Re: Newton's theory of "Universal Gravitation"
- From: srp@xxxxxxxxxxxx
- Date: Wed, 21 Nov 2007 11:28:29 -0800 (PST)
On 21 nov, 10:24, kingoleo <tahir.le...@xxxxxxxxx> wrote:
Newton's theory of "Universal Gravitation"
The Moon orbits around the Earth. Since its size does not appear to
change, its distance stays about the same, and hence its orbit must be
close to a circle.
It can be approximated to a circle for simple calculation needs, but
it is definitely not a circle. No stable orbit can be.
To keep the Moon moving in that circle--rather than
wandering off--the Earth must exert a pull on the Moon, and Newton
named that pulling force gravity.
Was that the same force which pulled all falling objects downward?
Supposedly, the above question occured to Newton when he saw an apple
falling from a tree. John Conduitt, Newton's assistant at the royal
mint and husband of Newton's niece, had this to say about the event
when he wrote about Newton's life:
In the year 1666 he retired again from Cambridge ... to his mother in
Lincolnshire & while he was musing in a garden it came into his
thought that the power of gravity (which brought an apple from a tree
to the ground) was not limited to a certain distance from earth, but
that this power must extend much further than was usually thought. Why
not as high as the Moon thought he to himself & that if so, that must
influence her motion & perhaps retain her in her orbit, whereupon he
fell a-calculating what would be the effect of that superposition...
( Keesing, R.G., The History of Newton's apple tree, Contemporary
Physics, 39, 377-91, 1998)
If it was the same force, then a connection would exist between the
way objects fell and the motion of the Moon around Earth, that is, its
distance and orbital period. The orbital period we know--it is the
lunar month, corrected for the motion of the Earth around the Sun,
which also affects the length of time between one "new moon" and the
next. The distance was first estimated in ancient Greece--see here and
here.
To calculate the force of gravity on the Moon, one must also know how
much weaker it was at the Moon's distance. Newton showed that if
gravity at a distance R was proportional to 1/R2 (varied like the
"inverse square of the distance"), then indeed the acceleration g
measured at the Earth's surface would correctly predict the orbital
period T of the Moon.
Newton went further and proposed that gravity was a "universal" force,
and that the Sun's gravity was what held planets in their orbits. He
was then able to show that Kepler's laws were a natural consequence of
the "inverse squares law" and today all calculations of the orbits of
planets and satellites follow in his footsteps.
Nowadays students who derive Kepler's laws from the "inverse-square
law" use differential calculus, a mathematical tool in whose creation
Newton had a large share. Interestingly, however, the proof which
Newton published did not use calculus, but relied on intricate
properties of ellipses and other conic sections. Richard Feynman,
Nobel-prize winning maverick physicist, rederived such a proof (as
have some distinguished predecessors); see reference at the end of the
section.
Here we will retrace the calculation, which linked the gravity
observed on Earth with the Moon's motion across the sky, two seemingly
unrelated observations. If you want to check the calculation, a hand-
held calculator is helpful.
Calculating the Moon's Motion
We assume that the Moon's orbit is a circle, and that the Earth's pull
is always directed toward's the Earth's center. Let RE be the average
radius of the Earth (first estimated by Erathosthenes)
RE= 6 371 km
The distance R to the Moon is then about 60 RE. If a mass m on Earth
is pulled by a force mg, and if Newton's "inverse square law" holds,
then the pull on the same mass at the Moon's distance would be 602 =
3600 times weaker and would equal
mg/3600
If m is the mass of the Moon, that is the force which keeps the Moon
in its orbit. If the Moon's orbit is a circle, since R = 60 RE its
length is
2 π R = 120 π RE
Suppose the time required for one orbit is T seconds. The velocity v
of the motion is then
v = distance/time = 120 π RE/T
(Please note: gravity is not what gives the Moon its velocity.
Quite an assertion. In Newtonian mechnics, the velocity of any
astronomical body is directly determined by the force of
gravity acting between it and its primary.
André Michaud
Whatever velocity the Moon has was probably acquired when it was
created. But gravity prevents the Moon from running away, and confines
it to some orbit.)
The centripetal force holding the Moon in its orbit must therefore
equal
mv2/R = mv2/(60 RE)
and if the Earth's gravity provides that force, then
mg/3600 = mv2/(60 RE)
dividing both sides by m and then multiplying by 60 simplifies things
to
g/60 = v2/RE = (120 π RE)2/(T2 RE)
Canceling one factor of RE , multiplying both sides by 60 T2 and
dividing them by g leaves
T2 = (864 000 π2 RE)/g = 864 000 RE (π2/g)
Providentially, in the units we use g ~ 9.81 is very close to π2 ~
9.87, so that the term in parentheses is close to 1 and may be
dropped. That leaves (the two parentheses are multiplied)
T2 = (864 000) (6 371 000)
With a hand held calculator, it is easy to find the square roots of
the two terms. We get (to 4-figure accuracy)
864 000 = (929.5)2 6 371 000 = (2524)2
Then
T ≅ (929.5) (2524) = 2 346 058 seconds
To get T in days we divide by 86400, the number of seconds in a day,
to get
T = 27.153 days
pretty close to the accepted value
T = 27.3217 days
The above looks like a simple and straightforward calculation.
However, it assumes something we nowadays accept without second
thought: that the pull of the Earth would be the same if all the mass
of the Earth were concentrated in its center.
It wasn't obvious to Newton, That falling apple... sure, there was
mass pulling it down, but there was also mass pulling it sideways in
all directions, pulls which largely canceled. Even if the sum-total of
all pulls pointed towards the center of the Earth, who was to say it
obeyed the same inverse-square law as a mass concentrated at a point?
Newton did not trust the above calculation until he proved to his
satisfaction that the Earth's attraction could always be replaced by
the one of a mass concentrated at its center.
Making a discovery often involves groping and guessing, before a
clear pattern emerges. We, who know that pattern and take it for
granted, may feel the discovery was obvious. But it need not have
appeared at first.
The Formula for the Force of Gravity
Newton rightly saw this as a confirmation of the "inverse square
law". He proposed that a "universal" force of gravitation F existed
between any two masses m and M, directed from each to the other,
proportional to each of them and inversely proportional to the square
of their separation distance r. In a formula (ignoring for now the
vector character of the force):
F = G mM/r2
Suppose M is the mass of the Earth, R its radius and m is the mass of
some falling object near the Earth's surface. Then one may write
F = m GM/R2 = m g
From this
g = GM/R2
The capital G is known as the constant of universal gravitation.
That is the number we need to know in order to calculate the
gravitational attraction between, say, two spheres of 1 kilogram each.
Unlike the attraction of the Earth, which has a huge mass M, such a
force is quite small, and the number G is likewise very, very small.
Measuring that small force in the lab is a delicate and difficult
feat.
It took more than a century before it was first achieved. Only in
1796 did Newton's countryman Henry Cavendish actually measure such
weak gravitational attraction, by noting the slight twist of a
dumbbell suspended by a long thread, when on of its weights was
attracted by the gravity of a third heavy object. A century later (as
already noted) the Hungarian physicist Roland Eötvös greatly improved
the accuracy of such measurements.
Gravity in our Galaxy (Optional)
Gravity obviously extends much further than the Moon. Newton
himself showed the inverse-square law also explained Kepler's laws--
for instance, the 3rd law, by which the motion of planets slows down,
the further they are from the Sun.
What about still larger distances? The solar system belongs to the
Milky Way galaxy, a huge wheel-like swirl of stars with a radius
around 100,000 light years. Being located in the wheel itself, we view
it edge-on, so that the glow of its distant stars appears to us as a
glowing ring circling the heavens, known since ancient times as the
Milky Way. Many more distant galaxies are seen by telescopes, as far
as one can see in any direction. Their light shows (by the "Doppler
effect") that they are slowly rotating.
Gravity apparently holds galaxies together. At least our galaxy
seems to have a huge black hole in its middle, a mass several million
times that of our Sun, with gravity so intense that even light cannot
escape it. Stars are much denser near the center of our galaxy, and
their rotation near their center suggests Kepler's third law holds
there, slower motion with increasing distance.
The rotation of galaxies away from their centers does not follow
Kepler's 3rd law--indeed, outer fringes of galaxies seem to rotate
almost uniformly. This observed fact has been attributed to invisible
"dark matter" whose main attribute is mass and therefore,
gravitational attraction (see link above). It does not seem to react
to electromagnetic or nuclear forces, and scientists are still seeking
more information about it.
Exploring Further
A site about the story that Newton's inspiration about the force of
gravity came from observing an apple drop from a tree.
A detailed article: Keesing, R.G., The History of Newton's apple tree,
Contemporary Physics, 39, 377-91, 1998
Richard Feynman's calculations can be found in the book "Feynman's
Lost Lecture: The Motion of Planets Around the Sun" by D. L Goodstein
and J. R. Goodstein (Norton, 1996; reviewed by Paul Murdin in Nature,
vol. 380, p. 680, 25 April 1996). The calculation is also described
and expanded in "On Feynman's analysis of the geometry of Keplerian
orbits" by M. Kowen and H. Mathur, Amer. J. of Physics, 71, 397-401,
April 2003.
An article in an educational journal about the subjects discussed
above: The great law by V. Kuznetsov. Quantum, Sept-Oct. 1999, p.
38-41.
.
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