Systems comprising several bodies and internal forces

Internal forces and external forces

When studying a system consisting of several objects (for example, two blocks connected by a rope, or a person in a lift), a distinction must be made between:

• internal forces: those exerted by the objects within the system on one another (these always occur in action-reaction pairs);
• external forces: those exerted by objects that do not belong to the system (weight, force of the ground, external tension, etc.).

An important consequence

In an isolated system (with no external forces), the sum of all internal forces is always zero, as each force cancels out its reaction… provided that the forces are summed over the entire system, never over a single object.

This explains why, for example, two people on roller skates pushing against each other each move off in opposite directions: the internal forces (the mutual pushes) cancel each other out at the level of the system as a whole, but each person accelerates individually according to their own mass.

Example: the taut rope

Two people are each pulling on a rope in opposite directions. The tension T that the rope exerts on person 1 is equal in magnitude to the tension T that it exerts on person 2 (the law of tension in an ideal string), in accordance with Newton’s third law as applied to the points of contact between the hand and the rope.

Example: riding a skateboard

If you are on a skateboard that is free to move and you push against a wall, the wall pushes back against you (reaction), and you move backwards with the skateboard. The wall, however, does not move because it is fixed to the Earth, which has an enormous mass: its acceleration a = F/m(Earth) is completely negligible.

Key points

When solving a problem involving several bodies, always mentally isolate each object, and apply the second law (F = m x a) only to the forces actually acting ON that object, never to those it exerts on the others.