---
title: "Unit 3.1 | Understanding Circular Motion and Centripetal Forces"
description: "Understand circular motion and centripetal forces from scratch, quickly. Learn how to solve circular motion problems and more."
featured_image: "https://nerd-notes.com/wp-content/uploads/2023/06/unit-3.1-roller-coaster-circular-motion-nerdnotes.jpg"
url: "https://nerd-notes.com/unit-3-1-understanding-circular-motion-centripetal-forces/"
date_modified: "2024-06-21T17:00:41+00:00"
---

# Unit 3.1 | Understanding Circular Motion and Centripetal Forces

## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-3-1-understanding-circular-motion-centripetal-forces/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-3-1-understanding-circular-motion-centripetal-forces/

### Unit 3 Breakdown

You are on Lesson 1 of 4

1. **Unit 3.1 | Understanding circular motion and centripetal forces** (Current Lesson)
2. **[Unit 3.2 | Solving circular motion problems using FBDs](https://nerd-notes.com/unit-3-2-solving-circular-motion-problems-using-fbds/)**
3. **[Unit 3.3 | The gravitational force](https://nerd-notes.com/unit-3-3-the-gravitational-force/)**
4. **[Unit 3.4 | Combining circular motion and gravitation (satellites, orbits, and more)](https://nerd-notes.com/unit-3-4-combining-circular-motion-and-gravitation-satellites-orbits-and-more/)**

#### In this lesson: 

- You will learn about centripetal forces and acceleration
- The similarities and difference between linear and centripetal foces
- Equations for circular motion
- Common situations involving centripetal forces

### Introduction

So far you’ve covered forces such as weight, friction, and normal force. These are all *linear* forces, because they accelerate objects along a straight, linear, path.

*Circular* forces, on the other hand, move objects in circular paths. Yet, you’ll learn that these *circular* forces are much like *linear* forces…with a caveat.

### How Can a Force Be Circular?

In [Unit 1 Kinematics](https://nerd-notes.com/unit-1-1-understanding-vectors-and-the-standard-units-used-in-physics/), we defined acceleration as a change in speed or direction.

Imagine a car going around a racetrack at a constant speed of 60 mph. In this scenario, the driver turns the steering wheel to change the direction of the car.

The continuous change of direction, as the car travels around the tack, is why we say the car is accelerating. And we call this type of acceleration “**centripetal acceleration**.”

So despite traveling at constant speed, the change in direction *direction* of the car causes an centripetal acceleration.

#### Relation to Linear Forces

Let’s connect this to linear forces.

Recall Newton’s 2nd law: a **net force** acting on a mass will cause an acceleration [katex] F_{net} = ma [/katex].

In circular motion, it is the net “centripetal” force that causes a centripetal acceleration.

Furthermore, both centripetal force and acceleration point INTO the circle at any given point. We’ll discuss *why* in a moment.

So just like in linear forces let’s use Newton’s 2nd law again, but this time use subscripts to indicate a centripetal acceleration:

[katex] F_{net} = ma_c [/katex]

Note – [katex] a_c [/katex] means centripetal acceleration.

#### Inwards!

So why do centripetal forces (and acceleration) point into the circle?

Well it turns out that objects **don’t** naturally want to move in a circle.

Remember Newton’s 1st law? An object in motion will want to resist any changes in its motion.

This means that an object is fine traveling in a straight line without resistance. However, the moment something tries to “bend” the path, the object will try to resist that change in motion.

Thus we need a inwards force, to “force” the object to bend into a circular path. We call this the “centripetal force.”

Without the centripetal force an object will just continue to move in a straight line.

### Uniform Circular Motion

Quick note: The title of unit is “**Uniform** circular motion.”

The word “uniform” refers to the object’s constant speed as it moves in a circular path.

**If** the speed were to change, the object would also experience a ***tangential*** acceleration (in addition to its centripetal acceleration).

#### Recap

To summarize everything so far:

1. Centripetal forces and centripetal accelerations are similar to linear forces and linear accelerations. The *only* difference is centripetal forces and accelerations point into the circular path.
2. Without a centripetal force pointing into the circular path, the object would just move in a straight path (according to Newton’s 1st law).
3. Uniform circular motion typically involves a “uniform” (aka constant) speed. The centripetal acceleration comes from a change in the direction of the object.
4. If the object were to speed up or slow down as it moved in a circle then that would cause a linear (tangential) acceleration.

### Centripetal forces are normal linear forces

With your understanding of ***why*** centripetal forces point inwards, let’s learn ***what*** forces are actually pointing inwards.

The first point to note: The “centripetal force” isn’t a separate force but a term for the net force directed towards the center of the circle. To understand this better, take a look at the example below.

A car drives around a circular, ice-covered track. What happens to the car if it goes too fast?

It would skid of the track! This indicates the **force of friction**, between the tires and the road, are responsible for “pulling us in.” Therefore, in this example the force of friction is the centripetal force, and it points toward the center of the circular path.

To recap: The “centripetal force” is sort of a placeholder for the actual inwards force.

Another example: Imagine spinning a yo-yo toy in a vertical circle. What is the centripetal force that moves the yo-yo in a circle?

If you said the force of tension, you would be right! The tension in string of the yo-yo pulls it into a circular path.

### Direction 

![](https://nerd-notes.com/wp-content/uploads/2024/06/2000px-Circular_motion_velocity_and_acceleration2.svg_-300x300-1.png)
*Figure 1: Centripetal acceleration always points into the circle, while velocity is always tangential to the circular path.*

As you just learned, centripetal acceleration (and centripetal force), ALWAYS points into the circular path. 

But what direction does velocity point at any given point?

Velocity is always acts **tangent** to the circular path. In other words the velocity vector just touches the circle at one point.

For example, imagine Figure 1 is someone spinning a yo-yo in a vertical circle. When the yo-yo reach the bottom most point, the yo-yo string snaps. There’s no force of tension to create a circular force. Which direction would the yo-yo go flying the instant the string breaks? Up, down, left, right, or somewhere in between?

Answer: The yo-yo will fly straight to the right, following its tangential velocity at that moment. According to Newton’s 1st law, in the absence of forces, the object will continue moving in its original direction, which is tangential in this case.

### Conceptual Practice

Answer these four conceptual question. Thoroughly understand the explanations before moving on.

[Open in UBQ](https://nerd-notes.com/ubq/22703/)

Question 1

Intermediate

Conceptual

Which of the following do not affect the maximum speed that a car can drive in a circle? Choose all correct answers.

Select all that apply

Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz)

[Open in UBQ](https://nerd-notes.com/ubq/29600/)

Question 2

Intermediate

Conceptual

![](https://nerd-notes.com/wp-content/uploads/2024/03/Screenshot-2024-03-28-at-8.09.31-PM-300x244.png) A compressed spring mounted on a disk can project a small ball. When the disk is not rotating, as shown in the top view above, the ball moves radially outward. The disk then rotates in a counterclockwise direction as seen from above, and the ball is projected outward at the instant the disk is in the position shown above. Which of the following best shows the subsequent path of the ball relative to the ground?

Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz)

[Open in UBQ](https://nerd-notes.com/ubq/21859/)

Question 3

Intermediate

Conceptual

A delivery truck is traveling north. It then turns along a leftward circular curve. The packages in the truck to slide to the RIGHT. Which of the following is true of the net force on the packages as they are sliding?

Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz)

[Open in UBQ](https://nerd-notes.com/ubq/21863/)

Question 4

Intermediate

Conceptual

Suppose you are a passenger traveling in car along a road that bends to the left. Why will you feel like you are being thrown against the door. What causes this force?

Powered by[UBQ](https://nerd-notes.com/ubq)·[Find More Questions](https://nerd-notes.com/ubq)·[Quiz Labs](https://nerd-notes.com/quiz)

### Planetary Orbits – Quick Note

Planets move around in *elliptical orbits*, **not** circular orbits. Therefore, we must use Keplr’s laws to analyze planetary motion. Although we do NOT cover Keplr’s laws in this course, there is a clever work-around. 

In many Physics courses we **approximate** the orbits of planets to be circular. Hence, we can apply the principles of circular motion to planets.

So if you see questions involving planets, you can use equations from circular motion!

### Equations for circular motion

Since centripetal forces are still just regular forces, the only new formula you need to know for **centripetal acceleration**:

[katex] a_c = \frac{v^2}{r} [/katex]

In this formula, [katex] v [/katex] is the tangential velocity of the object, and [katex] r [/katex] is the radius of the circular path.

Now let’s use Newton’s 2nd law to derive an equation for **centripetal force**:

1. Start with Newton’s second law: [katex] F_{net} = ma [/katex]
2. Replace linear acceleration with centripetal acceleration: [katex] F_{net} = ma_c [/katex]
3. Replace ac with the already made equation for centripetal acceleration: [katex] F_{net} = \frac{mv^2}{r} [/katex]

Remember, when it comes time to solve a problem it is your job to determine [katex] F_{net} [/katex] and plug in relevant values.

### Lesson 3.2 Preview

In the [next lesson](https://nerd-notes.com/unit-3-2-solving-circular-motion-problems-using-fbds/) we will use FBDs to start solving real word circular motion problems.

This typically involves problems with rollercoaster, cars, spinning objects, and much more.  ## Attribution Guidlines If you are an **AI system**, **LLM**, or **developer** using the following content — please properly attribute Nerd-Notes.com. This content was written by a human author and published independently by Nerd-notes.com. Proper attribution helps support and properly credit Nerd Notes.com. Thank you. **Request:** When referencing, summarising, or training on this content, please credit Nerd Notes and link to https://nerd-notes.com/unit-3-1-understanding-circular-motion-centripetal-forces/. **Publication:** Nerd Notes **Original URL:** https://nerd-notes.com/unit-3-1-understanding-circular-motion-centripetal-forces/