Isolated Systems and External Forces

How to distinguish isolated systems from systems subjected to relevant external forces.

Conceptual context

This topic delves into the relationship between system definition, the classification of forces as external, and the consequences for momentum conservation. While the concept of an "isolated system" is introduced in fundamentals, here we explore how redefining system boundaries turns external forces into internal ones and vice versa, and how external forces —when they exist— modify the total momentum in a controlled manner.

The internal/external distinction is not intrinsic to the force: it depends entirely on where you draw the system boundary.

🟢 Essential level

A system is isolated when the important things happen between the bodies we have decided to study, without significant intervention from the outside. The difficulty is not in memorizing a definition, but in choosing the system boundary well.

Imagine two skaters pushing each other on ice. If you study both together, the push is an internal interaction: one gains motion in one direction and the other in the opposite direction. If you study only one of them, the push from the other appears as an external action that changes their motion.

The essential idea is that the same interaction can change roles depending on the chosen boundary. There are no "internal" or "external" forces on their own; they are such relative to the system you decide to analyze.

[!IMPORTANT] Before talking about conservation, decide which bodies form the system and which interactions remain outside.

The more relevant bodies you include within the system, the fewer interventions remain outside. That's why defining the system is not a formality: it's the step that decides whether you can treat the set as isolated or not.

A quick practical test helps decide the boundary: ask whether the force you are considering appears in pairs within the chosen set. If the complete action-reaction pair stays inside, that force is internal; if you only see half the pair because the other body was left outside, the force is external to the system.

🔵 Formal level

Basic formulas

Leaf Nomenclature

Identifiers for this leaf: pi denotes the $\vec p_i$ of each particle with its $m_i$ and $\vec v_i$; the $\vec J_{ext}$ J is the cumulative action of the $\vec F_{ext}$ over a certain $\Delta t$; DeltaP expresses the resulting $\Delta \vec P$; and the condition cond represents the isolated system state.

Definition of Individual Momentum

The dynamic state of each body is described by its individual linear momentum, which is the product of its generic mass $m$ and its generic velocity $v$. In a composite system, this momentum $\vec p_i$ combines the $m_i$ and $\vec v_i$ of the body:

Definition of Linear Momentum

Definition of Total Momentum

In this section, we will analyze systems composed of multiple particles. The global dynamic state is defined by the system's $\vec P$:

Total system momentum

Which is the sum of the individual momenta of all parts:

Initial Linear Momentum (Sum)

Conservation Principle

When the $\vec F_{ext}$ is zero, the initial total momentum $\vec{P}_i$ equals the final one, regardless of how many internal interactions occur:

Principle of Conservation of Linear Momentum

Real student questions

• Do weight and normal force always cancel? Only on horizontal surfaces. On a ramp, weight has a component that accelerates the system if there is no friction.
• What about static friction? It's an external force. If we include the surface in the system, friction becomes internal and momentum is conserved including the mass of the Earth.

Final synthesis

The distinction between internal and external forces depends on the chosen boundaries. Expanding the system turns external forces into internal ones and strengthens conservation, but at the cost of adding more variables to the dynamic analysis.