---
title: "Gear Train & Idler Interactive Lab | Sandbox Physics"
description: "One extra gear. A different direction. Does an extra gear change speed or direction?"
canonical_url: "https://sandboxphysics.com/labs/gear-train-idler/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/gear-train-idler/"
last_modified: "2026-10-18"
---

M029 · Fixed-axis transmission

# Gear Train & Idler

Turn the input, change tooth counts and insert an idler. Follow real involute teeth, signed shaft speeds and the torque traded for speed.

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

Ideal rigid, lossless gearing at prescribed steady speeds. Involute flanks; schematic root joins, shafts and supports. No tooth stress, inertia or friction.

Physics tutorial

## Count meshes before counting speed

**Background** Rigid teeth of equal module constrain pitch-circle tangential velocities. Each external mesh reverses rotation.

**Why it matters** Connect visible teeth to signed speed, torque and power.

Start with the essentials

- **Focus question**: Does a larger idler make the output slower?
- **One-sentence intuition**: The idler ratio appears once in each direction and cancels from the final ratio.

### Core mathematical model

#### A single mesh

$$
N_a n_a+N_b n_b=0,\quad r_j=\frac{mN_j}{2}
$$

Pitch-circle surface velocities agree at the reference contact. Gear axes are fixed.

#### With and without an idler

$$
\frac{n_o}{n_1}=\begin{cases}-N_1/N_o&\text{direct}\\+N_1/N_o&\text{one idler}\end{cases}
$$

Changing the idler size changes its own speed and the layout, but not the output speed.

#### Steady external torques

$$
T_o=-\frac{T_1}{n_o/n_1},\quad T_{\mathrm{frame}}=-T_1-T_o
$$

The applied output torque opposes the power supplied by the input. The held frame absorbs the net reaction. The ratio stays defined at zero speed.

#### Power balance

$$
P_j=T_j\frac{2\pi n_j}{60},\quad P_1+P_o=0
$$

Zero frame speed gives zero frame power even with a nonzero reaction torque.

### Common difficulties

#### A slower shaft can carry more torque

**Typical misconception** Speed reduction destroys power.

**Better mental model** In this ideal steady model, torque increases as speed decreases. Real losses and transient inertia require additional physics.

#### Reference contact is not a material point

**Typical misconception** The pitch circles describe all actual tooth contact.

**Better mental model** Involute tooth contact travels along the line of action and generally includes sliding. The marked pitch point only verifies the transmission kinematics.

### Run the experiment

1. 01

   #### Add one gear

   Compare Direct pair with Insert an idler and scrub time.

   **What to observe:** The output reverses direction, while the magnitude of the end-to-end ratio stays the same.

2. 02

   #### Resize only the idler

   Change the middle tooth count while keeping the end gears fixed.

   **What to observe:** The middle shaft moves and its speed changes; output speed and torque remain fixed.

3. 03

   #### Trade speed for torque

   Increase the output tooth count and inspect the signed power row.

   **What to observe:** Output speed falls, compatible load torque grows, and ideal input/output power cancel.

Physical conclusion

Speed is a kinematic constraint. The displayed steady torques conserve power but do not predict acceleration.

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)