It is possible for objects to share similar motions with other objects, an example would be a car and the driver, both are traveling at the same speed and direction. When scenarios like this occur, we call it a coupled system. This section will help develop the necessary tools to identify coupled systems and the resulting constraints that arise from this coupling.

In coupled system we often have to think about which objects have the same motion or some known relative motion. Those objects that do move together can be grouped into one system with multiple objects inside and the acceleration of the whole will equal that of the individual pieces. They may also be constrained together in some way. Think of all the objects that move together, the constraints on their motions, and the Newton's 3rd law force pairs in the video below.

https://www.youtube.com/watch?v=t6dWQck6jkA

Mechanical advantage is measure of how much effort it to do something without a device, such as lifting a mass, divided by how much it takes to do that with the aid of a device, such as lifting the mass with a pulley system. Mechanical advantage tells us how much easier a job is with a tool than without. A ramp with a wheelbarrow, a bikes gears, a pulley system, theses are all examples of simple machines used to create mechanical advantage.

This video will help orient us towards what a pulley system is. We'll have to go deeper into the analysis but is a good start.

https://www.youtube.com/watch?v=LiBcur1aqcg

## Pre-lecture Study Resources

Read the BoxSand Introduction and watch the pre-lecture videos before doing the pre-lecture homework or attending class. If you have time, or would like more preparation, please read the OpenStax textbook and/or try the fundamental examples provided below.

### BoxSand Introduction

## Forces | Coupled Systems and Pulleys

Many mechanics problems involve two or more objects being connected together. An example could be two masses connected by a pulley, or one block stuck to another block, possible through friction. We call these coupled systems. These systems are often connected by third law force pairs, or just force pairs. And when any two objects are coupled together they must share something in common, such as the magnitude of a tension force or the magnitude of their kinematics variables.

### BoxSand Videos

### OpenStax Reading

### Fundamental examples

**Worked Examples**

Box pulled on top of another box

Box pushing on suspended second box

1. Consider the image below. The $7 \, kg$ mass rests on top of a frictionless table, and is connected to a massless string hung over a massless frictionless pulley. Find the magnitude of acceleration for both masses.

2. Consider the image below. The $7 \, kg$ mass rests on top of a frictionless table, and is connected to a massless string hung over a massless frictionless pulley. Find the magnitude of acceleration for both masses.

Short foundation building questions, often used as clicker questions, can be found in the clicker questions repository for this subject.

## Post-Lecture Study Resources

Use the supplemental resources below to support your post-lecture study.

### Practice Problems

Recommended example practice problems

BoxSand's Quantitative Practice Problems

BoxSand's Multiple Select Problems

Solutions to BoxSand Practice Problems

Universal Gravity Practice Problems

Solutions to Universal Gravity Practice Problems

Set 1: Problem 1 Problem 2 Problem 3

Set 2: Stacked Boxes

Set 3: Ropes and Pulleys

For additional practice problems and worked examples, visit the link below. If you've found example problems that you've used please help us out and submit them to the student contributed content section.

## Additional Boxsand Study Resources

### Additional BoxSand Study Resources

### Learning Objectives

## Summary

The goal of studying coupled systems is to apply Newton's laws of motion, with an emphasis on the 3rd law to situations with multiple objects. In these cases decisions about how to define the system(s) will be confronted.

## Atomistic Goals

Students will be able to...

- Analyze situations involving multiple objects and choose appropriate systems.
- Differentiate between external and internal forces.
- Use 3rd law force pairs to analyze coupled systems.
- Use constraints to analyze coupled systems.
- Choose coordinate systems relative to each other that simplify the analysis.
- Choose different systems (e.g. the system 1 and 2, or the combined 1 2 system) for the same physical situation and use the analysis from both to arrive at a solution.
- Identify the features of an ideal pulley.
- Demonstrate the ability to use a FBD and Newton's 2nd law to determine the mechanical advantage of a pulley system.

## Additional Study Resources

Use the supplemental resources below to support your post-lecture study.

### YouTube Videos

### Simulations

PhET Springs: A fun little applet that looks at springs. Don't worry about any of this normal mode stuff, just explore how changing one spring might effect another.

Atwoods Machine: Another Wiley interactive simulation directly applying concepts from Newton's 2nd Law;

For additional simulations on this subject, visit the simulations repository.

### Demos

### History

Oh no, we haven't been able to write up a history overview for this topic. If you'd like to contribute, contact the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

### Physics Fun

Oh no, we haven't been able to post any fun stuff for this topic yet. If you have any fun physics videos or webpages for this topic, send them to the director of BoxSand, KC Walsh (walshke@oregonstate.edu).

### Other Resources

Physics Classroom: Good introduction with examples, and it has a great lay out that will lead you through different coupled systems and works example problems to show you how objects might interact.

If you'd like to find out more about constraints this web page has some great examples. Specifically look at example 2 and 3;

## Resource Repository

This link will take you to the repository of other content on this topic.

## Problem Solving Guide

Use the Tips and Tricks below to support your post-lecture study.

### Assumptions

### Checklist

### Misconceptions & Mistakes

### Pro Tips

### Multiple Representations

Multiple Representations is the concept that a physical phenomena can be expressed in different ways.

### Physical

### Mathematical

### Graphical

### Descriptive

### Experimental