Rotational Kinematics: Definition & Equations

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  • 0:05 What Is Kinematics?
  • 1:07 Rotational Variables
  • 1:57 Equations
  • 2:41 Example Calculation
  • 4:02 Lesson Summary
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this lesson, you should be able to explain what kinematics is, give a rotational variable that corresponds to each linear kinematic variable, and solve problems using rotational kinematics equations. A short quiz will follow.

What Is Kinematics?

There are some parts of physics that are pretty intuitive and relate directly to our everyday experiences of life: velocity, weight, distance, height and so forth. But there are some parts of physics that aren't at all intuitive. For example, when I learned quantum mechanics, I realized that without the help of excellent teachers, I wouldn't have any hope in connecting any of it to the real world. It was all so abstract and on a scale we just don't experience.

Kinematics came before dynamics for a similar kind of reason. Anything kinetic involves motion. So kinematics is the study of the motion of objects, without any reference to the forces that cause that the motion. Forces and their impacts are more abstract than numbers of motion, like position, velocity and acceleration. The opposite of kinematics is dynamics, which is studying the motion of objects using forces. Both kinematics and dynamics can be studied for translational motion or (as we'll be doing in this lesson) for rotation.

In other videos, we talk about the kinematic quantities that affect translational motion. But you can study kinematics for rotation, too. We just replace a few of the variables.

Rotational Variables

In normal translational motion, there are five key variables: position, initial velocity, final velocity, acceleration and time. Now we need to look at variables for rotational motion. Each linear variable has a corresponding rotation one.

Position, x, is replaced with angle, theta, which is the position of an object around an axis of rotation, measured in radians.

Initial and final velocity are replaced with initial and final angular velocity, which is just how fast the object rotates, measured in radians per second.

Acceleration is replaced with angular acceleration, which describes the rate at which the angular velocity is changing, measured in radians per second per second, or radians per second squared.

And time is just time. It doesn't really matter what the object's doing -- the clock still ticks the same way.


There are some basic equations we can use for rotational kinematics:

Equations for Rotational Kinematics

Since velocity is change in position divided by time, angular velocity is change in angle divided by time. And since acceleration is change in velocity divided by time, angular acceleration is change in angular velocity divided by time, which is final angular velocity minus initial angular velocity, divided by time.

And just like the average velocity is the final velocity, plus the initial velocity divided by two, the average angular velocity is the initial angular velocity, plus the final angular velocity divided by two.

This is a common theme when moving to rotation -- replacing the variables in linear equations with angular variables. Once you do that, the equations pretty much always work just fine.

Example Calculation

Time to do an example problem. Let's say a merry-go-round is spinning at a rate of 5 radians per second. You want to get on, so you push it the other way until it slows to 0.5 radians per second. If it takes you 2 seconds to do this, what is the angular acceleration? And what was the average angular velocity during those two seconds?

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