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F net = ma c = m v 2 r . size 12{F rSub { size 8{ ital "net"} } = ital "ma" rSub { size 8{c} } =m { {v rSup { size 8{2} } } over {r} } } {}

The net external force on mass m size 12{m} {} is gravity, and so we substitute the force of gravity for F net size 12{F rSub { size 8{ ital "net"} } } {} :

G mM r 2 = m v 2 r . size 12{G { { ital "mM"} over {r rSup { size 8{2} } } } =m { {v rSup { size 8{2} } } over {r} } } {}

The mass m size 12{m} {} cancels, yielding

G M r = v 2 . size 12{G { {M} over {r} } =v rSup { size 8{2} } } {}

The fact that m size 12{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 size 12{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 size 12{T} {} into the equation. By definition, period T size 12{T} {} is the time for one complete orbit. Now the average speed v size 12{v} {} is the circumference divided by the period—that is,

v = r T . size 12{v= { {2π`r} over {T} } } {}

Substituting this into the previous equation gives

G M r = 2 r 2 T 2 . size 12{G { { ital "mM"} over {r rSup { size 8{2} } } } =m { {v rSup { size 8{2} } } over {r} } } {}

Solving for T 2 size 12{T rSup { size 8{2} } } {} yields

T 2 = 2 GM r 3 . size 12{T rSup { size 8{2} } = { {4π rSup { size 8{2} } } over { ital "GM"} } r rSup { size 8{3} } } {}

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

T 1  2 T 2  2 = r 1  3 r 2  3 . size 12{ { {T rSub { size 8{1} } rSup { size 8{2} } } over {T rSub { size 8{2} } rSup { size 8{2} } } } = { {r rSub { size 8{1} } rSup { size 8{3} } } over {r rSub { size 8{2} } rSup { size 8{3} } } } } {}

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 size 12{M} {} cancel.

Now consider what we get if we solve T 2 = 2 GM r 3 for the ratio r 3 / T 2 size 12{r rSup { size 8{3} } /T rSup { size 8{2} } } {} . We obtain a relationship that can be used to determine the mass M size 12{M} {} of a parent body from the orbits of its satellites:

r 3 T 2 = G 2 M . size 12{ { {r rSup { size 8{3} } } over {T rSup { size 8{2} } } } = { {G} over {4π rSup { size 8{2} } } } M} {}

If r size 12{r} {} and T size 12{T} {} are known for a satellite, then the mass M size 12{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 size 12{r rSup { size 8{3} } /T rSup { size 8{2} } } {} should be a constant for all satellites of the same parent body (because r 3 / T 2 = GM / 2 size 12{r rSup { size 8{3} } /T rSup { size 8{2} } = ital "GM"/4π rSup { size 8{2} } } {} ). (See [link] ).

It is clear from [link] that the ratio of r 3 / T 2 size 12{r rSup { size 8{3} } /T rSup { size 8{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 size 12{r} {} and T size 12{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 size 12{m} {} orbiting a larger mass M size 12{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:

    T 1  2 T 2  2 = r 1  3 r 2  3 , size 12{ { {T rSub { size 8{1} rSup { size 8{2} } } } over {T rSub { size 8{2} rSup { size 8{2} } } } } = { {r rSub { size 8{1} rSup { size 8{3} } } } over {r rSub { size 8{2} rSup { size 8{3} } } } } } {}

    where T size 12{m} {} is the period (time for one orbit) and r size 12{m} {} is the average radius of the orbit.

  • The period and radius of a satellite’s orbit about a larger body M size 12{m} {} are related by
    T 2 = 2 GM r 3 size 12{T rSup { size 8{2} } = { {4π rSup { size 8{2} } } over { ital "GM"} } r rSup { size 8{3} } } {}

    or

    r 3 T 2 = G 2 M . size 12{ { {r rSup { size 8{3} } } over {T rSup { size 8{2} } } } = { {G} over {4π rSup { size 8{2} } } } M} {}

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 . 08 × 10 3 km size 12{5 "." "08" times "10" rSup { size 8{3} } `"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.

Questions & Answers

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
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
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Unit 4 - uniform circular motion and universal law of gravity. OpenStax CNX. Nov 23, 2015 Download for free at https://legacy.cnx.org/content/col11905/1.1
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