The Comprehensive Guide to Circular Motion

By Brews ohare [CC BY-SA 3.0 (https://creativecommons.org/licenses/by-sa/3.0) or GFDL (http://www.gnu.org/copyleft/fdl.html)], from Wikimedia Commons

By Andrew Bennett

The motion of an object going around in circles is of special interest in AP physics 1. From a ball on a string to GPS satellites to charged particles moving through magnetic fields, examples of circular motion are everywhere!

Uniform Circular Motion Explained

Circular motion with a constant speed is called uniform circular motion. It is a frequent setup for intro-level high school physics classes. Since the speed is unchanging, many will incorrectly assume that the object is not accelerating. In fact, acceleration is a change in an object's velocity, which includes both speed and direction. The speed is constant, but the direction is not.

Objects traveling in a circle at a constant speed are always accelerating toward the center of the circle. (We call this centripetal acceleration.) They also have a velocity vector that is tangent to the circle. The amount of acceleration can be calculated by squaring the speed and dividing by the radius of the circle:

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Uniform Circular Motion Video

The magnitudes of these quantities are often made to be constant for first-year physics problems, but the directions necessarily change. At the end of this video, we discuss those changes in direction and how we could represent them in a graph.

Centripetal Forces in Circular Motion 

Any child who has ridden in a car (especially those with siblings) can tell you that when the car makes a turn to the left, everyone gets pushed to the right (so that you can justify smashing your brother or sister to your right). Yet there is no force pushing you to the right in this situation.

At one point in history, some people believed that the planets moved in their orbits because there were angels moving around behind the planets, beating their wings to drive the planets. The thinking was that the planets needed a force to push them forward as they orbit. Yet there is no force pushing or pulling the planets forward in their orbits.

Inertia's Role in Circular Motion

Inertia is the tendency of objects to keep moving at a constant speed and in a constant direction. It takes force to change an object's velocity. In the case of an object traveling in a circle, the force needs to point inward toward the center of the circle.

If I watched a car from above as it drove forward then began to move to the right, I could surmise that there was a force pushing or pulling it to the right. If it kept turning to the right, I would know that the force to the right was continuing, too. If the car kept turning to the right so that it made a big circle, then the force to the right would always point toward the center of that circle.

A passenger in that vehicle might end up further to the left on their seat than when they started without ever experiencing a force to the left. If we watched that passenger from above (maybe it's a convertible), we would see that they don't actually shift to the left. In fact, they shift to the right (relative to the ground). So, why do they end up on the left side of their seat? The car turns right and the passenger turns right, but the passenger doesn't turn quite as rapidly as the car does at first.

Centripetal Forces Definition

Forces that point toward the center of the circular path are centripetal forces or, more accurately, forces in the centripetal direction. Any force can point toward the center of an object's circular path. When a ball swings on a string, the tension in the string pulls the ball toward the center of the circle. A turning car experiences a frictional force from the road that acts toward the center of the circle. The Earth is pulled toward the Sun, which is at the center of the Earth's nearly circular path.

Centrifugal Forces Definition

There could also be forces that act away from the center of that circular path. These are sometimes called centrifugal forces. But these forces are not generated by that circular motion. When you start moving an object in a circle, there is no new "centrifugal force" that pushes it toward the outside of the circle. That perceived force is actually the inertia of the object making it difficult to change the direction of travel.

Fictitious Force Definition

In physics, we sometimes use a force called the "centrifugal force" in our calculations. This is what's known as a fictitious force.

If we solved a problem for someone riding in the car and used the car as the frame of reference, then we would need a force to explain why the person slides on their seat. In this case, we would treat the car as though it were still and would say that the ground was moving relative to the car. There is no object pushing or pulling the person in that direction; we simply need to make up a force to explain that motion.

This happens when we choose accelerating frames of reference, typically called "non-inertial frames of reference." In an entry-level physics class, you would not typically deal with fictitious forces in non-inertial frames of reference. So, many first-year physics teachers will tell their students that there is no "centrifugal force." If you are doing some physics homework, and this is the first you've heard about fictitious forces in non-inertial frames of reference, don't include a force called the "centrifugal force" in your calculations – you haven't learned how to do that correctly yet!

Applying Newton's 2nd Law of Motion

Now that we've discussed both forces and acceleration in the centripetal direction, we can also relate these two quantities using the equation from Newton's Second Law of Motion, that acceleration is equal to force divided by mass:

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How to Solve Circular Motion Force Problems

We can solve circular motion force problems in the same way that we solved with forces in the X-direction and the Y-direction. In this case, we'll be working in the centripetal direction:

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Since we know how to calculate the centripetal acceleration, we might also substitute that equation in for the centripetal acceleration here to get:

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Note that the left side of this equation will always be positive. This side represents the acceleration in the centripetal direction, which will always be toward the center of the circle for an object undergoing circular motion. The right side must be positive, too, which means that we must use the convention that forces toward the center of the circle are positive and forces away from the center of the circle are negative.

This video includes a general overview of forces in the centripetal and centrifugal directions. It also shows how to solve for unknown quantities using Newton's 2nd Law of Motion. Finally, you will learn to solve an example problem involving a pendulum at the bottom of its swing. This problem often shows up in physics textbooks as a person swinging from a rope (or Indiana Jones swinging from a whip or Tarzan swinging from a vine).

Additional Physics Tutorials

To learn more about circular motion, check out this playlist. To stay in the loop on my newest physics videos and posts, subscribe to my YouTube channel and blog.