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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.

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.

Questions & Answers

full meaning of GPS system
Anaele Reply
how to prove that Newton's law of universal gravitation F = GmM ______ R²
Kaka Reply
sir dose it apply to the human system
Olubukola Reply
prove that the centrimental force Fc= M1V² _________ r
Kaka Reply
prove that centripetal force Fc = MV² ______ r
Kaka
how lesers can transmit information
mitul Reply
griffts bridge derivative
Ganesh Reply
below me
please explain; when a glass rod is rubbed with silk, it becomes positive and the silk becomes negative- yet both attracts dust. does dust have third types of charge that is attracted to both positive and negative
Timothy Reply
what is a conductor
Timothy
hello
Timothy
below me
why below you
Timothy
no....I said below me ...... nothing below .....ok?
dust particles contains both positive and negative charge particles
Mbutene
corona charge can verify
Stephen
when pressure increases the temperature remain what?
Ibrahim Reply
what is frequency
Mbionyi Reply
define precision briefly
Sujitha Reply
CT scanners do not detect details smaller than about 0.5 mm. Is this limitation due to the wavelength of x rays? Explain.
MITHRA Reply
hope this helps
what's critical angle
Mahmud Reply
The Critical Angle Derivation So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
dude.....next time Google it
okay whatever
Chidalu
pls who can give the definition of relative density?
Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.
Chidalu
What is momentum
aliyu Reply
mass ×velocity
Chidalu
it is the product of mass ×velocity of an object
Chidalu
how do I highlight a sentence]p? I select the sentence but get options like copy or web search but no highlight. tks. src
Sean Reply
then you can edit your work anyway you want
Wat is the relationship between Instataneous velocity
Oyinlusi Reply
Instantaneous velocity is defined as the rate of change of position for a time interval which is almost equal to zero
Astronomy

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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