## Rotation Angle and Angular Velocity

Read this text. To understand circular or rotational motion, picture a spinning disk, such as the picture of a CD in Figure 6.2. This figure shows a CD with a line drawn from the center to the edge. All of the points along this line travel the same angle, in the same amount of time, as the CD spins. We call this the rotational angle, which is defined as $\Delta \theta=\frac{\Delta s}{r}$. We call the distance along the circumference traveled ($\Delta s$) the arc length, and we call the radius of the circular motion ($r$) the radius of curvature.

When describing angles, we often use the unit radian, abbreviated as rad. We define radians as 1 revolution = $2\pi$ rad. Radians are the standard unit for physics problems, but we can convert radians to the more familiar degrees for convenience. Pay attention to Table 6.1 for conversions between radians and degrees.

We define angular velocity (or rotational velocity), $\omega$ (the Greek letter omega), as the rate at which the angle changes while an object is rotating. We can write it as $\omega=\frac{\Delta \theta}{\Delta t}$, where $\Delta \theta$ is the change in angle and $\Delta t$ is the time it takes for the angle to change that amount.

We can relate angular velocity to linear velocity using the relation $v = r \omega$, with $r$ being the radius of curvature. Pay attention to the derivation of how angular velocity relates to linear velocity in equations 6.6, 6.7, 6.8, and 6.9.

### Angular Velocity

How fast is an object rotating? We define angular velocity $\omega$ as the rate of change of an angle. In symbols, this is

$\omega=\dfrac{\Delta \theta}{\Delta t}$

where an angular rotation $\Delta \theta$ takes place in a time $\Delta t .$ The greater the rotation angle in a given amount of time, the greater the angular velocity. The units for angular velocity are radians per second (rad/s).

Angular velocity $\omega$ is analogous to linear velocity $v$. To get the precise relationship between angular and linear velocity, we again consider a pit on the rotating CD. This pit moves an arc length $\Delta s$ in a time $\Delta t$, and so it has a linear velocity

$v=\dfrac{\Delta s}{\Delta t}$

From $\Delta \theta=\dfrac{\Delta s}{r}$ we see that $\Delta s=r \Delta \theta .$ Substituting this into the expression for $v$ gives

$v=\dfrac{r \Delta \theta}{\Delta t}=r \omega$

We write this relationship in two different ways and gain two different insights:

$v=r \omega \text { or } \omega=\dfrac{v}{r}.$

The first relationship in $v=r \omega$ or $\omega=\dfrac{v}{r}$ states that the linear velocity $v$ is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest $r$ ), as you might expect. We can also call this linear speed $v$ of a point on the rim the tangential speed. The second relationship in $v=r \omega$ or $\omega=\dfrac{v}{r}$ can be illustrated by considering the tire of a moving car. Note that the speed of a point on the rim of the tire is the same as the speed $v$ of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins-large $v$ means a large $\omega$, because $v=r \omega .$ Similarly, a larger-radius tire rotating at the same angular velocity $(\omega)$ will produce a greater linear speed $(v)$ for the car.

Figure 6.5 A car moving at a velocity $v$ to the right has a tire rotating with an angular velocity $\omega$. The speed of the tread of the tire relative to the axle is $v$, the same as if the car were jacked up. Thus the car moves forward at linear velocity $v=r \omega$, where $r$ is the tire radius. $A$ larger angular velocity for the tire means a greater velocity for the car.

#### Example 6.1 How Fast Does a Car Tire Spin?

Calculate the angular velocity of a 0.300 m radius car tire when the car travels at 15.0m/s (about 54km/h). See Figure 6.5.

#### Strategy

Because the linear speed of the tire rim is the same as the speed of the car, we have $v=15.0 \mathrm{~m} / \mathrm{s}$. The radius of the tire is given to be $r=0.300 \mathrm{~m}$. Knowing $v$ and $r$, we can use the second relationship in $v=r \omega, \omega=\dfrac{v}{r}$ to calculate the angular velocity.

#### Solution

To calculate the angular velocity, we will use the following relationship:

$\omega=\dfrac{v}{r}.$

Substituting the knowns,

$\omega=\dfrac{15.0 \mathrm{~m} / \mathrm{s}}{0.300 \mathrm{~m}}=50.0 \; \mathrm{rad} / \mathrm{s}.$

#### Discussion

When we cancel units in the above calculation, we get 50.0/s. But the angular velocity must have units of rad/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular velocity

$\omega=(15.0 \mathrm{~m} / \mathrm{s}) /(1.20 \mathrm{~m})=12.5 \; \mathrm{rad} / \mathrm{s}.$

Both $\omega$ and $v$ have directions (hence they are angular and linear velocities, respectively). Angular velocity has only two directions with respect to the axis of rotation -it is either clockwise or counterclockwise. Linear velocity is tangent to the path, as illustrated in Figure 6.6.

#### Take-Home Experiment

Tie an object to the end of a string and swing it around in a horizontal circle above your head (swing at your wrist). Maintain uniform speed as the object swings and measure the angular velocity of the motion. What is the approximate speed of the object? Identify a point close to your hand and take appropriate measurements to calculate the linear speed at this point. Identify other circular motions and measure their angular velocities.

Figure 6.6 As an object moves in a circle, here a fly on the edge of an old-fashioned vinyl record, its instantaneous velocity is always tangent to the circle. The direction of the angular velocity is clockwise in this case.