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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

Inflation is defined as the rise in price of a commodity.
Nureni Reply
It is defined as the rise in price of a commodity.
Nureni
the persistent rise in the prices of goods and services. or commodities.
SULEIMAN
pls the third and fourth law of supply
Yarouh
what's GDP?
Allen
gross domestic product
Annor
GDP stands for Gross Domestic Product
Sande
yes
HlobisileM
what us maxima and minima
MiXUP Reply
Maxima s below equilibrium. Whilst minima s above. Equilibrium
Afran
Wht is demand
Afran
is the willingness and the ability of a consumer to purchase goods at a given price and at a particular point in time.
Assan
Ohhk different question? Ask
Afran
why is the demand curve downwards sloppy?
Assan
3 Reasons.. 1... diminishing marginal utility 2... substitution effect 3...income effect
Harshita
thanks
Assan
Because of the negative or inverse relationship between price and quantity demanded
Afran
what is the law of diminishing returns states?
Assan
ohk
Assan
The law states that all other things being equall as much of variable factor(labour) is employed on fixed factor(land) the marginal product rises..attain a maximum and begins to fall.
Afran
What is income elasticity of demand
Afran
what is monetary policy
Edward
Monetary policy is an attempt to influence the economy by opera ting in such monetary variables
Afran
thanks
Edward
Wlcm
Afran
Wht is disutility?
Afran
is disutility? is rightly writing?
Yhlas
is it i wanna say
Yhlas
Yes please Disutility
Afran
what is macro economics?
Oyas
the branch of economics concerned with large-scale or general economic factors, such as interest rates and national productivity.
idk
in other words it is the study of the economic as a whole
idk
What is an Economic growth
Ayumo Reply
Economic growth is the process whereby the real per capita income of an economy increases over a long period of time.
Nureni
what is the generally accepted defination of economics and by who
IDY Reply
Economics is defined by Lionel Robbins as a social science which studies human behaviour as a relationship between ends and scarce means which have alternative uses
Tba
Importance of economic
Achike Reply
Helps in decision making
MP
I need like 5 importance
Achike
hi
Physcal
Hey
Muhammad
hellow dear.
juwel
hello
Al-ameen
Hello
MP
it helps an individual in rational decision making process
Assan
Fine and u
Buzabaryaho
how does it make individual in rational dicision making decisions
Annor
if an individual is faced with unlimited wants.
Assan
it also helps an individual in arranging their wants in order of their importance.
Assan
ohk
Annor
ok
Al-ameen
Hello guys
Radah
My name is Radah
Radah
Please what is a scale of preference used for?
Radah
it's use for arranging wants in order of their importance.
Assan
in other words when an individual is faced with unlimited wants,scale of preference would help the individual to select the most important wants.
Assan
Thanks
Radah
welcome
Assan
what is tourism
forgive Reply
Tourism is travel for pleasure or business
Yusuf
It is the commercial organization and operation of holidays and visits to places of interest.
Nureni
who is a price taker?
sam Reply
A price taker is a person or a company who have no control to dictate a prices of a goods or services
Unique
Someone who sets price
Nureni
In the trading world, a price taker is a trader who does not affect the price of the stock if he or she buys or sells shares.
Nureni
A price taker refers to a firm or an individual who sets the price of his good and services based on an external factor. In other words he cannot choose and set a price by himself. An example is a firm operating in perfect competition where prices are set through the price mechanism.
Tba
in a common and suitable sense state the law of diminishing returns
Unique Reply
The higher the satisfaction derived from a particular commodity,the lower the demand for it but that law doesn't match in some instances.
Nureni
state the features of an imperfect competitive market
Naomi
@NURENI instance like wat
Unique
imperfect competitive market involves large number of sellers and buyers price makers selling cost product differentiation free entry and exit of a firms
Unique
is economics a science
Alex Reply
yes. a social science.
Carlos
Yes of cause It uses scientific principles in its research. That is to say, analyzing data, making experiment as well as making deductions and drawing conclusions
Aziz
U can understand the scientific nature of economics by learning about the methods used by Abhijit Banerjee(indian) ,the nobel prize laureate 2019.
Harshita
it is considered as a social science
idk
Hence, economics is a science, a social science many can call it, or more appropriately, a young science
Taha
it can be called social science because of behaviour ,which is unpredictable.There r many theroies in economics which make economics a social science But some economic theories makes it science
Harshita
human behaviour*
Harshita
remember science derives from the root words "to know". With that being said most fields of study can be considered as a science or soft science, for they possess key knowledge to attaining understanding of our world.
Alexander
economics is a science cos it deals with human wants, desire or neads in order to satisfy them
Unique
according comparision of political science economic is science.
Hassan
what's the question?
Nureni Reply
question is answering for something to someone
Suheb
Discuss economics system
Henry Reply
discuss institutional system
Henry
Give 3 at most advantages and disadvantages of economics system and institutional syatem
Henry
Give the features characteristics of market or free enterprise
Henry
The structure of an economy is largely determined by the economic system which is a function of the economic ideology of the nation
Nureni
The economic system is grouped into 5 groups: 1: Pure market 2: Developed market 3: Centrally planned or Socialist 4: Mixed market and 5: Market Socialist Economic systems
Nureni
what is inflation
Prince Reply
Inflation is a sustained and general rise in the price of all goods and services of an economy
Tba
hello everyone , I'm New here, third degree price discrimination?
Saeed Reply
2nd degree price discrimination?
Saeed
hi
Kini
hi
Mitchel
Hi
MP
price paid by consumers after the sales tax is called?
Pinias Reply
why government impose price floor on certain products?
Pinias
how can black market be occurred when price ceiling is introduced?
Pinias
How can inflation affect goods and services?
Ph
When prices rise for energy, food, commodities, and other goods and services, the entire economy is affected
Joan
If inflation becomes too high the economy can suffer conversely, if inflation is controlled and at reasonable levels, the economy may prosper. With controlled, lower inflation, employment increases.
Joan
Is it necessary to make decision when it fails you
Evelin Reply
Pls when what fails u
MP
I think so
Kini
well i might naught know what you on about but i gotta tell you, it is necessary
Troy
yep
Ibe
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MP
yep
Ibe
good afternoon guy my name is sneh these any of you can do mathematics
HlobisileM
help me
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Yes I do
Sande
What help can we offer you.
Wahab
in analytical geometry
HlobisileM
chapter
HlobisileM
how to get the coordinates
HlobisileM
hello guys am Abdulai how is everyone doing
Abdulai
good evening guys
Alex
a(1,:3)b(t,0)c(p,-4)and corrdinate p
HlobisileM
pls send e mathematics question
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it has a drawing i can not send photos
HlobisileM
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kafui
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HlobisileM
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kafui
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Michael
Tnx
kafui
Difference between extinct and extici spicies
Amanpreet Reply
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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