Foundations — Centre of mass — derivation for common shapes (rod, triangle, semicircle, hemisphere)

Before you can read the parent derivations, you must own every symbol they throw at you. This page builds each one from nothing — plain words, then a picture, then why the topic needs it. Read top to bottom; each idea leans on the one above it.

---

0. The scale-and-average idea (the thing everything sits on)

The picture: think of a see-saw. Put a heavy kid at position $x_1$ and a light kid at $x_2$. The balance point is not the middle — it slides toward the heavy kid. That balance point is the weighted average, and it is exactly the centre of mass.

Why the topic needs it: every COM formula is this see-saw done with millions of tiny pieces. If you understand why the heavy side pulls the balance, you understand the whole chapter. See Centre of mass — definition and system of particles for the discrete version.

---

1. The symbols $m$, $M$, and $x_i$ (mass and position of a piece)

The picture: a row of beads on a wire. Each bead is a piece with its own mass $m_i$ sitting at its own spot $x_i$. Add all the bead masses → $M$.

The Greek capital sigma ==$\sum$== just means "add up all the things that follow it." So ${\sum }_i{m}_i{x}_i$ reads: "for every piece, multiply its mass by its position, then add all those products."

Why the topic needs it: the master formula $x_{cm}=\frac{\sum m_i{x}_i}{\sum m_i}$ is exactly the weighted average from step 0, written in symbols.

---

2. Coordinates and the vector $\vec{r}$ (where a point is, in any dimension)

The picture: an arrow drawn from the corner of your page (the origin) to a dot. Its horizontal shadow is $x$, its vertical shadow is $y$.

Why the topic needs it: the master formula is written once for a vector, ${\vec{r}}_{cm}=\frac{1}{M}\int \vec{r}dm$, and that single line secretly means three separate averages — one for $x$, one for $y$, one for $z$. Writing $\vec{r}$ instead of three formulas saves ink and shows the idea is dimension-blind.

---

3. Density: $\lambda$, $\sigma$, $\rho$ (how mass is spread out)

To turn a chunk into a mass number, you need to know how densely the mass is packed. There are three flavours depending on whether your body is a line, a sheet, or a solid.

The picture:

• A thin wire → mass smeared along a line → use $\lambda$.
• A flat sheet (a lamina) → mass smeared over an area → use $\sigma$.
• A solid lump → mass filling a volume → use $\rho$.

Why "density × size = mass": density answers "how much mass per unit of size?" Multiply by how much size you actually have, and you get mass. That's why: $dm=\lambda dl(\text{line}),dm=\sigma dA(\text{area}),dm=\rho dV(\text{volume}).$

---

4. The infinitesimal $dm$, $dx$, $dl$, $dA$, $dV$ (a "tiny piece")

The picture: take the rod and imagine cutting it into a huge number of paper-thin slices. One slice has width $dx$; the mass inside it is $dm$. The whole rod is these slices stacked back together.

Why the topic needs it: a continuous body has no separate beads to label with $i$. So we invent our own beads — the slices — each of mass $dm$. Now the row-of-beads idea from step 1 works again.

---

5. The integral $\int$ (adding up infinitely many tiny pieces)

The picture: the same stack of paper-thin slices — now we glue their contributions back together. The little numbers $0$ and $L$ (the limits) say where the slices start and stop.

Why the topic needs it: the master formula turns the bead sum into $x_{cm}=\frac{\sum m_i{x}_i}{\sum m_i}⟶x_{cm}=\frac{\int xdm}{\int dm}.$ Every shape in the parent note is just "pick the right $dm$, then run this integral." See Integration as continuous summation.

Recall Why does the density cancel for uniform bodies?

Because a constant (like $\lambda =M/L$) can be pulled out of both the top integral and the bottom integral — then it divides away. That's the reason a uniform body's COM is purely a geometric fact, equal to its centroid. ::: A constant factors out of numerator and denominator and cancels.

---

6. Symmetry (the shortcut that deletes half the work)

The picture: fold the semicircle along its vertical middle line — the two halves land perfectly on each other. For every bit of mass on the left there's an equal bit on the right, so their sideways pulls cancel: the COM's $x$-coordinate must be $0$.

Why the topic needs it: the parent note writes "by symmetry $x_{cm}=0$" for the semicircle and hemisphere. That single sentence replaces a whole integral. See Symmetry arguments in physics.

---

7. Centroid vs centre of mass (a naming clarification)

Why the topic needs it: all shapes in the parent note are uniform, so "find the COM" secretly means "find the centroid." The distinction only matters when density varies. Full comparison in Centroid vs centre of mass vs centre of gravity.

---

How these feed the topic

Each box is a symbol or idea you just learned; follow the arrows and you arrive at the master formula that powers every derivation in the parent topic.

---

Equipment checklist

Cover the answers and test yourself — if any line stumps you, re-read that section before tackling the derivations.

• What is a weighted average, in one sentence? ::: An average where each value counts in proportion to its mass (heavier values pull it toward themselves).
• What does ${\sum }_i{m}_i{x}_i$ tell you to do? ::: For every piece, multiply its mass by its position, then add all the products.
• What does the arrow in $\vec{r}$ mean, and how many numbers does it hold in 3D? ::: It's a position vector (arrow from origin to a point); in 3D it holds three numbers $(x,y,z)$.
• Give the units of $\lambda$, $\sigma$, $\rho$. ::: $\text{kg/m}$, ${\text{kg/m}}^2$, ${\text{kg/m}}^3$.
• Write $dm$ for a line, a sheet, and a solid. ::: $dm=\lambda dl$, $dm=\sigma dA$, $dm=\rho dV$.
• What does the symbol $dx$ physically represent? ::: The width of an infinitely thin slice, so thin that quantities are constant across it.
• How does $\int$ differ from $\sum$? ::: $\int$ adds infinitely many infinitesimal pieces; $\sum$ adds a finite list.
• Why does a constant density cancel out of $x_{cm}$? ::: It factors out of both numerator and denominator and divides away.
• What can a symmetry axis let you write down instantly? ::: The COM lies on it, so the off-axis coordinate is $0$.
• For a uniform body, centroid vs centre of mass? ::: They are the same point.