slope decelerates at g, stops, and slides back down.
Key skills tested
Resolving a force. A force P at an angle θ to a direction has components P cos θ along it and P sin θ perpendicular to it.
Inclined planes. The weight mg splits into mg sin α down the plane and mg cos α into it. Resolve parallel and perpendicular to the plane.
Normal reaction. Resolve perpendicular to the surface: R is not always mg or mg cos α. A pull upwards reduces R, and a push downwards increases it.
Friction. Friction is at most μR. When sliding, F = μR and it opposes the motion, so it reverses when the motion reverses.
Equations of motion. Along the plane, resultant = ma; perpendicular to it, the resultant is zero. Constant speed means a = 0.
Exact trigonometry. tan α = 3/4 gives sin α = 3/5 and cos α = 4/5; tan α = 5/12 gives 5/13 and 12/13. Keep these exact.
Will it move? Compare the force trying to move the particle with the greatest possible friction μR. On a slope alone, it slides if tan α is greater than μ.
Pulleys on planes. Write one equation for each particle. For the force on the pulley, add the two tensions using the angle between the parts of the string.
When forces change. Removing a force changes R, so friction and the acceleration change too.
Worked example
Question 3 from this chapter. A stone passes A at 8 m/s and slides to rest 10 m further on, across rough ice. Find (a) its deceleration, (b) the coefficient of friction, (c) the time to stop, and (d) explain why (b) does not depend on the stone's mass.
(a) Using v² = u² − 2as: 0 = 64 − 20a, so the deceleration is 3.2 m/s².
(b) The only horizontal force is friction, μmg, so μmg = m × 3.2 and μ = 3.2 ÷ 9.8 = 0.33.
(c) t = 8 ÷ 3.2 = 2.5 s.
(d) Friction and ma are both proportional to m, so the mass cancels.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2025, Q6: friction on a slope with a string pulling at an angle, where the tension changes the normal reaction, then whether the box moves, as in Q6 and Q10.
Where marks are lost
- R = mg cos α by habit. The most common error in January 2025 was ignoring the string's effect on the normal reaction.
- An essay instead of a calculation. To decide whether the box would move, a significant number wrote an explanation with no numbers.
- Friction that pushes. The final mark was often lost by students who thought friction would move the box up the plane. Friction only ever opposes motion.
- Tension without the weight. A few wrote the tension as 0.75 instead of 0.75mg, or left it as T.
Common questions
Is the normal reaction always mg cos α on a slope?
Only when no other force has a component perpendicular to the slope. A string at an angle changes it.
When is friction equal to μR?
When the particle is sliding, or on the point of sliding. Otherwise friction is just large enough to stop motion.
How do I show a particle slides down a slope?
Compare the component of weight down the slope with the greatest friction μR, and show the weight component is larger.
Where this chapter leads
- Mechanics 1 Chapter 7: Statics of a Particle: limiting friction when a particle is on the point of slipping.
- Mechanics 1 Chapter 6: Momentum and Impulse: friction slowing particles before and after a collision.