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

 Page 3 / 5
${F}_{\text{net}}={\text{ma}}_{\text{c}}=m\frac{{v}^{2}}{r}\text{.}$

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 is the stm
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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
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scanning tunneling microscope
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Rafiq
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 .?
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
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biomolecules are e building blocks of every organics and inorganic materials.
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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absolutely yes
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it is a goid question and i want to know the answer as well
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for teaching engĺish at school how nano technology help us
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NANO
how can I make nanorobot?
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what is fullerene does it is used to make bukky balls
are you nano engineer ?
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fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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