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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.

Rotation 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 $Δ θ Δ θ size 12{Δθ} {}$ to be the ratio of the arc length to the radius of curvature:

Δ θ = \frac{Δ s}{r} . Δ θ = \frac{Δ s}{r} . size 12{Δθ= {  {Δs}  over  {r} } "."} {}

6.1

The figure shows the back side of a compact disc. There is a scratched part on the upper right side of the C D, about one-fifth size of the whole area, with inner circular dots clearly visible. Two line segments are drawn enclosing the scratched area from the border of the C D to the middle plastic portion. A curved arrow is drawn between the two line segments near this middle portion and angle delta theta written alongside it.

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

Δ θ Δ θ size 12{Δθ} {}

in a time

Δ t Δ t size 12{Δt} {}

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

Δ θ Δ θ size 12{Δθ} {}

. The arc length

Δs Δs size 12{Δs} {}

is described on the circumference.

The arc length $Δ s size 12{Δs} {}$ is the distance traveled along a circular path as shown in Figure 6.3 Note that $r r size 12{r} {}$ 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 $r r size 12{r} {}$ . The circumference of a circle is $2π r 2π r size 12{2πr} {}$ . Thus for one complete revolution the rotation angle is

Δ θ = \frac{2π r}{r} = 2π . Δ θ = \frac{2π r}{r} = 2π . size 12{Δθ= {  {2πr}  over  {r} } =2π"."} {}

6.2

This result is the basis for defining the units used to measure rotation angles, $Δ θ Δ θ size 12{Δθ} {}$ to be radians (rad), defined so that

2π rad = 1 revolution. 2π size 12{2π" rad "=" 1 revolution."} {}

6.3

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

Degree Measures	Radian Measure
$30º 30º size 12{"30"°} {}$	$\frac{π}{6} \frac{π}{6} size 12{ { {π} over {6} } } {}$
$60º 60º size 12{"60"°} {}$	$\frac{π}{3} \frac{π}{3} size 12{ { {π} over {3} } } {}$
$90º 90º size 12{"90"°} {}$	$\frac{π}{2} \frac{π}{2} size 12{ { {π} over {2} } } {}$
$120º 120º size 12{"120"°} {}$	$\frac{2π}{3} \frac{2π}{3} size 12{ { {2π} over {3} } } {}$
$135º 135º size 12{"135"°} {}$	$\frac{3π}{4} \frac{3π}{4} size 12{ { {3π} over {4} } } {}$
$180º 180º size 12{"180"°} {}$	$π π size 12{π} {}$

Table 6.1 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 (

Δ θ Δ θ size 12{Δθ} {}

), but point 2 moves through a greater arc length

(Δ s) (Δ s) size 12{ left (Δs right )} {}

because it is at a greater distance from the center of rotation

(r) (r) size 12{ \( r \) } {}

If $Δ θ = 2 π Δ θ = 2 π size 12{Δθ=2π} {}$ rad, then the CD has made one complete revolution, and every point on the CD is back at its original position. Because there are $360º 360º size 12{"360"°} {}$ in a circle or one revolution, the relationship between radians and degrees is thus

2 π rad = 360º 2 π size 12{2π" rad"="360" rSup { size 8{ circ } } } {}

6.4

so that

1 rad = \frac{360º}{2π} \approx 57. 3º . 1 \approx 57. 3º . size 12{1" rad"= {  {"360" rSup { size 8{ circ } } }  over  {2π} } ="57" "." 3 rSup { size 8{ circ } } "."} {}

6.5

Angular Velocity

How fast is an object rotating? We define angular velocity $ω ω size 12{ω} {}$ as the rate of change of an angle. In symbols, this is

ω = \frac{Δ θ}{Δ t}, ω = \frac{Δ θ}{Δ t}, size 12{ω= {  {Δθ}  over  {Δt} } ","} {}

6.6

where an angular rotation $Δ θ Δ θ size 12{Δθ} {}$ takes place in a time $Δ t Δ t size 12{Δt} {}$ . 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 $ω ω size 12{ω} {}$ is analogous to linear velocity $v v size 12{v} {}$ . 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 $Δ s Δ s size 12{Δs} {}$ in a time $Δ t Δ t size 12{Δt} {}$ , and so it has a linear velocity

v = \frac{Δ s}{Δ t} . v = \frac{Δ s}{Δ t} . size 12{v= {  {Δs}  over  {Δt} } "."} {}

6.7

From $Δ θ = \frac{Δ s}{r} Δ θ = \frac{Δ s}{r} size 12{Δθ= { {Δs} over {r} } } {}$ we see that $Δ s = r Δ θ Δ s = r Δ θ size 12{Δs=rΔθ} {}$ . Substituting this into the expression for $v v size 12{v} {}$ gives

v = \frac{r Δ θ}{Δ t} = rω . v = \frac{r Δ θ}{Δ t} = rω . size 12{v= {  {rΔθ}  over  {Δt} } =rω"."} {}

6.8

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

v = rω orω = \frac{v}{r} . v = rω orω = \frac{v}{r} . size 12{v=rω``"or "ω= {  {v}  over  {r} } "."} {}

6.9

The first relationship in $v = rω orω = \frac{v}{r} v = rω orω = \frac{v}{r} size 12{v=rω``"or "ω= { {v} over {r} } } {}$ states that the linear velocity $v v size 12{v} {}$ is proportional to the distance from the center of rotation, thus, it is largest for a point on the rim (largest $r r size 12{r} {}$ ), as you might expect. We can also call this linear speed $v v size 12{v} {}$ of a point on the rim the tangential speed. The second relationship in $v = rω orω = \frac{v}{r} v = rω orω = \frac{v}{r} size 12{v=rω``"or "ω= { {v} over {r} } } {}$ 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 $v v size 12{v} {}$ of the car. See Figure 6.5. So the faster the car moves, the faster the tire spins—large $v v size 12{v} {}$ means a large $ω ω size 12{ω} {}$ , because $v = rω v = rω size 12{v=rω} {}$ . Similarly, a larger-radius tire rotating at the same angular velocity ( $ω ω size 12{ω} {}$ ) will produce a greater linear speed ( $v v size 12{v} {}$ ) for the car.

The given figure shows the front wheel of a car. The radius of the car wheel, r, is shown as an arrow and the linear velocity, v, is shown with a green horizontal arrow pointing rightward. The angular velocity, omega, is shown with a clockwise-curved arrow over the wheel.

Figure 6.5 A car moving at a velocity

v v size 12{v} {}

to the right has a tire rotating with an angular velocity

ω ω size 12{ω} {}

.The speed of the tread of the tire relative to the axle is

v v size 12{v} {}

, the same as if the car were jacked up. Thus the car moves forward at linear velocity

v = rω v = rω size 12{v=rω} {}

, where

r r size 12{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 $15.0 m/s 15.0 size 12{"15" "." 0`"m/s"} {}$ (about $54 km/h 54 size 12{"54"`"km/h"} {}$ ). See Figure 6.5.

Strategy

Because the linear speed of the tire rim is the same as the speed of the car, we have $v = 15.0 m/s . v = 15.0 m/s . size 12{v} {}$ The radius of the tire is given to be $r = 0.300 m . r = 0.300 m . size 12{r} {}$ Knowing $v v size 12{v} {}$ and $r r size 12{r} {}$ , we can use the second relationship in $v = rω, ω = \frac{v}{r} v = rω, ω = \frac{v}{r} size 12{v=rω,``ω= { {v} over {r} } } {}$ to calculate the angular velocity.

Solution

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

ω = \frac{v}{r} . ω = \frac{v}{r} . size 12{ω= {  {v}  over  {r} } "."} {}

6.10

Substituting the knowns,

ω = \frac{15.0 m/s}{0.300 m} = 50 . 0 rad/s. ω = \frac{15.0}{0.300} = 50 . 0 size 12{ω= {  {"15" "." 0" m/s"}  over  {0 "." "300"" m"} } ="50" "." 0" rad/s."} {}

6.11

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

ω = (15.0 m/s) / (1.20 m) = 12 . 5 rad/s. ω = (15.0 size 12{ω= \( "15" "." 0`"m/s" \) / \( 1 "." "20"`m \) ="12" "." 5`"rad/s."} {}

6.12

Both $ω ω size 12{ω} {}$ and $v v size 12{v} {}$ 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.

The given figure shows the top view of an old fashioned vinyl record. Two perpendicular line segments are drawn through the center of the circular record, one vertically upward and one horizontal to the right side. Two flies are shown at the end points of the vertical lines near the borders of the record. Two arrows are also drawn perpendicularly rightward through the end points of these vertical lines depicting linear velocities. A curved arrow is also drawn at the center circular part of the record which shows the angular velocity.

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.

Ladybug Revolution

Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.

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Figure 6.7