# 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:

$\text{Δ}\theta =\frac{\text{Δ}s}{r}\text{.}$

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
$\text{30º}$ $\frac{\pi }{6}$
$\text{60º}$ $\frac{\pi }{3}$
$\text{90º}$ $\frac{\pi }{2}$
$\text{120º}$ $\frac{2\pi }{3}$
$\text{135º}$ $\frac{3\pi }{4}$
$\text{180º}$ $\pi$

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

Inflation is defined as the rise in price of a commodity.
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
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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
Afran
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Assan
3 Reasons.. 1... diminishing marginal utility 2... substitution effect 3...income effect
Harshita
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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
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Afran
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Monetary policy is an attempt to influence the economy by opera ting in such monetary variables
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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
Economic growth is the process whereby the real per capita income of an economy increases over a long period of time.
Nureni
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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
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Helps in decision making
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if an individual is faced with unlimited wants.
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it also helps an individual in arranging their wants in order of their importance.
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Please what is a scale of preference used for?
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.
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Tourism is travel for pleasure or business
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It is the commercial organization and operation of holidays and visits to places of interest.
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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
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The higher the satisfaction derived from a particular commodity,the lower the demand for it but that law doesn't match in some instances.
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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
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yes. a social science.
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idk
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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.
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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
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Inflation is a sustained and general rise in the price of all goods and services of an economy
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2nd degree price discrimination?
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price paid by consumers after the sales tax is called?
why government impose price floor on certain products?
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Ph
When prices rise for energy, food, commodities, and other goods and services, the entire economy is affected
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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.
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