---
title: "Planetary Gears & Two Inputs Interactive Lab | Sandbox Physics"
description: "Hold one part. Change the whole gearbox. How does holding a different member change the gearbox?"
canonical_url: "https://sandboxphysics.com/labs/planetary-gears/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/planetary-gears/"
last_modified: "2026-10-18"
---

M029 · Moving-axis transmission

# Planetary Gears & Two Inputs

Hold the ring, sun or carrier, then unlock two inputs. See three planets spin and orbit while the same tooth constraint balances every speed and power port.

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

One ideal simple planetary set with three equally spaced planets. Prescribed steady speeds and external port torques; no bevel differential, clutch dynamics, backlash or efficiency model.

Physics tutorial

## The frame moves too

**Background** A planet has two motions: spin around its own shaft and orbital motion with the carrier.

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

Start with the essentials

- **Focus question**: Why does changing the held member change the ratio?
- **One-sentence intuition**: Subtract carrier rotation first; in that frame every planet axis is fixed.

### Core mathematical model

#### Assembly geometry

$$
N_r=N_s+2N_p,\quad r_j=\frac{mN_j}{2}
$$

Equal module and concentric sun/ring fix the planet-center distance. The module here is two millimetres.

#### One speed constraint

$$
N_s(n_s-n_c)+N_r(n_r-n_c)=0
$$

The relation is nonsingular even when all three members turn together.

#### Absolute planet spin

$$
n_p=n_c-\frac{N_s}{N_p}(n_s-n_c)
$$

The white material-point marker uses fixed-frame spin, not just spin relative to its carrier.

#### Compatible external torques

$$
(T_s,T_r,T_c)=q\,(N_s,N_r,-N_s-N_r)
$$

The multiplier is fixed by the chosen drive torque. The held port supplies its reaction.

#### Signed power at every port

$$
P_j=T_j\frac{2\pi n_j}{60},\quad P_s+P_r+P_c=0
$$

Power is in watts for torque in newton metres and speed in revolutions per minute. Positive enters the set.

### 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

   #### Hold the ring

   Play the default set and compare gold sun, blue planets and coral carrier.

   **What to observe:** The carrier turns more slowly in the same direction. The stationary ring carries torque with zero power.

2. 02

   #### Change the held member

   Select Carrier held, then Sun held. Use the front view to read tooth motion.

   **What to observe:** With the carrier held the ring reverses. With the sun held the ring drives the carrier.

3. 03

   #### Use two inputs

   Choose Both inputs together, then Opposing inputs. Change the second input speed.

   **What to observe:** Equal inputs rotate as a rigid assembly. Opposing inputs can stop the carrier while the teeth continue to move.

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)