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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     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 are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, College physics arranged for cpslo phys141. OpenStax CNX. Dec 23, 2014 Download for free at http://legacy.cnx.org/content/col11718/1.4
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