# 6.5 Satellites and kepler’s laws: an argument for simplicity  (Page 3/5)

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${F}_{\text{net}}={\text{ma}}_{\text{c}}=m\frac{{v}^{2}}{r}\text{.}$

The net external force on mass $m$ is gravity, and so we substitute the force of gravity for ${F}_{\text{net}}$ :

$G\frac{\text{mM}}{{r}^{2}}=m\frac{{v}^{2}}{r}\text{.}$

The mass $m$ cancels, yielding

$G\frac{M}{r}={v}^{2}\text{.}$

The fact that $m$ cancels out is another aspect of the oft-noted fact that at a given location all masses fall with the same acceleration. Here we see that at a given orbital radius $r$ , all masses orbit at the same speed. (This was implied by the result of the preceding worked example.) Now, to get at Kepler’s third law, we must get the period $T$ into the equation. By definition, period $T$ is the time for one complete orbit. Now the average speed $v$ is the circumference divided by the period—that is,

$v=\frac{2\pi r}{T}\text{.}$

Substituting this into the previous equation gives

$G\frac{\text{M}}{r}=\frac{{\mathrm{4\pi }}^{2}{r}^{2}}{{T}^{2}}\text{.}$

Solving for ${T}^{2}$ yields

${T}^{2}=\frac{{4\pi }^{2}}{\text{GM}}{r}^{3}\text{.}$

Using subscripts 1 and 2 to denote two different satellites, and taking the ratio of the last equation for satellite 1 to satellite 2 yields

This is Kepler’s third law. Note that Kepler’s third law is valid only for comparing satellites of the same parent body, because only then does the mass of the parent body $M$ cancel.

Now consider what we get if we solve ${T}^{2}=\frac{{4\pi }^{2}}{\text{GM}}{r}^{3}$ for the ratio ${r}^{3}/{T}^{2}$ . We obtain a relationship that can be used to determine the mass $M$ of a parent body from the orbits of its satellites:

$\frac{{r}^{3}}{{T}^{2}}=\frac{G}{{4\pi }^{2}}M\text{.}$

If $r$ and $T$ are known for a satellite, then the mass $M$ of the parent can be calculated. This principle has been used extensively to find the masses of heavenly bodies that have satellites. Furthermore, the ratio ${r}^{3}/{T}^{2}$ should be a constant for all satellites of the same parent body (because ${r}^{3}/{T}^{2}=\text{GM}/{4\pi }^{2}$ ). (See [link] ).

It is clear from [link] that the ratio of ${r}^{3}/{T}^{2}$ is constant, at least to the third digit, for all listed satellites of the Sun, and for those of Jupiter. Small variations in that ratio have two causes—uncertainties in the $r$ and $T$ data, and perturbations of the orbits due to other bodies. Interestingly, those perturbations can be—and have been—used to predict the location of new planets and moons. This is another verification of Newton’s universal law of gravitation.

## Making connections

Newton’s universal law of gravitation is modified by Einstein’s general theory of relativity, as we shall see in Particle Physics . Newton’s gravity is not seriously in error—it was and still is an extremely good approximation for most situations. Einstein’s modification is most noticeable in extremely large gravitational fields, such as near black holes. However, general relativity also explains such phenomena as small but long-known deviations of the orbit of the planet Mercury from classical predictions.

## The case for simplicity

The development of the universal law of gravitation by Newton played a pivotal role in the history of ideas. While it is beyond the scope of this text to cover that history in any detail, we note some important points. The definition of planet set in 2006 by the International Astronomical Union (IAU) states that in the solar system, a planet is a celestial body that:

1. is in orbit around the Sun,
2. has sufficient mass to assume hydrostatic equilibrium and
3. has cleared the neighborhood around its orbit.

what us maxima and minima
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
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
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
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
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
Helps in decision making
MP
I need like 5 importance
Achike
hi
Physcal
Hey
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
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.
Assan
Thanks
welcome
Assan
what is tourism
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?
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
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
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?
Discuss economics system
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
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?
2nd degree price discrimination?
Saeed
hi
Kini
hi
Mitchel
Hi
MP
price paid by consumers after the sales tax is called?
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
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
Kk
MP
yep
Ibe
how can the demand side approach solve unemployment
demand solves unemployment when it is addressed with supply you can't just expect demand to work alone without supply the two are interconnected
Nureni
You have to apply the concept of aggregate demand
Tba
That is apply demand side policies to boost aggregate demand hence increasing need for labour and decreasing unemployment(more people get jobs)
Tba
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
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