Moment of inertia of - rod (about end, centre), disk, ring, sphere (solid, hollow), cylinder: Foundations

Before you can read the parent note, you need to own every symbol it fires at you. This page builds each one from nothing, in the order that lets the next one make sense. Nothing here assumes you have seen the topic before.

0. The picture everything lives in

Every formula in the parent topic is about ONE setup: a solid body, and a straight line through space called the axis, that the body spins around. Fix that picture in your head first.

Figure — Moment of inertia of - rod (about end, centre), disk, ring, sphere (solid, hollow), cylinder

Why does the topic need an axis at all? Because "how hard to spin" has no answer until you say spin around what. The same rod is easy to spin one way and hard another. The axis is the reference for every distance we are about to measure.

1. Mass — $m$ and $M$

Picture: a heavy bowling ball vs a beach ball of the same size — the bowling ball has more mass.
Why the topic needs it: moment of inertia is the rotational cousin of mass. To build the cousin, we first need mass itself. Big $M$ is the grand total; little $m_{i}$ is the mass of the $i$ -th tiny chunk when we chop the body up.

The little subscript $i$ just means "the $i$ -th piece" — piece number 1, 2, 3, … We chop the body into many pieces and label them.

2. Distance from the axis — $r$

This is the single most important symbol in the chapter, and the one people get wrong.

Picture: stand a pencil (the axis) upright. Hold a marble out to the side. $r$ is how far the marble is sideways from the pencil — not the slanted distance, the straight-across distance.
Why perpendicular? When the body spins, each piece travels a circle. The radius of that circle is exactly this perpendicular distance. A piece directly above the axis (small sideways gap) makes a tiny circle; a piece far out makes a huge circle.

3. Angle and angular speed — $θ$ , $ω$

Picture: a clock hand. $θ$ = where the hand points now; $ω$ = how fast the hand sweeps.
Key fact everyone on the body shares one $ω$ : in a rigid body, every piece turns through the same angle in the same time. The near pieces and far pieces all have the same $ω$ — but the far ones move faster in ordinary speed, because their circle is bigger.

That last sentence is the seed of the whole chapter. Let us make "move faster" precise.

4. Linear speed $v$ , and the bridge $v = ω r$

Why this formula? In one full turn a piece travels the circumference $2 π r$ . If it does $ω / (2 π)$ turns per second, its speed is $ω / (2 π) \times 2 π r = ω r$ . The $2 π$ s cancel — clean.
Picture: on a merry-go-round the kid at the rim ( $r$ big) is flung fast; the kid near the pole ( $r$ small) barely moves. Same $ω$ , different $v$ , because $v = ω r$ .
Why the topic needs it: this is the link that turns a spinning problem into a kinetic-energy problem, which is where $r^{2}$ is born (next section).

5. Kinetic energy — $K E = \frac{1}{2} m v^{2}$ — and WHERE $r^{2}$ COMES FROM

Now watch the birth of the whole chapter. Take one piece $m_{i}$ at distance $r_{i}$ . Its speed is $v_{i} = ω r_{i}$ . Its energy: $K E_{i} = \frac{1}{2} m_{i} v_{i}^{2} = \frac{1}{2} m_{i} (ω r_{i})^{2} = \frac{1}{2} m_{i} ω^{2} r_{i}^{2}$

Add up every piece. The $\frac{1}{2}$ and $ω^{2}$ are the same for all pieces, so pull them out: $K E = \frac{1}{2} ω^{2} (\sum_{i} m_{i} r_{i}^{2})$

The bracket $\sum_{i} m_{i} r_{i}^{2}$ is exactly what we name $I$ . See Rotational Kinetic Energy for the full story of this substitution.

6. The sum symbol $\sum$ and the integral $\int$

Picture: $\sum$ is stacking real bricks and adding their weights. $\int$ is pouring sand — you cannot count grains, so you add up continuously.
Why the topic needs $\int$ : a rod or disk is not a few beads; it is continuous matter. To add up $r^{2}$ over a continuous body we replace $\sum m_{i} r_{i}^{2}$ by $\int r^{2} d m$ , where $d m$ is the mass of one infinitesimal sliver.

7. Density — $λ$ , $σ$ , $ρ$ — turning geometry into $d m$

To do the integral we must express the sliver's mass $d m$ using the body's shape. That is what density does.

Picture: $λ$ = grams per centimetre of a wire; $σ$ = grams per square centimetre of a sheet of paper; $ρ$ = grams per cubic centimetre of a metal block.
Why the topic needs them: the integral $\int r^{2} d m$ cannot be done until $d m$ is written in terms of a coordinate you can integrate over ( $d x$ , $d r$ , $d θ$ ). Density is the converter, and it is uniform (same everywhere) for every standard body in this chapter.

8. The shape symbols — $L$ , $R$

Do not confuse $R$ (a fixed property of the body) with $r$ (the moving distance-to-axis of a sliver, which sweeps from $0$ up to $R$ as you scan across the body).

9. The two theorems the parent leans on

The parent note checks its answers with two shortcuts. You do not need to prove them here, only to know what they claim.

One more idea the topic quietly uses: the Radius of Gyration, a single distance $k$ with $I = M k^{2}$ — "if all the mass sat at $k$ , you'd get the same $I$ ."

Prerequisite map

Equipment checklist

Recall Are you ready? (cover the answers)

What is the axis of rotation, in one line? ::: The fixed straight line the body spins around; points on it don't move. What exactly does $r$ measure? ::: The perpendicular (shortest, right-angle) distance from a mass piece to the axis — NOT to the centre. Why does $r$ appear squared in $I$ ? ::: One power from $v = ω r$ , a second from squaring $v$ in $\frac{1}{2} m v^{2}$ . What is $ω$ and who shares it on a rigid body? ::: Angular speed in rad/s; every piece of a rigid body shares the same $ω$ . State the bridge between $v$ and rotation. ::: $v = ω r$ . What do $λ$ , $σ$ , $ρ$ mean and give their $d m$ ? ::: mass/length ( $d m = λ d x$ ), mass/area ( $d m = σ d A$ ), mass/volume ( $d m = ρ d V$ ). Why replace density using $M$ at the end? ::: So the final $I$ contains only measurable totals $M$ , $R$ or $L$ . Difference between $R$ and $r$ ? ::: $R$ is the body's fixed radius; $r$ is the running distance-to-axis of a sliver, sweeping $0 \to R$ . State the Parallel Axis Theorem. ::: $I = I_{c m} + M d^{2}$ , with $d$ the gap between parallel axes. Why can't you compute $I$ without naming an axis? ::: "Hard to spin" is undefined until you say spin about what — the axis sets every $r$ .