Objects traveling in a circle at a constant speed are considered to be in uniform circular motion (UCM). Recall velocity has both magnitude and direction, so even though the magnitude of the tangential velocity remains the same in UCM, the direction is constantly changing. For this special case of motion the acceleration points towards the center of the circle which means the net force does as well.

Check out this trailer video from OpenStax about forces on objects in UCM.

https://www.youtube.com/watch?v=iP4RYkt_ur4

## 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 | Uniform Circular Motion

An object undergoes Uniform Circular Motion (UCM) when it moves around a circle of constant radius at a constant speed. When an object undergoes UCM its x and y components of position and velocity are always changing. This makes them more difficult to deal with than, if instead, we choose the direction pointing towards the center of the circle. The radial direction, and the direction that points around the circle, called the tangential direction, provide a simplification in the analysis. Objects undergoing UCM have a net force pointing towards the center of the circle - entirely in the radial direction. The acceleration is also thus in the radial direction, according to Newton's 2nd Law, and is equal to $a_r = \frac{v^2}{r} $, where v is the speed and r is the radius.

## Key Equations and Infographics

Now, take a look at the pre-lecture reading and videos below.

### BoxSand Videos

## Required Videos

## Suggested Supplemental Videos

### OpenStax Reading

OpenStax Section 6.3 | Centripetal Force

**Warning**: careful consideration must be made when using the term centripetal force, as it implies an actual force, separate from others, in the radial direction. The term centripetal is referring to the net effect of all the real forces when an object is in UCM. In the case of UCM the net force is in the radial direction and that net force (from real forces) is called the centripetal force. Often students will describe a centripetal force as a source rather than a description of the phenomena, to avoid this make sure to describe what type of force is causing the centripetal force i.e. a normal force on a roller coaster, a friction force between a car and a race track etc.

OpenStax Section 6.4 | Fictitious Forces and Non-inertial Frames: The Coriolis Force

### Fundamental examples

1. A 10kg object travels in a circle of diameter 3.0 meters at a constant speed. If it takes the object 8.2 seconds to complete 2 full revolutions, what is the approximate net force in the radial direction? Disregard any effects from gravity.

a. 18 N

b. 70 N

c. 35 N

d. 8.8 N

e. Constant speed, there is no net force in the radial direction.

2. A 6 kg object is attached to a string that can only withstand a maximum force of tension of 200 N. If the object is swung around in a circle of radius 3.0 meters, what is the maximum speed that the object can have before the string breaks? Disregard any effects from gravity.

a. 100 m/s

b. 10 m/s

c. 20 m/s

d. 400 m/s

e. Not enough information.

CLICK HERE for solutions.

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

**Worked Examples**

Loop The Loop

https://media.oregonstate.edu/media/t/0_okppmxkh

Recommended example practice problems

BoxSand's Quantitative Practice Problems

BoxSand's Multiple Select Problems

Solutions to BoxSand Practice Problems

Orbital Motion Practice Problem

Solutions to Orbital Motion Practice Problems

Set 1: Problems 2-15

Set 2: Ball On a String *Notation Warning: They use "x" to label the radial direction. We suggest not doing that and labeling it as the radial r-direction.

Set 3: problem 1 and solution 1, problem 2 and solution 2

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 is to study systems where the net force is perpendicular to the velocity such as the case in uniform circular motion (UCM), where objects travel around a circle at a constant speed. UCM is also an excellent place to study the effects of reference frames that have fictitious forces arise.

## Atomistic Goals

Students will be able to...

- Identify systems that exhibit uniform circular motion.
- Draw physical representations from multiple directions and choose the perspective that helps the analysis the most.
- Draw a FBD that only includes real forces.
- Identify which forces are responsible for keeping the object traveling in a circle.
- Define centripetal force.
- Show that the direction of the net force, and thus acceleration, are perpendicular to the direction of velocity during UCM.
- Show that the direction of the net force and acceleration point towards the center of the circle during UCM.
- Identify the radially inward direction and align the coordinate system with it.
- Identify the tangential direction.
- Realize that UCM is a situation where the speed of the object affects the magnitude of the acceleration (v2/r) and thus net force.
- (UPMF) Explain the nature of fictitious forces that arise in UCM and inertia's role in the effect.
- Show how period, frequency, speed, and distance are related.

### YouTube Videos

Lo-fi UCM demo to chill/study to:

### Simulations

The Ladybug Revolution!

The Return of the Ladybug Revolution: part II

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

The Circular Motion page by PhysClips, an Australia physics education project has excellent clips with animation that describe most of UCM including the directions of some very important vectors. Make sure to watch the Uniform Circular Motion clip and the Centripetal Acceleration clip under Circular Motion in the module. Don't forget to take notes! *Aside: remember that a differential or dv implies a very small change in velocity in this case and the a is just the average acceleration, which is still a vector.

The Physics Classroom UCM page offers a concise short description worth reading in addition to the textbook above.

**Centrifugal force explained!** The Physics Classroom does a good job of explaining this pseudo force.

## 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

A pretty good video showing how to solve UCM problems with insights into why certain assumptions/conclusions are made. Beware, there are no vector symbols and the units are "sloppy."

### Misconceptions & Mistakes

- If an object is moving in a circle, the misconception that since the speed is constant the net force must be zero. There must always be a net force towards the center.
- The net force in the radial direction is always equal to m v
^{2}/r. - Many people believe that if an object is accelerating then its speed must be changing. This is not true. Speed is merely the magnitude of a vector. The direction of a vector can change without changing its magnitude. This means I can change my velocity, i.e. accelerate, but I don't have to change my speed.
- The misconception that you must always choose the tangential direction when choosing your axis.
- Warning: careful consideration must be made whenever using the term centripetal force, as it implies an actual force, separate from others, in the radial direction. The term centripetal is referring to the net effect of all the real forces when an object is in UCM. In the case of UCM, the net force is in the radial direction and that net force (from real forces) is called the centripetal force.
- Another issue people have is that they believe that since, in uniform circular motion, the speed of the object stays constant then the object must be in equilibrium. They then assume that another force is balancing out the inward forces. This is not true. For an object to change its direction, it must have a non-zero net force.
- The difference between Centripetal and Centrifugal

### Pro Tips

- Often times circular motion can be tricky to visualize. Drawing the snapshot in time of the scenario for different viewing angles can help clarify the directions of forces.
- Any object that travels in uniform circular motion always takes the same amount of time to move completely around the circle so you can use the period and the circumference to find the tangential velocity.
- Labeling the radial direction of your FBD with $\hat{r}$ will help avoid making the mistake of thinking that the acceleration in that direction is linear when translating the FBD to Newton's 2
^{nd}law.

### Multiple Representations

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

### Physical

### Mathematical

The Physics classroom has a great walk through for Uniform Circular Motion.

### Graphical

College-Physics does a number of good plots that look at Uniform Circular Motion.

### Descriptive

Cory is on a marry-go-round with a radius of 1.5m. Andrew is spinning the marry-go-round so that Cory is moving at 3 m/s. Cory is barely able to hold on to the marry-go-round due to his radial acceleration. Andrew is a Jerk…

### Experimental

A wonderful video with objects undergoing Uniform Circular motion

https://www.youtube.com/watch?v=yyDRI6iQ9Fw

Also, Wolfram Alpha does a spectacular analysis of tether-ball. (This does require you to sign up and download an add on)