Systems of objects

Sometimes it is easier to treat several connected objects as one system. Think of a train: when the engine speeds up, every coach speeds up with it.

Two boxes joined by a string, on a smooth floor, with Box 1 pulled by a force F. The string's pull, called tension (T), pulls Box 2 forwards and Box 1 backwards. Because it acts inside the system, it is an internal force, and it cancels out when you look at the whole system.

So, look at the boxes and string together. Only the external forces matter. Along the floor, that is just F, so:

a = F/(m₁ + m₂).

The pair moves like one object with the total mass. Up and down, the total weight (m₁ + m₂)g is balanced by the floor's normal force N₁ + N₂.

Combined loads. A tractor's pull F gives a harrow alone an acceleration a₁ (so its mass m₁ = F/a₁), and a trolley alone a₂ (m₂ = F/a₂). Hitched together: a = F/(m₁ + m₂) = a₁a₂/(a₁ + a₂).

All three laws together:

• The snake-boat race: 95 oarsmen row backwards and 5 forwards, each with 200 N. The water pushes the boat with (95 – 5) × 200 = 18,000 N forward.
• A bullet in wood: a 50 g bullet stopping from 100 m s⁻¹ in 50 cm needs a force of 500 N.
• A penalty kick: an 800 N kick sends a 0.4 kg ball to 30 m s⁻¹. The foot touches the ball for only 0.015 s.

How far do Newton's laws go? They work for everyday objects, for planets and for stars. They need changing only in extreme cases: very close to hugely massive objects, at speeds close to the speed of light, and at the scale of atoms.