Parallel axis theorem — I = I_CM + Md² — proof: Foundations

Before you can follow the parent proof, you need to already own its alphabet. Below, each symbol comes with plain words → a picture → why the topic needs it, ordered so every new one leans only on the ones before it.

1. A rigid body, and chopping it into pieces

To do any physics with a shape, we pretend it is built from tiny bricks.

Figure — Parallel axis theorem — I = I_CM + Md² — proof

Add up all the chunk masses and you recover the whole: $\sum_{i} m_{i} = M$

2. Locating a chunk: coordinates $x_{i}, y_{i}$

To say where a chunk sits we need an address system.

3. Perpendicular distance $r_{i}$ and why we square it

By the Pythagoras rule on the address $(x_{i}, y_{i})$ with the axis through the origin: $r_{i}^{2} = x_{i}^{2} + y_{i}^{2}$

4. Moment of inertia $I$ : the star of the show

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

See Moment of inertia — definition for the full story of this quantity.

5. Center of mass (CM): the balance point

The address of the CM is the average of all chunk addresses, weighted by mass: $x_{C M} = \frac{\sum _{i} m _{i} x _{i}}{M}, y_{C M} = \frac{\sum _{i} m _{i} y _{i}}{M}$

See Center of mass — definition and computation for how to compute this.

6. Parallel axes and the distance $d$

If the second axis pierces the $x y$ -plane at address $(a, b)$ while the CM-axis sits at the origin, then by Pythagoras: $d^{2} = a^{2} + b^{2}$

7. How the foundations feed the theorem

Read top to bottom: chunks give us mass and addresses → addresses give distances → distances give $I$ → the CM kills the cross terms → parallel-axis geometry supplies $d$ → the theorem falls out.

8. A tiny worked sanity-check

9. Where these tools show up next

Radius of gyration — repackages $I = M k^{2}$ ; needs the same $I$ and $M$ .
Rotational kinetic energy — uses $I$ as the spin analogue of mass.
Perpendicular axis theorem — another axis-shifting shortcut built on $\sum m_{i} r_{i}^{2}$ .
Physical pendulum — a direct customer of the parallel-axis theorem.
Back to the full proof: parent (Hinglish).

Equipment checklist

Recall Self-test: can you explain each in one plain sentence?

What is a rigid body? ::: A solid whose parts never move relative to each other — it spins as one frozen shape. What does $m_{i}$ mean? ::: The mass of the $i$ -th tiny chunk the body is imagined to be sliced into. What does $\sum_{i}$ do? ::: Adds a quantity over every chunk $i$ — shorthand for a huge chain of plus-signs. What is $M$ ? ::: The total mass, $\sum_{i} m_{i}$ — what a scale reads. What do $x_{i}, y_{i}$ represent? ::: The sideways and upward address of chunk $i$ from a chosen origin. Why can we ignore $z_{i}$ ? ::: Distance to a vertical axis depends only on sideways position, not on height along the axis. What is $r_{i}$ and why squared? ::: The perpendicular distance from a chunk to the axis; $r_{i}^{2} = x_{i}^{2} + y_{i}^{2}$ , and inertia uses the square so far mass counts far more. What is the moment of inertia $I$ ? ::: A number, $\sum_{i} m_{i} r_{i}^{2}$ , measuring how hard the body is to start or stop spinning about an axis. What is $\int r^{2} d m$ ? ::: The continuous version of $\sum m_{i} r_{i}^{2}$ for a smooth body — same idea, infinitely fine chunks. What is the center of mass? ::: The mass-weighted average position — the point where the body balances. Why do $\sum m_{i} x_{i}$ and $\sum m_{i} y_{i}$ vanish in the proof? ::: Because the origin is placed at the CM, where the mass-weighted position averages to zero. What are parallel axes? ::: Two lines pointing the same direction that never meet. What is $d$ ? ::: The perpendicular (shortest, right-angle) distance between the two parallel axes.