M10.5MechanicsStretch

Connected Particles

When two objects are connected — a car towing a trailer, or a lift and its load — they move together with the same acceleration. You can analyse the whole system at once, or each particle separately to find the connecting force.

30 min Forces and Motion

Edexcel AS Level Maths: 10.5 Connected ParticlesCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Recognise that connected particles share an acceleration
Analyse the system as a whole
Analyse each particle to find the tension/thrust
Apply this to tow-bars and lifts

Shared acceleration

Connected particles (joined by a string, tow-bar or in contact) move with the $\textbf{same acceleration}$ . The connection exerts equal and opposite forces (tension or thrust) on the two parts.

Whole-system method

To find the acceleration, treat both particles as a $\textbf{single object}$ . The internal connecting forces cancel, so you use the total mass and the external driving force.

F_{\text{driving}} = (m_1 + m_2)\,a

Treat the system as one mass to find

a

Example 1A 1000 kg car tows a 500 kg trailer with a 3000 N driving force (no resistance). Find a.

a = \dfrac{3000}{1000 + 500}

Answer

2

m/s²

Single-particle method

To find the $\textbf{connecting force}$ (tension in the tow-bar), apply $F = ma$ to just $\textbf{one}$ particle, using the acceleration found above.

Example 1For the trailer (500 kg, a = 2 m/s²), find the tow-bar tension.

1On the trailer, the only forward force is the tension:

T = m a = 500 \times 2

Answer

T = 1000

Tip — System method → acceleration; single-particle method → the connecting force.

Formula recap

a = \dfrac{F}{m_1 + m_2}

Whole-system acceleration.

T = m a \ (\text{one particle})

Connecting force.

\text{same } a \text{ for both}

Connected ⟶ shared acceleration.

Common mistakes to avoid

Including the internal tension when using the whole-system method.

Internal forces cancel for the system; use only external forces and total mass.

Using total mass when finding the tension on one particle.

For the connecting force, apply F = ma to a single particle.

Key takeaways

Connected particles share the same acceleration.
Whole-system (total mass, external force) gives the acceleration.
Analysing one particle gives the connecting tension/thrust.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

All 6 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.