# Rotation angle and angular velocity

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• Define arc length, rotation angle, radius of curvature and angular velocity.
• Calculate the angular velocity of a car wheel spin.

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.

## Rotation angle

When objects rotate about some axis—for example, when the CD (compact disc) in [link] 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     $\text{Δ}\theta$ to be the ratio of the arc length to the radius of curvature:

$\text{Δ}\theta =\frac{\text{Δ}s}{r}\text{.}$ 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 Δ θ size 12{Δθ} {} in a time Δ t size 12{Δt} {} . The radius of a circle is rotated through an angle Δ θ size 12{Δθ} {} . The arc length Δs size 12{Δs} {} is described on the circumference.

The arc length     $\phantom{\rule{0.25em}{0ex}}\text{Δ}s$ is the distance traveled along a circular path as shown in [link] 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

$\text{Δ}\theta =\frac{2\pi r}{r}=2\pi \text{.}$

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

$2\pi \phantom{\rule{0.25em}{0ex}}\text{rad}=\text{1 revolution.}$

A comparison of some useful angles expressed in both degrees and radians is shown in [link] .

Comparison of angular units
$\text{30º}$ $\frac{\pi }{6}$
$\text{60º}$ $\frac{\pi }{3}$
$\text{90º}$ $\frac{\pi }{2}$
$\text{120º}$ $\frac{2\pi }{3}$
$\text{135º}$ $\frac{3\pi }{4}$
$\text{180º}$ $\pi$ Points 1 and 2 rotate through the same angle ( Δ θ size 12{Δθ} {} ), but point 2 moves through a greater arc length Δ s size 12{ left (Δs right )} {} because it is at a greater distance from the center of rotation ( r ) size 12{ $$r$$ } {} .

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

$2\pi \phantom{\rule{0.25em}{0ex}}\text{rad}=\text{360º}$

so that

$1\phantom{\rule{0.25em}{0ex}}\text{rad}=\frac{\text{360º}}{2\pi }\approx \text{57.}3º\text{.}$

## 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

$\omega =\frac{\text{Δ}\theta }{\text{Δ}t}\text{,}$

where an angular rotation $\text{Δ}\theta$ takes place in a time $\text{Δ}t$ . 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 $\text{Δ}s$ in a time $\text{Δ}t$ , and so it has a linear velocity

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