Foundations — Applications — mass, centre of mass, moments of inertia

Before you can compute a single moment of inertia, you must be able to read the notation without stumbling. This page introduces each symbol one at a time, in an order where every new thing is built from the previous thing — never used before it is defined and drawn.

0. The stage: a flat plate sitting in the plane

Everything happens on a plate — a thin flat shape (a "lamina"). We lay it flat on a sheet of paper and draw two number lines through it: a horizontal one and a vertical one.

Why we need this. Every later symbol ($\rho$, $dA$, $M_x$, $I_y$…) is a rule that reads off a point's $x$ and $y$, both measured from the origin. If you cannot point to where the origin, $x$ and $y$ live in the picture, none of the formulas mean anything. You will meet these axes again the moment you set up integration limits in Double Integrals over General Regions.

1. The region $R$ — the shape we sum over

Look at the triangle in the figure below: $R$ is the entire shaded interior. When we write ${\iint }_R$, the little $R$ underneath is telling us "only add up tiles that live inside this shape."

Why we need it. A real plate is not the whole infinite plane — it is a bounded shape. $R$ names that shape so the integral knows where to start and stop. Describing $R$ as inequalities (like $0\le x\le 1,0\le y\le 1-x$) is exactly the skill of Double Integrals over General Regions.

2. The tiny tile and its area $dA$

Here is the heart of the whole subject. We cannot handle a spread-out plate all at once, so we chop it into a grid of tiny rectangular tiles.

Why the word "element"? $dA$ is the building element — the brick. The whole plate is millions of these bricks. An integral is the machine that adds all the bricks back together as they shrink to zero size.

3. The integral signs $\int$ and $\iint$

${\iint }_R(\text{something})dA={\lim }_{\text{tiles}\to 0}{\sum }_{\text{all tiles in }R}(\text{something at that tile})\times (\text{tile area})$

Why two signs and not one? A plate is two-dimensional: to cover it you sweep in both the $x$ direction and the $y$ direction. Each direction contributes one $\int$. Doing them one after the other (inner then outer) is the whole method of Double Integrals over General Regions; a solid body needs three signs, i.e. Triple Integrals.

4. Density $\rho$ — how heavy each tile is

The Greek letter $\rho$ (say "roh", the curly-p letter) is our name for density.

The mass of ONE tile is then

$dm=\rho (x,y)dA⟸\underset{\rho }{\underbrace{\text{(mass per area)}}}\times \underset{dA}{\underbrace{\text{(area of tile)}}}.$

Why we need $\rho$. Without it, a large tile and a small tile would count equally, and a heavy region would count the same as a light one. Density is what turns geometry (area) into physics (mass).

5. Distance from an axis — the "lever arm"

The last raw ingredient is distance. Three distances matter, and each is just a coordinate you already own.

Why $\sqrt{x^2+y^2}$? That is the Pythagoras theorem: the straight hop is the hypotenuse of a right triangle whose two legs are $x$ (across) and $y$ (up). Square each leg, add, square-root — you get the diagonal. The picture above draws exactly this triangle. (Squaring also throws away any minus sign, which is why $r$ is always $\ge 0$.)

Why the sign matters. Imagine a plate that straddles the $y$-axis, equal and opposite. If we used $|x|$ everywhere, the two halves would add and push the balance point off-centre — wrong. Using signed $x$, the left half's negative contributions subtract from the right half's positive ones, and the balance point lands correctly in the middle. Keep this in mind whenever a region crosses an axis.

6. First power vs second power — the whole plot in one line

Now every symbol is defined, the three recipes read cleanly. Watch only the power of the coordinate:

Because there are two axes, the moment and the inertia each come in two labelled versions — one per axis. Here is the notation the parent topic uses, defined now so it never appears unannounced:

• Power $0$ (multiply by $1$) → mass $m$. You are just counting stuff.
• Power $1$ (multiply by the signed coordinate) → moment $M_x$ or $M_y$; divide by $m$ to get the balance point. First power because balance is about leverage, which is linear — and signed so opposite sides can cancel.
• Power $2$ (multiply by coordinate²) → moment of inertia $I_x,I_y,I_0$. Second power because spin energy grows with speed², and speed grows with radius. See Rotational Kinetic Energy and Angular Momentum for that $\frac{1}{2}I{\omega }^2$ story.

Why divide by $m$ for the balance point? The moment $M_y$ is "total sideways leverage". A single point mass $m$ placed at the balance point $(\bar{x},\bar{y})$ must reproduce that same leverage, so $m\bar{x}=M_y$, giving $\bar{x}=M_y/m$. When $\rho$ is constant it cancels and the balance point becomes the pure-geometry centroid — the subject of Centroids and the Pappus Theorems.

7. The bar notation $\bar{x},\bar{y}$

8. A word about polar tiles (preview)

Sometimes a round region is far easier in polar coordinates, where a point is named not by $(x,y)$ but by its distance from the origin and a direction.

In these coordinates the tile is not a rectangle but a curved wedge, and its area is

$dA=rdrd\theta (\text{the extra }r\text{ is mandatory}),$

where $dr$ is a tiny step outward and $d\theta$ a tiny turn. That extra $r$ is the Jacobian factor; the full story lives in Polar Coordinates and the Jacobian and its general version Change of Variables and Jacobians. For now just log the symbols $r$ and $\theta$ so they are not a shock later.

Prerequisite map

Read it bottom-up: the origin anchors the axes, axes give you points, points build the region and the lever arms, the region gets chopped into tiles, density turns tile-area into tile-mass, the integral adds them, and the power you raise the coordinate to selects mass vs moment vs inertia.

Equipment checklist

Test yourself — cover the right side and answer each aloud.

This is the full toolkit assumed by the parent topic. Once every checklist line is automatic, the formulas there read like sentences.