---
title: "Moment-of-Inertia Designer Interactive Lab | Sandbox Physics"
description: "Which distances matter to rotational inertia? Distances from the rotation axis you choose. Each mass adds its mass times the square of that distance, so moving a bead twice as far out makes its share four times larger. Placing the axis some distance from the center of mass adds the total mass times that distance squared."
canonical_url: "https://sandboxphysics.com/labs/inertia-designer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/inertia-designer/"
last_modified: "2026-10-22"
---

M051 · Rotation and mass distribution

# Moment-of-Inertia Designer

**Which distances matter to rotational inertia?** Distances from the rotation axis you choose. Each mass adds its mass times the square of that distance, so moving a bead twice as far out makes its share four times larger. Placing the axis some distance from the center of mass adds the total mass times that distance squared.

Build a three-dimensional rotor from a ring-shaped carrier and two symmetric pairs of beads. Drag the beads or the bearing in the coordinate view to set the starting layout while the total mass stays fixed, then apply a calibrated torque. Watch the bearing, the moving mass centre and the signed vectors along the axis. The parts of the moment of inertia, the angular speed and the work update together, and a faint curve shows a centred uniform disk of the same mass and radius for comparison.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

The rotor is an ideal rigid, flat body turning in a horizontal plane on a fixed bearing, driven by a pure couple. Each bead carries one quarter of the total bead mass. The ring-shaped carrier is uniform and the spokes are massless. Changing the layout resets the starting state instead of reshaping a spinning rotor. Under constant torque the motion is solved exactly, and the work and angular-impulse balances are checked independently.

Physics tutorial

## Distance from the axis is the design variable

**Background** The moment of inertia weights each bit of mass by the square of its perpendicular distance from the chosen rotation axis. Balanced layouts with the same total mass can have many different moments of inertia.

**Why it matters** OpenStax University Physics sections 10.5, 10.7 and 10.8 connect the inertia of spread-out mass, applied torque and rotational work. Here all three come from the same rotor, which you can rearrange.

Start with the essentials

- **Focus question**: How can you change angular acceleration without changing total mass or torque?
- **One-sentence intuition**: Pick the axis first: a bead moved twice as far from it contributes four times as much, and moving the bearing adds the parallel-axis term.

### Core mathematical model

#### Moment of inertia about the bearing

$$
I=\int r_\perp^2\,\mathrm{d}m=I_{\mathrm{CM}}+Md^2
$$

The rotor is balanced, so its mass centre sits at the centre of the carrier. The bearing offset is measured from that centre.

#### Uniform annulus and bead pairs

$$
I_{\mathrm{CM}}=\frac{M(1-f)}2(R^2+R_{\mathrm{hole}}^2)+\frac{Mf}2(r_x^2+r_y^2)
$$

The total bead mass is shared equally among four beads. The spokes that hold them have no mass.

#### Rotation under a constant couple

$$
\dot\omega=\frac\tau I,\qquad\theta(t)=\omega_0t+\frac{\tau t^2}{2I}
$$

During this trial only the angular speed changes; the mass distribution and the bearing stay fixed.

#### Work and angular impulse, checked separately

$$
\Delta K=\tau\Delta\theta,\qquad\Delta L=\tau t,\qquad K=\frac12I\omega^2
$$

Work and angular impulse use the same signed torque, but work builds up with the angle turned and angular impulse with the time elapsed.

### Common difficulties

#### Same mass, different inertia

**Typical misconception** Only the total mass determines rotational response.

**Better mental model** Compare a layout with the beads close in and one with them at the rim, under the same torque. Once each mass is weighted by its squared distance from the axis, the totals differ.

#### A moment of inertia needs an axis

**Typical misconception** A body has one moment of inertia, whatever the axis.

**Better mental model** Shift the bearing and look at the parallel-axis contribution it adds.

#### Rearranging the beads starts a new run

**Typical misconception** Dragging a bead during playback conserves angular momentum.

**Better mental model** Changing the layout restarts this rigid-body trial. Reshaping a body while it spins would need a separate model for the work done by internal forces.

### Run the experiment

1. 01

   #### Design a fast response

   Move the bead pairs close to the axis and complete the torque trial.

   **What to observe:** A smaller moment of inertia gives a larger angular acceleration.

2. 02

   #### Compare a rim and a disk

   Use the rim preset and compare it with the disk reference curve.

   **What to observe:** The same total mass placed farther from the axis speeds up more slowly.

3. 03

   #### Move the bearing

   Drag the red marker without changing the mass or the torque.

   **What to observe:** The moment of inertia gains a parallel-axis term.

4. 04

   #### Brake, then reverse

   Set a positive initial spin and a negative torque.

   **What to observe:** The motor’s work first takes energy out; after the spin reverses, it puts energy back in.

Physical conclusion

A moment of inertia belongs to a mass distribution and a particular axis together. With the bearing fixed, a pure torque gives an exact way to compare designs.

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)