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A Level Maths: Connected Particles
A Level Maths Connected Particles
Ever open a mechanics paper, spot two blocks linked by a piece of string, and think, here we go again? Well, that’s exactly our topic today: connected particles.
It looks messy, but it’s still the same F = ma you’ve been using since Year 12.
Only now there are two bodies sharing one story — and one string keeping them honest.
Once you see that, the fog lifts a bit.
🔙 Previous topic:
“Return to forces before connecting multiple objects.”
🧭 Before We Start
📏 Everything you learned last year still matters: diagrams, directions, signs, and a cool head.
The difference in Year 13 is that the objects talk to each other through tension. Sometimes they’re on a slope, sometimes there’s friction, sometimes both.
🧠 Honestly, if the question doesn’t hand you a diagram, that’s your first job.
I once had a student stare at numbers for five minutes, then mutter “I need to draw this” — and solve it in forty seconds.
So yes, draw first. Actually — wait — draw carefully. Every arrow should mean something.
⚙️ What “Connected” Really Means
Light, inextensible string — sounds fancy, but it just means no stretch, no slack.
⚙️ Same tension everywhere.
📏 Same acceleration for both bodies.
That’s it. Everything else is just two F = ma lines and a bit of linking algebra.
Simple idea, sneaky detail.
📘 Example 1 – Train and Hanging Mass
Picture this: a 500 kg train on a 30° slope tied over a pulley to a 1000 kg hanging mass.
Resistance on the train is 100 N.
Find the acceleration and the tension.
Sketch it first. Slope, pulley, weights, tension, resistance.
Now the maths.
Hanging mass: 1000g − T = 1000a
Train: T − 100 − 500g sin 30 = 500a
Add them: 1000g − 100 − 500g sin 30 = 1500a
→ a = (1000g − 100 − 500g sin 30)/1500 ≈ 4.83 m/s²
Then T = 1000g − 1000(4.83) ≈ 4970 N.
🧠 Seems fine — heavier mass drags down, train climbs.
Actually, pause — 4.8 m/s² is about half of g, so the numbers feel right.
✅ Always write both F = ma equations before touching a calculator. It keeps signs straight.
📘 Example 2 – Car and Caravan (Smooth Road)
A 900 kg car pulls a 700 kg caravan. Driving force 2400 N. No resistance.
Car: 2400 − T = 900a
Caravan: T = 700a
Add → 2400 = 1600a → a = 1.5 m/s²
Then T = 700 × 1.5 = 1050 N.
See it? That’s the trick.
Two equations, one acceleration. No mystery.
⚙️ Example 3 – Car and Caravan (With Resistance)
Now let’s add reality.
Car 1100 kg, caravan 900 kg, driving force 3000 N, resistances 300 N and 450 N.
Find a and T.
Method 1 – Each Object
Car: 3000 − 300 − T = 1100a
Caravan: T − 450 = 900a
Add → 2250 = 2000a → a = 1.125 m/s²
Then T = 450 + 900(1.125) ≈ 1463 N.
🧠 Reasonable — smaller acceleration because friction’s eating force.
Actually, I remember trying a toy-car demo once — same pattern, the trailer lagged just slightly before matching pace.
Method 2 – Whole System
Total mass = 2000 kg
Total resistance = 750 N
Driving = 3000 N
F = ma → 3000 − 750 = 2000a → a = 1.125 m/s².
Then T − 450 = 900(1.125) → T = 1463 N.
Anyway — both methods are fine.
Pick whichever clicks quicker in the exam.
📏 Example 4 – With Friction on a Table
A 6 kg block on a rough table (μ = 0.5) tied over a pulley to a 4 kg hanging mass.
Find: frictional force, acceleration, tension.
F = μR = 0.5 × (6g) = 29.4 N.
Table block: T − 29.4 = 6a
Hanging block: 4g − T = 4a
Add → 4g − 29.4 = 10a → a = 0.98 m/s²
Then T = 29.4 + 6(0.98) ≈ 35.3 N.
🧠 Slower acceleration — friction’s doing its job.
Right, and does that number feel small enough? Yeah, just under one m/s². You can almost see it creeping across the bench.
I once tried this with Lego beams — didn’t budge until I added a washer as weight. Real physics, same maths.
❗ Common Mistakes
- Mixing directions; pick “up slope = positive” and commit.
- Forgetting mg sin θ and mg cos θ.
- Losing the resistance term halfway through.
- Thinking tension always pulls forward — it flips sides.
- Skipping units.
✅ Write “positive direction → ____” at the top. Examiners love that line.
Actually, no joke — it’s worth a method mark all by itself.
🧠 Teacher Insight
Every connected-particle question is two F = ma equations sharing the same a and T.
That’s all. The algebra never kills you — it’s the setup.
I tell my classes: slow down before you write anything. Ten seconds of sketching saves ten marks of panic.
And if friction appears? Fine. Swap in μR and carry on.
Now — don’t rush the last line; that’s where people drop the sign.
📘 Real-World Connection
Think cranes, lifts, ski-lifts, trailers. Same physics, different paint job.
I had a student once who worked at a kart track; he said when the lead kart pulled off too hard, the trailer jolt felt exactly like the tension change we’d modelled.
That’s when he finally believed F = ma.
Right then — so next time you see a lift cable, that’s this topic doing its job quietly.
🚀 Next Steps
If this still feels wobbly, our 3-day A Level Maths course walks through every setup — slopes, pulleys, friction, the works.
🚀 Each lesson sounds just like this: live commentary, scribbles, honest pauses.
Anyway — that’s enough numbers for now.
Go draw one more pulley sketch tonight; it’ll stick better than you think.
✅ Quick Recap Table
Concept | Formula / Rule | Quick Reminder |
Newton’s 2nd Law | F = ma | One per object |
Shared acceleration | a₁ = a₂ | String taut = same a |
Tension | Equal throughout | Opposite directions |
Friction | F = μR | Opposes motion |
Slope forces | mg sin θ / mg cos θ | Resolve early |
System method | Combine equations | Quicker in exams |
Author Bio
S. Mahandru • Head of Maths, Exam.tips
S. Mahandru is Head of Maths at Exam.tips. With over 15 years of experience, he simplifies complex calculus topics and provides clear worked examples, strategies, and exam-focused guidance.
🧭 Next topic:
“Continue by exploring connected particles in detail.”