A system is any collection of objects you choose to analyze together — a single particle, a block, an entire rocket, or several unrelated objects lumped together for convenience. The boundary is a choice, not a fact of nature, and that choice has consequences: a system's properties come from the interactionsbetween the objects inside it.
Systems aren't sealed off from the world, either — energy or mass can cross a system's boundary through interactions with the environment. Keeping track of what's "inside" versus "outside" your chosen system is a habit you'll lean on for the rest of this course, especially once momentum and energy transfer enter the picture.
For any object or system with a symmetrical mass distribution, the center of mass lies somewhere on its lines of symmetry. A uniform ruler's center of mass sits at its midpoint; a uniform disk's center of mass sits at its geometric center. No calculation required — symmetry alone tells you where to look.
For a system made of individual point masses, the center of mass along any axis is a mass-weighted average of position — each mass's position, weighted by how much of the total mass it represents.
Try it directly below: adjust three point masses and their positions and watch the center of mass respond in real time.
Three point masses on a 1D axis. Adjust each mass and position — the triangle marks the system's center of mass, updating live from x(cm) = Σmᵢxᵢ / Σmᵢ.
Three masses sit on the x-axis: 2 kg at x = 0 m, 3 kg at x = 2 m, and 5 kg at x = 6 m. Find the system's center of mass.
Real objects aren't a handful of discrete point masses — they're continuous distributions of mass. To find the center of mass of a continuous, nonuniform solid, replace the sum with an integral, treating the object as a collection of infinitesimal mass elements dm:
For a rod or other linear (1D) object, the linear mass density λ describes how mass is distributed along its length — and it's defined as a derivative: the rate of change of mass with respect to position.
If you're given a density function, the object's total mass comes from integrating that density over the object's length, area, or volume — one dimension for a rod, two for a plate, three for a solid.
Explore this directly below — compare a uniform rod (where symmetry alone gives you the answer) against a rod that gets denser toward one end (where you genuinely need the integral).
A 6 m rod with linear mass density λ(x). Toggle between a uniform rod (λ = k, constant) and a rod that gets denser toward one end (λ = kx). Darker shading means more mass packed into that stretch of rod.
For a uniform rod, x(cm) always lands at the midpoint (L/2) — no integration needed, symmetry alone tells you.
A rod of length L = 4 m has linear mass density λ(x) = 3x (kg/m), measured from one end at x = 0. Find the rod's total mass and its center of mass.
Once you know where a system's center of mass is, you can model the entire system as a single object located at that point — this is exactly the "object model" you first met all the way back in Lesson 1.2, now placed on solid mathematical footing.