0.5 4.6 kepler’s laws  (Page 3/3)

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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.

Section summary

• Kepler’s laws are stated for a small mass $m$ orbiting a larger mass $M$ in near-isolation. Kepler’s laws of planetary motion are then as follows:

Kepler’s first law

The orbit of each planet about the Sun is an ellipse with the Sun at one focus.

Kepler’s second law

Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.

Kepler’s third law

The ratio of the squares of the periods of any two planets about the Sun is equal to the ratio of the cubes of their average distances from the Sun:

where $T$ is the period (time for one orbit) and $r$ is the average radius of the orbit.

• The period and radius of a satellite’s orbit about a larger body $M$ are related by
${T}^{2}=\frac{{4\pi }^{2}}{\text{GM}}{r}^{3}$

or

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

Problem exercises

A geosynchronous Earth satellite is one that has an orbital period of precisely 1 day. Such orbits are useful for communication and weather observation because the satellite remains above the same point on Earth (provided it orbits in the equatorial plane in the same direction as Earth’s rotation).

Unreasonable Results

(a) Based on Kepler’s laws and information on the orbital characteristics of the Moon, calculate the orbital radius for an Earth satellite having a period of 1.00 h. (b) What is unreasonable about this result? (c) What is unreasonable or inconsistent about the premise of a 1.00 h orbit?

a) $5\text{.}\text{08}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}\text{km}$

b) This radius is unreasonable because it is less than the radius of earth.

c) The premise of a one-hour orbit is inconsistent with the known radius of the earth.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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 ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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