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High School Physics: Homework Help Resource22 chapters | 280 lessons

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Lesson Transcript

Instructor:
*Gerald Lemay*

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

In this lesson, we define tangential velocity in terms of the radius of a circle and the time needed to make one complete revolution of the circle. We then extend this definition to include frequency and angular frequency.

From physics, we define a **vector** as a quantity having both magnitude and direction. For example, **velocity** is a vector where the magnitude is the **speed**. And speed is distance divided by time. What about the direction? For **tangential velocity**, we are describing the motion along the edge of a circle and the direction at any given point on the circle is always along the tangent line. Because of this understanding of tangential direction, we use the words ''tangential velocity'', but our equation will be calculating the speed.

Imagine a circle of radius *r*. Sitting on the edge of the circle, **how far have we gone in one complete revolution**? Answer: a distance given by 2Ï€*r*. Sure, this distance is the same as the circumference of the circle. Also, we don't have to be on the edge of the circle. We can talk about the tangential velocity of any point *r* units from the center of a circle.

Are you familiar with the concept of a **period of revolution**? This is the time required to make one complete revolution. We usually use the letter *T* for the period.

We are now ready to give an equation for the tangential velocity.

As an equation, the tangential velocity is the distance, 2Ï€*r*, divided by the time, *T*. Thus,

A point on the circle moves a distance 2Ï€*r* in a time *T*.

We can extend our equation by looking at a few ideas. These concepts include the angular speed, Ï‰, and the frequency, *f.*

- The
**angular speed**, Ï‰, is a speed of rotation. It measures the number of radians (or degrees) per second. Note, other names for angular speed include angular frequency and circular frequency. - Another measure of rotation per second is the
**frequency**,*f*. There is an important difference, however. The frequency is the number of cycles per second. A**cycle**refers to one complete revolution where we are back at the starting point.

For the example, if we do 2 complete revolutions in 1 second, the frequency is 2 cycles per second.

How are Ï‰ and *f* related?

- The angular frequency is 2Ï€ times the frequency.

Ï‰ is usually in radians per second. Every time we complete 2Ï€ radians of rotation, we have made one complete cycle around the circle. Thus, Ï‰ divided by 2Ï€ is the number of cycles per second. The number of cycles per second is the frequency. Thus, Ï‰/(2Ï€) = *f* and Ï‰ = 2Ï€ *f.*

Frequency, *f,* and period, *T,* are also related.

- The frequency and the period are reciprocals of each other.

Continuing with our example, if we do 2 revolutions in 1 second, then the time to do one complete revolution is 1/2 second. The period is 1/2 second and the frequency is 2 cycles per second. (Note, we usually use **hertz** for the units of cycle per second.) Thus, *T* and *f* are reciprocals of each other. As equations,

and

Now, we are ready to develop another form for the tangential velocity equation.

Consider the following:

In the first line, we start with our earlier equation for tangential velocity with distance (2Ï€ *r*) over time for one complete revolution (*T*). Then we write dividing by *T* as multiplying by 1 / *T*.

In the second line, the 1 / *T* is replaced with the frequency, *f.*

Next,

In the first line we re-arrange 2Ï€*r* *f* to be 2Ï€*f* *r.*

In the second line, we replace 2Ï€*f* with Ï‰.

Thus, another form for the tangential velocity is *v* = Ï‰ *r.*

Example: An object is rotating at an angular speed, Ï‰, of 30 radians/sec about a point 2 meters away. **What is the object's tangential velocity**?

- Use the equation,
*v*= Ï‰*r*. - Substitute 30 radians/sec for Ï‰ and 2 meters for
*r*. - Simplify.

Let's quickly review! For **tangential velocity**, we are describing the motion along the edge of a circle and the direction at any given point on the circle is always along the tangent line. As an **equation**, tangential velocity is:

the distance, 2Ï€*r*, divided by the time, *T*.

In order to understand this equation and successfully calculate it, we need to grasp the concepts of angular speed, omega and the frequency *f*.

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High School Physics: Homework Help Resource22 chapters | 280 lessons

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