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.

Introduction and Rotational Angle

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 $\Delta t$ .

A circle of radius r and center O is shown. A radius O-A of the circle is rotated through angle delta theta about the center O to terminate as radius O-B. The arc length A-B is marked as delta s.

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.

Degree Measures	Radian Measure
$30^{\circ}$	$\dfrac{\pi}{6}$
$60^{\circ}$	$\dfrac{\pi}{3}$
$90^{\circ}$	$\dfrac{\pi}{2}$
$120^{\circ}$	$\dfrac{2 \pi}{3}$
$135^{\circ}$	$\dfrac{3 \pi}{4}$
$180^{\circ}$	$\pi$

Table6.1 Comparison of Angular Units

A circle is shown. Two radii of the circle, inclined at an acute angle delta theta, are shown. On one of the radii, two points, one and two are marked. The point one is inside the circle through which an arc between the two radii is shown. The point two is on the cirumfenrence of the circle. The two arc lengths are delta s one and delta s two respectively for the two points.

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
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