---
title: "Reaction-Wheel Balance Interactive Lab | Sandbox Physics"
description: "The rod stops falling. The wheel keeps spinning. Can a spinning wheel stabilize a stationary rod?"
canonical_url: "https://sandboxphysics.com/labs/reaction-wheel-balance/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/reaction-wheel-balance/"
last_modified: "2026-10-19"
---

Control & mechanics · Momentum storage

# Reaction-Wheel Balance

Balance a pivoted rod by accelerating its flywheel. Change wheel radius, available torque or initial spin, and see why a stable rod can leave a fast-spinning rotor behind.

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

One-plane rigid rod and centered uniform disk. Ideal torque-speed envelope and gravity-compensated angle feedback. No swing-up, momentum unloading, impacts or out-of-plane gyroscopic motion.

Physics tutorial

## A steady rod can hide a spinning wheel

**Background** Upright is an unstable equilibrium. Feedback must use the actual motion and an actuator with finite authority.

**Why it matters** Follow the apparatus, the requested actuation and the mechanical work as one experiment.

Start with the essentials

- **Focus question**: Why can the pole balance while the rotor keeps its angular momentum?
- **One-sentence intuition**: A motor applies equal and opposite torques. The rod can settle while angular momentum remains stored in the wheel; gravity supplies external torque about the pivot.

### Core mathematical model

#### Body and rotor dynamics

$$
I_b\ddot\theta=G\sin\theta-\tau,\qquad J\dot\nu=\tau,\qquad\dot\phi=\nu
$$

The rotor angle and speed are absolute; the shaft sees the difference between rotor and rod speeds. Body inertia includes rod and wheel-center translation, excluding rotor axial spin.

#### Unsaturated angle feedback

$$
\tau_d=G\sin\theta+I_b(\lambda^2\theta+2\lambda\dot\theta),\qquad\ddot\theta+2\lambda\dot\theta+\lambda^2\theta=0
$$

The second relation holds only while the motor delivers the requested torque in the active upright neighborhood.

#### Power at the relative shaft

$$
E=\tfrac12 I_b\dot\theta^2+\tfrac12J\nu^2+G(\cos\theta-1),\qquad\dot E=\tau(\nu-\dot\theta)
$$

Motor work can be positive or negative. This is mechanical shaft work; electrical consumption and losses are omitted.

### Common difficulties

#### Feedback is not unlimited

**Typical misconception** Turning feedback on guarantees recovery from every release.

**Better mental model** Compare the limit presets. These are local balancing controllers with finite drive capacity, not swing-up algorithms.

#### Absolute and relative rotation

**Typical misconception** A steady rod means the flywheel has stopped.

**Better mental model** Read the absolute rotor angular momentum and motor-relative speed separately.

### Run the experiment

1. 01

   #### Release

   Drag the gold initial-tilt handle, then play.

   **What to observe:** Follow the rod and signed motor demand.

2. 02

   #### Find the limit

   Choose Low speed ceiling, then compare a larger wheel.

   **What to observe:** Look for separation between requested and delivered actuation.

3. 03

   #### Audit the energy

   Go to the end and export the record. Compare standard and fine integration.

   **What to observe:** Mechanical energy change follows signed motor work.

Physical conclusion

A local stabilizing law needs sufficient actuator authority. A saturated actuator cannot deliver an ideal control command.

Primary theory: [MIT · Cart-Pole](https://underactuated.mit.edu/acrobot.html) · [ETH Zürich · 1D reaction-wheel model, Section II](https://idsc.ethz.ch/content/dam/ethz/special-interest/mavt/dynamic-systems-n-control/idsc-dam/Research_DAndrea/Cubli/cubliCDC13.pdf)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into 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)