Foundations — Rotational kinetic energy = ½Iω²

This page assumes nothing. Before you can trust $K=\frac{1}{2}I{\omega }^2$ (see the parent topic), you must own every symbol inside it. We build them one at a time, each picture leaning on the last.

1 — A point mass: $m$

Picture the smallest thing we can: one dot, one particle. It has no shape, no spin — just a lump of matter. Everything else on this page is built by gluing many such dots together.

Why the topic needs it: the parent note "chops the body into tiny pieces $m_1,m_2,\dots$." Those pieces are point masses. If you don't picture a single dot first, the sum ${\sum }_i{m}_i$ is meaningless.

2 — Speed $v$ and the kinetic energy of a moving dot

Look at the figure. Two identical dots (same $m$), but the right one moves twice as fast. Its arrow is twice as long — yet its energy bar is four times taller. That is the meaning of the little ${}^2$ (the "square"): the energy grows with speed multiplied by itself, so doubling $v$ quadruples $K$.

Why the topic needs it: $\frac{1}{2}m{v}^2$ is the single brick the whole formula is built from. The parent literally writes $K={\sum }_i\frac{1}{2}{m}_i{v}_i^2$. See Kinetic energy of a particle.

3 — The axis, the radius $r$, and going in circles

Now we stop letting the dot fly free and pin the body to a fixed line — the axis of rotation.

The figure shows the spinning body from the front. The axis is the dot in the middle (it points straight at you, out of the page). Three particles sit at different radii $r_1<r_2<r_3$. Each sweeps its own circle. The rim particle traces the biggest loop; the inner one barely moves.

Why the topic needs it: the moment of inertia is ${\sum }_i{m}_i{r}_i^2$ — it lives or dies on $r$. No radius, no formula. This is the geometric heart of Moment of inertia.

4 — Angle $\theta$, radians, and angular velocity $\omega$

Here is the magic that makes the whole crowd manageable: they all turn together.

The key insight, drawn in the figure: in one tick of the clock, every particle sweeps the same angle $\Delta \theta$ — the inner one and the rim one turn through identical wedges. So they share one single $\omega$. That is what "rigid body" means: no particle races ahead of another.

But the rim particle covers a longer arc in that same wedge (its circle is bigger). Longer arc in the same time = faster speed. This gives the bridge equation:

$v=r\omega$

Why: differentiate $s=r\theta$ with respect to time and $r$ is constant, so $\frac{ds}{dt}=r\frac{d\theta }{dt}$, i.e. $v=r\omega$. See Angular velocity.

5 — The summation sign $\sum$

Why the topic needs it: total energy is the sum of each dot's energy. $\sum$ is just "add them all" written compactly. When you see ${\sum }_i{m}_i{r}_i^2$, read it as "for each particle, multiply its mass by its radius squared, then add everything."

6 — Moment of inertia $I$: bundling geometry into one number

Now watch the parent's derivation assemble from the bricks above:

$K={\sum }_i\frac{1}{2}{m}_i{v}_i^2={\sum }_i\frac{1}{2}{m}_i(r_i\omega {)}^2=\frac{1}{2}{\omega }^2{\sum }_i{m}_i{r}_i^2$

The step-by-step why lives in the parent note; here we only name the leftover.

The figure contrasts two bodies of the same total mass: a disc (mass spread all the way in) and a hoop (mass parked at the rim). The hoop's mass has bigger $r$, and because $r$ is squared, its $I$ is larger — it is harder to spin and stores more energy at the same $\omega$. Distribution, not just amount, is what $I$ captures.

Why the topic needs it: $I$ is the whole reason $K=\frac{1}{2}I{\omega }^2$ looks like $\frac{1}{2}m{v}^2$. It is the "$m$ in costume." Deep dive: Moment of inertia.

7 — Putting on the costume: the final analogy

Every symbol on the right is now something you built with your own hands. That is the whole foundation the parent topic stands on.

Prerequisite map

Equipment checklist