---
title: "Scotch Yoke Interactive Lab | Sandbox Physics"
description: "A pin goes around. A slot draws a sine. How does a slot turn a circle into a sine wave?"
canonical_url: "https://sandboxphysics.com/labs/scotch-yoke/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/scotch-yoke/"
last_modified: "2026-10-20"
---

Mechanism design · A mechanical sine wave

# Scotch Yoke

Turn a crank through a real open yoke slot. Follow its horizontal projection, inspect the phase between displacement and velocity, and see why the end of a stroke has zero speed but maximum acceleration.

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

Rigid planar yoke, ideal vertical slot and horizontal guides. Exact harmonic kinematics and massless quasistatic signed load transfer. No contact pressure, wear, inertia, compliance or motor-sizing model.

Physics tutorial

## A circle projects into a sine wave

**Background** A pin can transmit motion through a slot without a connecting rod.

**Why it matters** The apparatus and the quantitative record share the same exact geometric state.

Start with the essentials

- **Focus question**: Why is acceleration largest where the yoke stops?
- **One-sentence intuition**: The vertical slot removes the pin’s vertical coordinate from the output motion. Its horizontal coordinate is a cosine.

### Core mathematical model

#### Projection constraint

$$
x=r\cos\theta,\qquad y_{\rm pin}=r\sin\theta
$$

The yoke slides horizontally while the pin slides vertically inside it.

#### Harmonic motion

$$
\dot x=-r\omega\sin\theta,\qquad\ddot x=-\omega^2x
$$

The input angular speed is constant. Stroke is twice the crank radius.

#### Quasistatic virtual power

$$
\tau_{\rm in}=-F_x\frac{\mathrm dx}{\mathrm d\theta},\qquad\tau_{\rm in}\omega+F_x\dot x=0
$$

No inertia or frictional power is included.

### Common difficulties

#### Prescribed motion

**Typical misconception** A smooth-looking animation proves smooth forces.

**Better mental model** Input speed is imposed. Dynamic inertia, impacts, friction and flexibility are omitted. Geneva boundary acceleration is discontinuous.

#### Signed load

**Typical misconception** A constant external load must always resist motion.

**Better mental model** A constant signed force or torque can supply or absorb power depending on output velocity.

### Run the experiment

1. 01

   #### Follow the pin

   Play one turn, then scrub the cycle or drag the gold pin in the plan.

   **What to observe:** The input pin, output and graph cursors move together.

2. 02

   #### Inspect a stroke end

   Compare Mid-stroke and Stroke end. Read velocity and acceleration together.

   **What to observe:** At an end the yoke changes direction; zero velocity does not mean zero acceleration.

3. 03

   #### Check the boundary

   Reverse the external force and export the cycle.

   **What to observe:** The motion is prescribed; reversing force reverses torque, not the trajectory.

Physical conclusion

Kinematics specifies the motion; it does not establish contact forces or a real motor’s ability to produce it.

Theory & apparatus: [Carnegie Mellon · Geneva Wheel](https://www.cs.cmu.edu/~rapidproto/mechanisms/chpt8.html) · [Purdue · Scotch Yoke](https://www.physics.purdue.edu/demos/display_page.php?item=1S-30)

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)