Kepler's third Law
We have already derived Kepler’s third law for the particular case of circular orbits. Equation (13.12) shows that the period of a satellite or planet in a circular orbit is proportional to the 3 2 power of the orbit radius. Newton was able to show that this same relationship holds for an elliptical orbit, with the orbit radius r replaced by the semi-major axis a:
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Since the planet orbits the sun, not the earth, we have replaced the earth’s mass mE in Eq. (13.12) with the sun’s mass mS. Note that the period does not depend on the eccentricity e. An asteroid in an elongated elliptical orbit with semi-major axis a will have the same orbital period as a planet in a circular orbit of radius a. The key difference is that the asteroid moves at different speeds at different points in its elliptical orbit (Fig. 13.19c), while the planet’s speed is constant around its circular orbit.
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Frequently Asked Questions
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Ans: In a small time interval dt, the line
from the sun S to the planet P turns through an angle du. The area swept out is
the colored triangle with height r, base length r du, and area dA = 1
2 r2 du in
. The rate at which area is swept out, view more..
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Ans: First consider the elliptical orbits described in Kepler’s first law. Figure 13.18 shows the geometry of an ellipse. The longest dimension is the major axis, with half-length a; this half-length is called the semi-major axis. view more..
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Ans: The name planet comes from a Greek word meaning “wanderer,” and indeed the
planets continuously change their positions in the sky relative to the background
of stars. One of the great intellectual accomplishments of the 16th and 17th centuries
was the threefold realization that the earth is also a planet, that all planets
orbit the sun, and that the apparent motions of the planets as seen from the earth
can be used to determine their orbits precisely view more..
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Ans: We have already derived Kepler’s third law for the particular case of circular
orbits. Equation (13.12) shows that the period of a satellite or planet in a circular
orbit is proportional to the 3
2 power of the orbit radius. view more..
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Ans: We have assumed that as a planet or comet orbits the sun, the sun remains absolutely
stationary. This can’t be correct; because the sun exerts a gravitational
force on the planet, the planet exerts a gravitational force on the sun of the same
magnitude but opposite direction. In fact, both the sun and the planet orbit around
their common center of mass view more..
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Ans: We have stated without proof that the gravitational interaction between two
spherically symmetric mass distributions is the same as though all the mass of
each were concentrated at its center. Now we’re ready to prove this statement.
Newton searched for a proof for several years, and he delayed publication of the
law of gravitation until he found one view more..
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Ans: We start by considering a ring on the surface of a shell , centered on
the line from the center of the shell to m. We do this because all of the particles
that make up the ring are the same distance s from the point mass m. view more..
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Ans: Any spherically symmetric mass distribution can be thought of as a combination
of concentric spherical shells. Because of the principle of superposition of forces,
what is true of one shell is also true of the combination. So we have proved half
of what we set out to prove: that the gravitational interaction between any spherically
symmetric mass distribution and a point mass is the same as though all the
mass of the spherically symmetric distribution were concentrated at its center. view more..
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Ans: We assumed at the beginning that the point mass m was outside the spherical
shell, so our proof is valid only when m is outside a spherically symmetric mass
distribution. view more..
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Ans: Because the earth rotates on its axis, it is not precisely an inertial frame of reference.
For this reason the apparent weight of a body on earth is not precisely equal
to the earth’s gravitational attraction, which we will call the true weight w
0 of
the body. Figure 13.26 is a cutaway view of the earth, showing three observers.
Each one holds a spring scale with a body of mass m hanging from it. view more..
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Ans: In 1916 Albert Einstein presented his general theory of relativity, which included
a new concept of the nature of gravitation. In his theory, a massive object actually
changes the geometry of the space around it view more..
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Ans: Think first about the properties of our own sun. Its mass M = 1.99 * 1030 kg
and radius R = 6.96 * 108
m are much larger than those of any planet, but compared
to other stars, our sun is not exceptionally massive view more..
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Ans: The first expression for escape speed in Eq. (13.29) suggests that a body of
mass M will act as a black hole if its radius R is less than or equal to a certain
critical radius. view more..
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Ans: At points far from a black hole, its gravitational effects are the same as those
of any normal body with the same mass. If the sun collapsed to form a black
hole, the orbits of the planets would be unaffected. But things get dramatically
different close to the black hole. view more..
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Ans: If light cannot escape from a black hole and if black holes are small
. how can we know that such things exist? The answer is
that any gas or dust near the black hole tends to be pulled into an accretion disk
that swirls around and into the black hole, rather like a whirlpool view more..
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Ans: HERE ISA SUMMARY OF GRAVITATION , FOR QUICK REVISION view more..
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Ans: Many kinds of motion repeat themselves over and over: the vibration of
a quartz crystal in a watch, the swinging pendulum of a grandfather
clock, the sound vibrations produced by a clarinet or an organ pipe,
and the back-and-forth motion of the pistons in a car engine. This kind of motion,
called periodic motion or oscillation view more..
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Ans: n. A body with mass m rests on a frictionless horizontal guide system, such
as a linear air track, so it can move along the x-axis only. The body is attached to
a spring of negligible mass that can be either stretched or compressed. The left
end of the spring is held fixed, and the right end is attached to the body. The
spring force is the only horizontal force acting on the body; the vertical normal
and gravitational forces always add to zero view more..
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