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Mechanics · Forces & Newton's laws

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Connected particles

Why two bodies joined by a light inextensible string share one acceleration and one tension, the two-step method that runs every problem — whole system for the acceleration, a single body for the internal force — and why the connecting force is internal to the system but external to each body.

Mechanics · Forces & Newton's laws

Connected particles

Why two bodies joined by a light inextensible string share one acceleration and one tension, the two-step method that runs every problem — whole system for the acceleration, a single body for the internal force — and why the connecting force is internal to the system but external to each body.

Why it works

Tow a trailer, stack two blocks, hang a load from a lift: in each case two bodies move together, joined by a string or by contact. Two facts unlock all of them.
  • Same acceleration. A light, inextensible string can't stretch, so the two
bodies move as one rigid unit — they have the same acceleration. (Bodies simply in contact and staying in contact share an acceleration too.)
  • Same tension. A light string has no mass, so the tension is the **same
all the way along it** — the trailer pulls back on the car exactly as hard as the car pulls forward on the trailer. There is one number TT, not two.400 kg1200 kgDR2RT*A car of mass 1200 kg1200\,\text{kg} tows a 400 kg400\,\text{kg} trailer (not accurately drawn). The driving force DD acts on the car; resistances oppose each body; the tow rope carries one tension TT, pulling each vehicle towards the other.*

The two-step method. This is the whole topic:
  1. Whole system for the acceleration. Treat both bodies as one object of the
combined mass. The tension is internal — it pulls forwards on one body and equally backwards on the other, so it cancels and never appears. Only the external forces (driving force, resistances, weights) drive the system: external resultant=(m1+m2) a.\text{external resultant} = (m_1 + m_2)\,a.
  1. One body for the internal force. To find the tension or contact force, now
look at a single body and apply F=maF = ma to it alone, using the acceleration from step 1. On that one body the connecting force is external, so it appears — and the body's own mass (not the total) is what multiplies aa.

The classic mistake is using the total mass when finding the tension. The tension acts on one body, so it must be balanced against that body's mass times the shared acceleration.

Why "internal" matters. Add the equations for the two bodies and the tension terms cancel — that's algebraically why step 1 can ignore it, and the proof that the same TT acts on both. Connected-particle problems are really one acceleration solved twice: once for the pair, once for a part.ABP*Block AA is pushed by a force PP and presses on block BB (not accurately drawn). The push travels through the contact: AA pushes BB forward, and by Newton's third law BB pushes back on AA — equal and opposite, on different bodies.*