About this chapter
Dynamics connects forces to motion through Newton's second law, F = ma. The questions are about cars towing trailers, crates on cranes, lifts, trains and particles over pulleys, and each one rewards a clear force diagram and one equation of motion per particle.
This chapter covers force diagrams, Newton's second law, vertical motion and lifts, forces as vectors, connected particles, pulleys, Newton's third law, what happens when a string breaks or goes slack, and the modelling words light, inextensible and smooth. Several questions finish with a second stage of motion found with suvat.
The ten questions
- Q1 (6 marks): three forces as vectors on a 0.5 kg particle: the unknown force, and the size and direction of the acceleration.
- Q2 (6 marks): a car's acceleration, its driving force, and the time to stop after the engine is switched off.
- Q3 (6 marks): a car and trailer braking: the deceleration and the force in the towbar.
- Q4 (6 marks): two crates lifted by a crane: the acceleration and the tension in each rope.
- Q5 (6 marks): a ball thrown up against air resistance: its deceleration, height, and speed on return.
- Q6 (10 marks): two particles over a pulley: the acceleration and tension, then the motion after one hits the ground.
- Q7 (10 marks): a lift carrying two crates stacked on each other: its acceleration and the forces between them.
- Q8 (10 marks): an engine pulling a carriage: the acceleration, the coupling tension, and later motion.
- Q9 (10 marks): two forces whose resultant is parallel to i + 2j: the unknown, the acceleration, and the velocity later.
- Q10 (12 marks): a particle on a table pulled by two particles hanging in a chain over a pulley.
Key skills tested
Force diagrams. Mark every force: weight mg, normal reaction R, tension T, thrust and resistance. Mass is in kg and weight in newtons.
Newton's second law. The resultant force in the direction of the acceleration equals ma. g goes in the weight, never in the ma term.
Vertical motion and lifts. Upwards, T − mg = ma. Decelerating upwards means the acceleration acts downwards. The floor pushes up on a passenger with a force R.
Forces as vectors. Add the force vectors and set the total equal to m times the acceleration vector. Give a magnitude when one is asked for, not a vector.
Connected particles. Use the whole system for the acceleration, since internal forces cancel, then one part for the tension or thrust. A towbar can push as well as pull.
Pulleys. With a light inextensible string over a smooth pulley, the tension and the acceleration are the same on both sides.
Newton's third law. B pushes on A with the same size force as A pushes on B, in the opposite direction.
When the motion changes. If a string breaks or goes slack, find the new acceleration. The final speed of one stage starts the next.
Modelling. Light means the same tension throughout. Inextensible means the same acceleration. A smooth pulley has no friction.
Worked example
Question 2 from this chapter. A car of mass 900 kg speeds up from 10 m/s to 25 m/s over 150 m against a resistance of 450 N. Find (a) the acceleration and (b) the driving force. (c) The engine is then switched off: find the time taken to stop.
(a) Using v² = u² + 2as: 625 = 100 + 300a, so a = 1.75 m/s².
(b) By F = ma along the road: D − 450 = 900 × 1.75 = 1575, so D = 2025 N, or 2030 N to 3 significant figures.
(c) With only the resistance acting, the deceleration is 450 ÷ 900 = 0.5 m/s². Then 0 = 25 − 0.5t, so t = 50 s.
Seen on real papers
Every question in our booklets is original. These are the recent papers where each question type has appeared.
- January 2025, Q1(a): forces as vectors, F = ma, and the size of the acceleration, as in Q1 and Q9.
- January 2025, Q7: particles connected over a pulley, then a second stage of motion after the set-up changes, as in Q6 and Q10.
Where marks are lost
- A vector instead of a magnitude. The most common error in January 2025 was leaving the acceleration as a vector when its size was asked for.
- g in the wrong place. Using m instead of 2m or 3m, leaving g out of the weight, or putting g into the ma term were the most common errors in the pulley question.
- One equation for two stages. In January 2025 many did not realise the motion had two parts and used one equation for the whole of it.
- Tonnes instead of kilograms. In a June 2022 towing question, a significant number used masses in tonnes and could score at most 4 of the 6 marks.
Common questions
Why use the whole system first?
The tension or thrust between the parts cancels, so one equation gives the acceleration straight away.
Does g go in the ma term?
Never. g only appears in weights, as mg.
How do I handle a lift that is slowing down while going up?
Its acceleration is downwards, so take a as negative with up as positive, or take down as the positive direction.
Where this chapter leads
- Mechanics 1 Chapter 5: Forces and Friction: F = ma on rough surfaces and inclined planes.
- Mechanics 1 Chapter 6: Momentum and Impulse: forces acting over short times in collisions.