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Jepler°s second observation was that the planets do not go around the sun at
a uniform speed, but move faster when they are nearer the sun and more sÌowly
when they are farther from the sun, in precisely this way: Suppose a planet
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Z7
2 Z7
Fig. 7-2. Kepler's law of areas.
1s observed at any two successive times, let us say a week apart, and that the
radius vector* is drawn to the planet for each observed position. The orbital are
traversed by the planet during the week, and the two radius vectors, bound a
certain plane area, the shaded area shown in Fig. 7-2. If two similar observations
are made a week apart, at a part of the orbit farther from the sun (where the
planet moves more slowly), the similarly bounded area is exactly the same as in
the first case. So, in accordance with the second law, the orbital speed of each
planet is such that the radius “sweeps out” equal areas in equal times.
tPinally, a third law was discovered by Kepler much later; this law is of a
diferent category from the other two, because it deals not with only a single
planet, but relates one planet to another. 'This law says that when the orbital
period and orbit size of any two planets are compared, the periods are proportional
to the 3/2 power of the orbit sỉze. In this statement the period is the tỉme interval
1t takes a planet to go completely around ïits orbit, and the size is measured by
the length of the greatest diameter of the elliptical orbit, technically known as
the major axis. More simply, If the planets went in circles, as they nearly do, the
time required to go around the circle would be proportional to the 3/2 power of
the diameter (or radius). Thus Kepler”s three laws are:
I. Each planet moves around the sun in an ellipse, with the sun at one focus.
TL. The radius vector from the sun to the planet sweeps out equal areas in
equal intervals of time.
TH. The squares of the periods of any two planets are proportional to the ceubes
of the semimajor axes of their respective orbits: 7' œ a3⁄2.
—* A radius vector is a line drawn from the sun to any point in a planet”s orbit.
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7-3 Development of dynamics
While Kepler was discovering these laws, Galileo was studying the laws of
motion. The problem was, what makes the planets go around? (In those days,
one of the theories proposed was that the planets went around because behind
them were invisible angels, beating their wings and driving the planets forward.
You will see that this theory is now modifiedL It turns out that in order to keep the
planets going around, the invisible angels must fly in a diÑerent direction and they
have no wings. Otherwise, it is a somewhat similar theoryl) Galileo discovered a
very remarkable fact about motion, which was essential for understanding these
laws. That is the principle oŸ 7nmerf2a—lŸ something is moving, with nothing
touching it and completely undisturbed, it will go on forever, coasting at a
uniform speed in a straight line. (W2, does i% keep on coasting? We do not
know, but that is the way it is.)
Newton modifed this idea, saying that the only way to change the motion
of a body is to use ƒorce. lf the body speeds up, a force has been applied #n
the direction oƒ motion. On the other hang, 1f its motion is changed to a new
đircction, a force has been applied s7deuazs. Newton thus added the idea that
a Íforce is needed to change the speed ør ¿he direcfion of motion of a body. For
example, IŸ a stone is attached to a string and is whirling around in a circle, i%
takes a force to keep 1È in the circle. We have to pull on the string. In fact, the
law is that the acceleration produced by the force is inversely proportional to the
mass, or the force is proportional to the mass times the acceleration. The more
massive a thing is, the stronger the force required to produce a given acceleration.
(The mass can be measured by putting other stones on the end of the same string
and making them go around the same circle at the same speed. In this way 1
1s found that more or less force is required, the more massive object requiring
more force.) The brilliant idea resulting from these considerations is that no
tangeniial force is needed to keep a planet in its orbit (the angels do not have to
ñy tangentially) because the planet would coast in that direction anyway. lÝ there
were nothing at all to disturb it, the planet would go of in a sứraight line. But
the actual motion deviates from the line on which the body would have gone if
there were no force, the deviation beiïng essentially at right angles to the motion,
not in the direction of the motion. In other words, because of the principle of
inertia, the force needed to control the motion of a planet arownd the sun is not
a force around the sun but #øouard the sun. (TỶ there is a force toward the sun,
the sun might be the angel, oŸ coursel)
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7-4 Newton?s law of gravitation
tFrom his better understanding of the theory of motion, Newton appreciated
that the sun could be the seat or organization of forces that govern the motion of
the planets. NÑewton proved to himself (and perhaps we shall be able to prove it
soon) that the very fact that equal areas are swept out in equal times is a precise
sign post of the proposition that all deviations are precisely rœd¿al—that the law
Of areas 1s a direct consequence of the idea that all of the forces are directed
exactly £ouard the sun.
Next, by analyzing Kepler's third law it is possible to show that the farther
away the planet, the weaker the forces. If two planets at diferent distances
from the sun are compared, the analysis shows that the forces are inversely
proportional to the squares of the respective distances. With the combination
of the two laws, Newton concluded that there must be a force, inversely as the
square of the distance, directed in a line between the two obJects.
Being a man of considerable feeling for generalities, NÑewton supposed, of
course, that this relationship applied more generally than just to the sun holding
the planets. It was already known, for example, that the planet Jupiter had
moons going around it as the moon of the earth goes around the earth, and
Newton felt certain that each planet held its moons with a force. He already knew
of the force holding us on the earth, so he proposed that this was a unớuersal
ƒorce—that cuerWthing pulls cueruthing cls.
The next problem was whether the pull of the earth on its people was the
“same” as its pull on the moon, 1.e., inversely as the square of the distance. If
an objJect on the surface of the earth falls 16 feet in the first second after it 1s
released from rest, how far does the moon fall in the same time? We might say
that the moon does not fall at all. But if there were no force on the moon, 1§
would go of in a straight line, whereas it goes in a circle instead, so it really
ƒalls ín Írom where i% would have been ïf there were no force at all. We can
calculate from the radius of the moon's orbit (which is about 240,000 miles) and
how long it takes to go around the earth (approximately 29 days), how far the
mmoon moves in its orbit in 1 second, and can then calculate how far it falls in