<< Chapter < Page Chapter >> Page >
  • 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     Δ θ size 12{Δθ} {} to be the ratio of the arc length to the radius of curvature:

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

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.
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} {} .

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.
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     Δ s size 12{Δs} {} is the distance traveled along a circular path as shown in [link] Note that 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 size 12{r} {} . The circumference of a circle is r size 12{2πr} {} . Thus for one complete revolution the rotation angle is

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

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

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

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

Comparison of angular units
Degree Measures Radian Measure
30º size 12{"30"°} {} π 6 size 12{ { {π} over {6} } } {}
60º size 12{"60"°} {} π 3 size 12{ { {π} over {3} } } {}
90º size 12{"90"°} {} π 2 size 12{ { {π} over {2} } } {}
120º size 12{"120"°} {} 3 size 12{ { {2π} over {3} } } {}
135º size 12{"135"°} {} 4 size 12{ { {3π} over {4} } } {}
180º size 12{"180"°} {} π size 12{π} {}
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.
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 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 size 12{"360"°} {} in a circle or one revolution, the relationship between radians and degrees is thus

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

so that

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

Angular velocity

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

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

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

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Physics subject knowledge enhancement course (ske). OpenStax CNX. Jan 09, 2015 Download for free at http://legacy.cnx.org/content/col11505/1.10
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Physics subject knowledge enhancement course (ske)' conversation and receive update notifications?