Rotation Angle and Angular Velocity

In Kinematics, we studied motion along a straight line and introduced such concepts as displacement, velocity, and acceleration. Two-Dimensional Kinematics dealt with motion in two dimensions. Projectile motion is a special case of two-dimensional kinematics in which the object is projected into the air, while being subject to the gravitational force, and lands a distance away. In this chapter, we consider situations where the object does not land but moves in a curve. We begin the study of uniform circular motion by defining two angular quantities needed to describe rotational motion.

Rotational Angle

When objects rotate about some axis -for example, when the CD (compact disc) in Figure 6.2 rotates about its center -each point in the object follows a circular arc. Consider a line from the center of the CD to its edge. Each pit used to record sound along this line moves through the same angle in the same amount of time. The rotation angle is the amount of rotation and is analogous to linear distance. We define the rotation angle $\Delta \theta$ to be the ratio of the arc length to the radius of curvature:

$\Delta \theta=\dfrac{\Delta s}{r}.$

Figure 6.2 All points on a CD travel in circular arcs. The pits along a line from the center to the edge all move through the same angle $\Delta \theta$ in a time $\Delta t$.

Figure 6.3 The radius of a circle is rotated through an angle $\Delta \theta$. The arc length $\Delta \mathrm{s}$ is described on the circumference.

The arc length $\Delta s$ is the distance traveled along a circular path as shown in Figure 6.3 Note that $r$ is the radius of curvature of the circular path.

We know that for one complete revolution, the arc length is the circumference of a circle of radius $r$. The circumference of a circle is $2 \pi r$. Thus for one complete revolution the rotation angle is

$\Delta \theta=\dfrac{2 \pi r}{r}=2 \pi$

This result is the basis for defining the units used to measure rotation angles, $\Delta \theta$ to be radians (rad), defined so that

$2 \pi \mathrm{rad}=1 \text { revolution. }$

A comparison of some useful angles expressed in both degrees and radians is shown in Table 6.1.

Table6.1 Comparison of Angular Units

Figure 6.4 Points 1 and 2 rotate through the same angle $(\Delta \theta)$, but point 2 moves through a greater arc length $(\Delta s)$ because it is at a greater distance from the center of rotation $(r)$.

If $\Delta \theta=2 \pi$ rad, then the CD has made one complete revolution, and every point on the $\mathrm{CD}$ is back at its original position. Because there are $360^{\circ}$ in a circle or one revolution, the relationship between radians and degrees is thus

$2 \pi \operatorname{rad}=360^{\circ}$

so that

$1 \mathrm{rad}=\dfrac{360^{\circ}}{2 \pi} \approx 57.3^{\circ}$

Source: Rice University, https://openstax.org/books/college-physics/pages/6-1-rotation-angle-and-angular-velocity
This work is licensed under a Creative Commons Attribution 4.0 License.

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.

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.

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.