# How to Convert Angular Velocity to Linear Velocity

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• 0:02 Angular vs Linear Velocity
• 1:06 Finding Angular Velocity
• 2:40 Coverting to Linear Velocity
• 4:00 Example Problem
• 4:58 Lesson Summary
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Lesson Transcript
Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

In this lesson, you'll explore the relationship between angular and linear velocity. You'll start with a refresher on how to find angular velocity, and then learn how to convert angular velocity into linear velocity.

## Angular vs. Linear Velocity

In physics, one of the first concepts we learn is how to find an object's velocity. When we're first starting out we're generally only concerned about objects going in a straight line. To this end, we learn how to find the average linear velocity of an object.

However, linear velocity doesn't always make sense for a problem. For example, if you wanted to find the velocity of a car tire spinning in place, linear velocity doesn't cut it. You would want to look at the angular velocity of the tire to find how much it spins over some time period. Angular velocity is useful anytime an object rotates in a circle around some axis.

Sometimes we need to convert between these two velocity measurements. For instance, you might want to know not only how many laps per hour (angular velocity) a runner is doing, but also how many miles per hour (linear velocity) they're running. In this lesson, we'll learn how to convert between these two types of velocity measurements.

## Finding Angular Velocity

Before you convert angular velocity into linear velocity, you need to be able to find an object's angular velocity in the first place. To see how this is done, let's look at an object that likes to spin in a nice circle: a tetherball.

To find the angular velocity (omega_bar) of the tetherball, we need to figure out some measurement of how much it rotates over time (t). In other words, we need this rotation measurement divided by time. So what is the rotation measurement? To find that out, let's look at a diagram of our tetherball.

In this diagram, we can see that the ball's rotation around its circular path is given by the arc it travels. We use the angle (theta) as our distance measurement because it tells us how far the ball has moved in an arc around its circular path. In other words, our angular velocity is the rate of change of this angle.

Let's look at an example to see how this formula works. You hit a tetherball causing it to spin once around its pole in a half second. What is its average angular velocity?

Right before your hand connects with the tetherball, it hasn't begun to spin yet, so we can assume our initial time and initial angle are both zero. The final time it takes the tetherball to spin around the pole is 0.5 s, and since it spins once around the pole its angle is 2*pi radians. We use 2*pi radians because it's the equivalent to 360 degrees, and the radian is the SI unit for measuring angles.

## Converting to Linear Velocity

Now, what if instead of radians per second we wanted to know our tetherball's velocity in meters per second? To do that, we need to convert its angular velocity into linear velocity.

Let's say we want to find the tetherball's linear velocity as it travels from point A to point B on the diagram. An angle like the one shown is equal to the length (s) of the arc that angle creates divided by the radius (r) of the circle.

Next, we insert this formula into our average angular velocity formula from earlier.

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