9709 Mechanics: Connected Particles and Pulleys, Solved

Connected-particle problems appear regularly in Cambridge 9709 Mechanics. Two masses joined by a string over a pulley, a block on a table pulled by a hanging weight, a car towing a trailer: the settings change, but the method does not. Students who lose marks usually have the right idea but an untidy diagram, a sign error or a missing force.

This guide gives a method that works every time, then applies it to five original examples of increasing difficulty.

Where it sits

• Component: Paper 4, Mechanics, syllabus section 4.4 (Newton's laws of motion), 2026–2027 syllabus.
• Paper 4: 1 hour 15 minutes, 50 marks, 6–8 structured questions; 40% of the AS Level and 20% of the A Level (where Mechanics is in your route).
• g: the syllabus states that the approximate value g = 10 m s⁻² is expected in this component.

What the syllabus says

You should be able to apply Newton's laws to particles of constant mass under constant forces, which may include friction, tension in an inextensible string and thrust in a connecting rod. You should also be able to solve problems modelled as connected particles, for example "particles connected by a light inextensible string passing over a smooth pulley, or a car towing a trailer by means of either a light rope or a light rigid tow-bar".

The modelling words matter: - Light string or rod: its mass is ignored. - Inextensible: both particles have the same magnitude of acceleration. - Smooth pulley: the tension is the same on both sides.

The five-step method

1. Draw a large diagram with every force on each particle: weight, tension, normal reaction, friction, driving force, resistance.
2. Choose the direction of motion for each particle and mark the acceleration a on both.
3. Write F = ma for each particle separately, along its direction of motion.
4. Solve simultaneously. Adding the equations often eliminates the tension.
5. Check by substituting back into the other equation.

Example 1: two masses over a pulley (original)

Particles of mass 5 kg and 3 kg are connected by a light inextensible string over a smooth fixed pulley and released from rest. Find the acceleration and the tension.

• 5 kg (moves down): 50 − T = 5a
• 3 kg (moves up): T − 30 = 3a

Adding: 20 = 8a → a = 2.5 m s⁻². Then T = 30 + 3(2.5) = 37.5 N. Check: 50 − 37.5 = 12.5 = 5 × 2.5 ✓

Extension: the force on the pulley from the string is 2T = 75 N, acting vertically downwards.

Example 2: table and hanging mass (original)

A 4 kg block rests on a smooth horizontal table. It is attached by a light inextensible string, passing over a smooth pulley at the edge of the table, to a 1 kg particle hanging freely. The system is released from rest. Find the acceleration and the tension. The 1 kg particle hits the floor after falling 0.5 m; find its speed at that moment.

• 4 kg: T = 4a
• 1 kg: 10 − T = a

Adding: 10 = 5a → a = 2 m s⁻², T = 8 N. Speed: v² = u² + 2as = 0 + 2(2)(0.5) = 2 → v = 1.41 m s⁻¹ (3 s.f.)

A frequent follow-up asks what happens next: once the string goes slack, the block continues at constant speed on a smooth table, or decelerates on a rough one.

Example 3: a rough table (original)

Repeat with a 4 kg block on a rough table (coefficient of friction 0.2) and a 2 kg hanging particle.

Normal reaction on the block R = 40 N, so maximum friction F = 0.2 × 40 = 8 N. - 4 kg: T − 8 = 4a - 2 kg: 20 − T = 2a

Adding: 12 = 6a → a = 2 m s⁻², T = 16 N. Check: 16 − 8 = 8 = 4 × 2 ✓

Always check that motion occurs: here 20 N > 8 N, so the system moves. If the hanging weight were less than the maximum friction, the system would stay at rest.

Example 4: car and trailer (original)

A car of mass 1200 kg tows a trailer of mass 400 kg along a straight horizontal road using a light rigid tow-bar. The car's driving force is 2000 N. Resistances are 300 N on the car and 100 N on the trailer. Find the acceleration and the tension in the tow-bar.

Whole system: 2000 − 300 − 100 = 1600a → a = 1 m s⁻². Trailer: T − 100 = 400(1) → T = 500 N. Check (car): 2000 − 300 − 500 = 1200 = 1200 × 1 ✓

If the car brakes, the trailer may push on the tow-bar. The force then becomes a thrust, which is why the syllabus mentions a rigid tow-bar rather than a rope.

Example 5: an inclined plane (original)

A 3 kg particle rests on a smooth plane inclined at 30° to the horizontal. It is connected by a light inextensible string over a smooth pulley at the top of the plane to a 2 kg particle hanging freely. Find the acceleration and the tension.

The component of weight down the plane is 3 × 10 × sin 30° = 15 N. - 2 kg (moves down): 20 − T = 2a - 3 kg (moves up the plane): T − 15 = 3a

Adding: 5 = 5a → a = 1 m s⁻², T = 18 N.

On a rough plane, remember that friction always opposes the direction of motion. The acceleration up the plane is therefore different from the acceleration down it, a point the syllabus notes explicitly.

Mistakes that cost marks

1. Using one equation for the whole system and then trying to find the tension from it. You need an equation for an individual particle.
2. Forgetting friction or resistance on one of the particles.
3. Using g = 9.8 or 9.81 when the question expects 10.
4. Assuming the tension equals the hanging weight. It does not when the system accelerates.
5. Missing the slack-string phase in multi-stage questions.
6. Giving answers to 2 significant figures when 3 are expected.

How to practise

• Generate a 9709 Paper 4 topical PDF on Newton's laws from Kambridge Topicals.
• Sit full Mechanics papers from the 9709 past papers library under the 75-minute limit.
• Use the 9709 grade thresholds guide to set a target score for Paper 4.

Frequently asked questions

Should I always use g = 10? In 9709 Mechanics, the syllabus states that the approximate value 10 m s⁻² is expected, unless a question says otherwise.

Do I need to resolve forces on inclined planes in every question? Only along and perpendicular to the plane, which is usually simplest for particles on slopes.

What is the difference between tension and thrust? Tension pulls along a string or rod; thrust pushes along a rod. A string can only be in tension.

Make mechanics methodical

Dr. Rizwan Jahangir, PhD in Mathematics (Sabancı University), teaches mechanics with the same disciplined routine every time: diagram, equations, solve, check.

Sources: Cambridge International AS & A Level Mathematics 9709 syllabus for examination in 2026–2027: section 4.4 and the Paper 4 assessment overview.

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