Moment of inertia I = Σmᵢrᵢ² — concept: Foundations

This page assumes you have seen nothing. Before we can even read $I = \sum_{i} m_{i} r_{i}^{2}$ out loud, we must earn every symbol in it: the summation sign, the little subscript $i$ , what $m_{i}$ and $r_{i}$ mean, what "perpendicular distance to an axis" looks like, and the two tools ( $v = ω r$ and $K E = \frac{1}{2} m v^{2}$ ) that the derivation leans on. We build them one at a time.

1. What is an "axis of rotation"? (the picture everything hangs on)

Before distances, before mass, we need the thing we measure distance from.

Figure — Moment of inertia I = Σmᵢrᵢ² — concept

Look at the figure. The violet vertical line is the axis. The blue dot sits right on it — when the disk spins, that dot never moves. The magenta dot sits off to the side — it rides around in a circle. The orange dot is even farther out, so its circle is bigger.

Why the topic needs this: moment of inertia is always about a chosen axis. Change the line, and every distance changes, so $I$ changes. There is no "moment of inertia" without first pointing at an axis.

2. Mass $m$ — the "how much stuff" number

The picture: think of $m$ as the weight of a bag of marbles — it doesn't care where the bag is, just how full it is.

Why the topic needs it: mass is the raw "amount of stuff" that resists motion. Spinning-resistance must start from ordinary mass, then get modified by geometry.

3. The subscript $i$ and many particles $m_{i}$

Real objects are not one lump — imagine chopping the body into tiny pieces and labelling them $1, 2, 3, \dots$

The picture: a body dotted into little coloured balls, each with its own tag.

Why the topic needs it: different pieces of an object sit at different distances from the axis, so we must treat them separately — hence we tag each one with $i$ .

4. The summation sign $\sum$

Why the topic needs it: an object has thousands of pieces. Writing every " $+$ " would fill a page. $\sum$ lets us say "add over all pieces" in one stroke.

5. Perpendicular distance $r_{i}$ — the subtle one

This is the symbol students most often get wrong, so we give it its own figure.

Read the figure carefully:

The violet line is the axis.
The orange dashed segment drops from the mass perpendicular (at a right angle, shown by the little square) onto the axis. Its length is $r_{i}$ — the radius of the circle that mass will sweep.
The grey dotted line is the (longer) distance to the origin. That is the wrong thing to use.

Why the topic needs it: the size of a piece's circle — how far it must swing — is exactly this perpendicular distance. That is what makes far-out mass so hard to spin.

6. Angular velocity $ω$ and the link $v = ω r$

To connect spinning-speed to ordinary speed, we need one more idea.

Now the key bridge. A piece at perpendicular distance $r$ rides a circle of radius $r$ . In one second the whole body turns through angle $ω$ , so the piece travels an arc of length $ω r$ . That arc length per second is its ordinary straight-line speed $v$ :

The figure shows two dots on the same spinning arm: the inner (magenta) traces a small circle slowly, the outer (orange) traces a big circle fast — but both complete a turn in the same time, so their $ω$ is identical while their $v$ differ.

Why the topic needs it: this is the exact spot where distance $r$ sneaks into the physics. See Angular velocity ω for the full story.

7. Kinetic energy $K E = \frac{1}{2} m v^{2}$ — the foundation we trust

The picture: a fast heavy truck carries a lot of $K E$ ; a slow marble almost none. The " $\frac{1}{2}$ " is a bookkeeping constant, and $v^{2}$ says energy grows with the square of speed.

Why the topic needs it: this is the rock-solid linear fact the whole derivation stands on. When we plug $v = ω r$ into this $v^{2}$ , the square lands on $r$ — and that is precisely why $I$ contains $r^{2}$ and not $r$ . See Kinetic energy ½mv².

Recall Why the

square on $r$ (and not just $r$ ) is inevitable Because $K E$ carries $v^{2}$ , and $v = ω r$ means $v^{2} = ω^{2} r^{2}$ . The square is inherited — it was never a free choice. Contrast this with Centre of mass (first moment Σmr), which uses $r$ to the first power because it comes from a different question (balance point, not energy).

8. Putting the symbols together

Now — and only now — every symbol in the parent formula is earned:

$I = \sum_{i} m_{i} r_{i}^{2}$

Reading it aloud with everything we built:

Where each part came from:

$\sum_{i}$ — add over all pieces (§4)
$m_{i}$ — each piece's mass (§3)
$r_{i}$ — perpendicular distance to the chosen axis (§5)
the square — inherited from $v^{2}$ in $K E$ via $v = ω r$ (§6, §7)

The parent note then feeds $I$ into Rotational kinetic energy ½Iω², Angular momentum L = Iω, Torque τ = Iα, and refines it with the Parallel axis theorem.

Prerequisite map

Equipment checklist

Test yourself — cover the right side and answer each before revealing.

What is the axis of rotation, in one sentence?

The fixed straight line a body spins around; points on it don't move, all others sweep circles about it.

What does

m

mean and its unit?

Mass — the amount of matter — measured in kilograms (

kg

What is the subscript

i

m_{i}

A counter labelling each piece of the body: piece 1, piece 2, and so on.

Expand

\sum_{i} x_{i}

in plain words.

Add up

x_{1} + x_{2} + x_{3} + \dots

, one term for every piece

i

What exactly is

r_{i}

The perpendicular (right-angle) distance from piece

i

to the axis line — NOT the distance to the origin.

A mass sits far along the axis but right on the line — what is its

r_{i}

Zero, so it contributes nothing to

I

What is

ω

and what do all pieces of a rigid body share?

Angular velocity in

rad/s

; every piece shares the same

ω

State the bridge equation linking

v

and

ω

v = ω r

— bigger perpendicular distance means bigger straight-line speed.

Write kinetic energy and name its crucial feature.

K E = \frac{1}{2} m v^{2}

; the crucial feature is the

v^{2}

(energy grows with the square of speed).

Why does

r

appear squared in

I

Because

K E

has

v^{2}

and

v = ω r

, so

v^{2} = ω^{2} r^{2}

— the square lands on

r