Defining a System

In physics, a system is any collection of objects that you choose to study together. The boundary you draw around that collection is called the system boundary. Everything inside the boundary is part of the system; everything outside is the surroundings (or environment). This might sound like a bookkeeping trick, but it is one of the most powerful ideas in mechanics — choosing your system carefully can turn a complicated problem into a straightforward one. A system can be as small as a single electron or as large as a galaxy. Once you have defined a system, every force acting on any object inside it falls into one of two categories. Internal forces are forces that objects within the system exert on each other — for example, the tension in a rope between two skaters holding hands. External forces are forces exerted on the system by objects in the surroundings — for example, the ground pushing up on one of the skaters. This distinction is crucial: Newton's third law guarantees that internal forces always come in equal and opposite pairs, so they cancel when you consider the system as a whole. The total momentum of a system changes only if a net external force acts on it. Internal forces, no matter how large, cannot change the system's total momentum. This principle underpins the law of conservation of momentum and allows us to define and locate the centre of mass — the single point that moves as if all the system's mass were concentrated there and all external forces acted upon it.

Key takeaways
  • Drawing the system boundary
  • Identifying internal forces
  • Identifying external forces
  • Why the distinction matters: an analogy
  • Flexible system choice: worked strategy

Drawing the system boundary

Choose which objects belong to your system before you write a single equation. Draw an imaginary closed surface (the system boundary) around those objects. This is often called a free-body boundary. There is no single 'correct' boundary — your choice is a tool, and a clever choice simplifies the algebra. For instance, if two trolleys collide, you can define the system as 'both trolleys together', eliminating the collision force from your equations entirely.

Identifying internal forces

Any force whose action and reaction both lie inside the system boundary is internal. Examples: gravitational attraction between two planets in a binary system; the normal force between two books stacked on each other (if both are in the system); tension in a string connecting two blocks sliding together. By Newton's third law, every internal force F⃗ is paired with an equal and opposite force −F⃗, also internal. When you sum all forces on the system, internal forces sum to zero — they contribute nothing to the net force on the system.

Identifying external forces

Any force exerted on a system object by something outside the system boundary is external. Examples: gravity from the Earth acting on a ball (if the Earth is not in your system); air resistance; a hand pushing a trolley. Only external forces can accelerate the centre of mass or change total momentum. When solving problems, list every external force, find their vector sum F⃗_net, and apply Newton's second law to the whole system: F⃗_net = M·a⃗_cm.

Why the distinction matters: an analogy

Think of a football team as the system. Arguments between teammates (internal conflicts) don't change the team's overall position in the league — only outside opponents (external forces) do. Likewise, internal forces between particles may transfer energy or momentum from one particle to another inside the system, but they cannot alter the system's total momentum. This is why a person cannot lift themselves off the ground by pulling on their own shoelaces — that pull is an internal force.

Flexible system choice: worked strategy

Different choices of system give different equations — all equally valid. (A) If you want the acceleration of just one block, define a single-block system; the rope tension becomes external. (B) If you want to avoid finding tension, include both blocks in one system; tension becomes internal and vanishes from the net-force equation. Always state your system choice explicitly at the start of a solution. Examiners and markers look for this declaration as evidence of clear physical reasoning.

The Key Equation

F⃗net, ext=M a⃗cm\vec{F}_{\text{net, ext}} = M\,\vec{a}_{\text{cm}}
  • \vec{F}_{\text{net, ext}} — Net external force (N (newtons)): The vector sum of every force exerted on the system by objects outside the system boundary. Internal forces cancel and do not appear here.
  • M — Total mass of system (kg (kilograms)): The scalar sum of the masses of every object inside the system boundary: M = m₁ + m₂ + … + mₙ.
  • \vec{a}_{\text{cm}} — Acceleration of centre of mass (m s⁻² (metres per second squared)): The acceleration of the single point that represents the system's average mass-weighted position. Only external forces determine this quantity.

Worked Example

Problem. Two ice skaters, Alice (mass 50 kg) and Bob (mass 70 kg), stand at rest on frictionless ice. Alice pushes Bob with a force of 140 N for 0.5 s. (a) Define the system as 'Alice + Bob'. What is the net external force on this system, and what happens to the system's total momentum? (b) Now define the system as 'Bob alone'. What is the net external force on Bob, and what is his velocity after the push?

vBob=1.0 m s−1,Δpsystem=0N ⋅ sv_{\text{Bob}} = 1.0 \text{ m s}^{-1}, \quad \Delta p_{\text{system}} = 0 \text{N}\,\cdot\,\text{s}

Answer. Part (a): Net external horizontal force = 0 N; total momentum remains 0 kg m s⁻¹. Part (b): Bob's velocity = 1.0 m s⁻¹ in the direction of the push.

Figure
Diagram
System Boundary: Internal vs External ForcesSystem boundary (dashed)A50 kgB70 kgInternal: F_ABInternal: F_BA = −F_ABW_AN_AW_BN_BF_ext(external)Internal forces (cancel in sum)External forceWeight/Normal
Origins · 1 minIsaac Newton and how this was discovered
IN
Isaac Newton
1643–1727 · English

Newton's three laws of motion, published in his 1687 masterwork Philosophiæ Naturalis Principia Mathematica, established the framework for defining systems of particles. His third law — that every force has an equal and opposite reaction — is the direct reason internal forces cancel within a system. Newton also laid the mathematical groundwork for the concept of the centre of mass, treating extended bodies as if their mass were concentrated at a single point when calculating gravitational effects. Newton reportedly kept his greatest discoveries — calculus, the law of gravitation, and the nature of light — largely private for nearly 20 years before colleagues persuaded him to publish. The Principia was only written after astronomer Edmond Halley personally financed its printing and badgered Newton into completing it.

  • Centre of Mass — Once a system is defined, the centre of mass is the key point whose motion is governed solely by the net external force — it is the direct application of system thinking.
  • Conservation of Momentum — When the net external force on a system is zero, total momentum is conserved — this law is a direct consequence of correctly identifying that internal forces cancel within a system.
  • Newton's Third Law — Newton's third law is the reason internal forces always come in equal-and-opposite pairs and therefore sum to zero for any defined system.
  • Free-Body Diagrams — A free-body diagram is essentially a visual representation of a chosen system boundary, showing only the external forces acting on that system — directly linked to the concept of defining a system.