Angular Speed: Definition, Formula & Problems

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  • 0:03 What Is Angular Speed?
  • 0:57 What Are Radians?
  • 1:15 Some Practice Problems
  • 3:33 Angular Speed vs.…
  • 4:08 Lesson Summary
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Lesson Transcript
Instructor: Sharon Linde

Sharon has a Masters of Science in Mathematics

Have you ever heard of angular speed? Would you like to know how it is calculated, or the difference between angular speed and angular velocity? Do you need to solve problems involving rotation? Read on for more information on angular speed.

What Is Angular Speed?

We hear about speed in a lot of different contexts, from driving to how fast a ball is pitched. When we talk about angular speed, it simply refers to how quickly an object is rotating. It's defined as the change in angle of the object per unit of time. The formula for angular speed is:

Angular speed formula

where omega is angular speed in radians per second, theta is the angle turned through, and t is the duration of the rotation. Notice anything off about that formula? If it looks and sounded Greek to you, you're right. That's how the formula is typically written. To make things a bit easier, we're going to use a slightly modified version for the rest of this lesson. We'll replace omega with w and theta with a (for angle), which gives us the following equation:

w = a / t

What Are Radians?

When calculating angular speed, it's important to know that the angle is generally measured in radians. Radians are a way of measuring angles where a right angle is defined as pi/2 radians. One full revolution would contain about 6.28 radians.

Some Practice Problems

Let's start off with something small and easy, like a car tire, and progress to spinning the entire earth.

Problem #1: What is the angular velocity of a car tire that rotates through 150 radians in 10 seconds while it's driving down the highway?

As stated in the problem, a = 150 radians, and t = 10 seconds. When we substitute these known quantities into our equation, w = a / t, we see that:

w = 150 radians / 10 seconds = 15 radians/second, which is our answer.

Problem #2: Through what angle would the car tire from our previous problem turn in exactly 17.4 seconds?

Here, we know both the angular speed and the duration of the rotation using information found in both the first and second problems. When we apply these known quantities to our equation, w = a / t, we find that:

15 radians/second = a / 17.4 seconds

Solving for a, we get a = (15 radians/second) * (17.4 seconds) = 261 radians

Problem # 3: Now let's try something a little harder. We're going to calculate the angular speed of the entire earth while it spins around its axis exactly one time.

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