### Angular Velocity

How fast is an object rotating? We can answer this question by using the concept of angular velocity. Consider first the angular speed $(\omega )$ is the rate at which the angle of rotation changes. In equation form, the angular speed is

6.2$$\omega =\frac{\text{\Delta}\theta}{\text{\Delta}t}\text{,}$$

which means that an angular rotation $(\Delta \theta )$ occurs in a time, $\text{\Delta}t$. If an object rotates through a greater angle of rotation in a given time, it has a greater angular speed. The units for angular speed are radians per second (rad/s).

Now let’s consider the direction of the angular speed, which means we now must call it the angular velocity. The direction of the angular velocity is along the axis of rotation. For an object rotating clockwise, the angular velocity points away from you along the axis of rotation. For an object rotating counterclockwise, the angular velocity points toward you along the axis of rotation.

Angular velocity (ω) is the angular version of linear velocity **v**. Tangential velocity is the instantaneous linear velocity of an object in rotational motion**. **To get the precise relationship between angular velocity and tangential velocity, consider again a pit on the rotating CD. This pit moves through an arc length $$(\Delta s)$$ in a *short* time $$(\Delta t)$$ so its tangential *speed* is

6.3
$$v=\frac{\text{\Delta}s}{\text{\Delta}t}\text{.}$$

From the definition of the angle of rotation, $\text{\Delta}\theta =\frac{\text{\Delta}s}{r}$, we see that $\text{\Delta}s=r\text{\Delta}\theta $. Substituting this into the expression for *v *gives

$$v=\frac{\text{r}\text{\Delta}\theta}{\text{\Delta}t}=r\omega \text{.}$$

The equation $v=r\omega $ says that the tangential speed *v* is proportional to the distance *r* from the center of rotation. Consequently, tangential speed is greater for a point on the outer edge of the CD (with larger *r*) than for a point closer to the center of the CD (with smaller *r*). This makes sense because a point farther out from the center has to cover a longer arc length in the same amount of time as a point closer to the center. Note that both points will still have the same angular speed, regardless of their distance from the center of rotation. See Figure 6.4.

Now, consider another example: the tire of a moving car (see Figure 6.5). The faster the tire spins, the faster the car moves—large $\omega $ means large *v* because $v=r\omega $. Similarly, a larger-radius tire rotating at the same angular velocity, $\omega $, will produce a greater linear (tangential) velocity, **v,** for the car. This is because a larger radius means a longer arc length must contact the road, so the car must move farther in the same amount of time.

However, there are cases where linear velocity and tangential velocity are not equivalent, such as a car spinning its tires on ice. In this case, the linear velocity will be less than the tangential velocity. Due to the lack of friction under the tires of a car on ice, the arc length through which the tire treads move is greater than the linear distance through which the car moves. It’s similar to running on a treadmill or pedaling a stationary bike; you are literally going nowhere fast.

### Tips For Success

Angular velocity **ω** and tangential velocity **v** are vectors, so we must include magnitude and direction. The direction of the angular velocity is along the axis of rotation, and points away from you for an object rotating clockwise, and toward you for an object rotating counterclockwise. In mathematics this is described by the right-hand rule. Tangential velocity is usually described as up, down, left, right, north, south, east, or west, as shown in Figure 6.6.

### Watch Physics

#### Relationship between Angular Velocity and Speed

This video reviews the definition and units of angular velocity and relates it to linear speed. It also shows how to convert between revolutions and radians.

Grasp Check

For an object traveling in a circular path at a constant speed, would the linear speed of the object change if the radius of the path increases?

- Yes, because tangential speed is independent of the radius.
- Yes, because tangential speed depends on the radius.
- No, because tangential speed is independent of the radius.
- No, because tangential speed depends on the radius.