A powerful aspect of interaction diagrams is that the circle need not enclose only a single object: one can circle a group of objects and treat it as a single system. This move greatly simplifies problems in which several bodies move together, because it allows us to ignore all internal forces and focus only on what the system exchanges with the outside.
Principle — System rule
If several objects are circled together, the squiggles internal to the circle (between objects inside the system) do not count. Only the squiggles that cross the boundary of the circle count: these are the forces external to the system.
The deep reason is Newton’s third law. Every internal squiggle represents an action–reaction pair: both twin forces act on bodies that are inside the system, so their sum is zero and cannot influence the motion of the system as a whole. Only interactions with the outside remain relevant, namely the squiggles that cut across the boundary of the circle.
Example — The Bremen Town Musicians
A donkey, a dog, a cat and a rooster are stacked on top of one another.
Circling all four animals as a single system, the three internal constraint reactions (rooster–cat, cat–dog, dog–donkey) stay inside the boundary and cancel out. Only the four weights and the floor’s reaction cross the boundary.
Circling the whole system, the constraint reactions between the animals are internal and cancel out. Only the four weight forces (exiting towards the Earth) and the floor’s reaction remain. Imposing equilibrium of the system along the vertical, the floor’s reaction must balance the sum of all the weights:
The result is intuitive — the floor holds up the total weight of the stack — but the interaction diagram obtains it without ever having to calculate the individual reactions between one animal and the next. This is the great advantage of choosing the system’s boundary well.
Tip
The boundary can be redrawn as needed, depending on what one is looking for. If instead one wants to find the constraint reaction between the cat and the dog, it suffices to circle just the cat and the rooster: the squiggles leaving that boundary give the forces on the sub-system, among which the reaction that was previously internal now appears.
Links
Topics: Dinamica Concepts: Terza legge di Newton · Forza normale · Forza peso Skills: Diagramma di interazione
Related exercises: Problema — Vero o falso su forze e moto · Vero o falso su leggi di Newton · Esercizio svolto — due scatole in equilibrio