Center of Mass

The center of mass is simply the balance point of an object: the spot where you could rest it on your fingertip and it wouldn't tip. It's where gravity acts as if all the weight were gathered in one place, which is why engineers need it for anything that must not tip over, from cranes to airplanes. Start with something you know, a seesaw. A heavy kid close to the middle can balance a light kid far out, because what matters is weight × distance (that product is called a moment), and the balance point is where the moments on each side cancel. The formula $\bar x = \frac{1}{A}\int x\,[f(x) - g(x)]\,dx$ looks busy, but it's just $x$ (how far over a thin strip is), $[f(x) - g(x)]\,dx$ (how much plate that strip holds), the integral (add up distance × amount for every strip), and $\frac1A$ (divide by the total to turn it into an average). Bear with it; it's a weighted average and nothing more.

A plate works the same way, just with infinitely many tiny pieces of weight. So cut the region into thin vertical strips, the same strips we used for area. A strip at position $x$ has area $[f(x) - g(x)]\,dx$, and since it's thin, its own balance point is halfway up, at height $\tfrac12[f(x) + g(x)]$.

• The moment about the y-axis adds up position × area: $M_y = \int x\,[f(x) - g(x)]\,dx$.
• The moment about the x-axis adds up height × area: $M_x = \int \tfrac12[f(x) + g(x)]\,[f(x) - g(x)]\,dx = \int \tfrac12\left(f^2 - g^2\right)dx$.

Then $\bar x = M_y/A$ and $\bar y = M_x/A$. Think about why dividing by $A$ makes sense: $M_y$ is a total of position × area, and dividing a total by the amount gives an average. So $\bar x$ is literally the average $x$-position of the plate. The density would appear on top and bottom and cancel for a uniform plate, so the center of mass depends only on the shape. That's why it's also called the centroid.

Use symmetry. If the region is symmetric about a line, the centroid lies on that line, since the moments on either side cancel. A semicircle centered on the y-axis has $\bar x = 0$ without any integration.

Pappus's theorem ties this lesson back to volumes: rotating a region about an axis gives a solid of volume $V = 2\pi\bar r A$, where $\bar r$ is the distance from the centroid to the axis. In words, the volume is the area times the distance the centroid travels around.

An aircraft's center of mass must sit ahead of its wings' lift or it pitches up uncontrollably, so loading crews compute it before every flight. Cranes, robot arms and ship hulls are all designed around where the mass balances.