Chapter 14GRAVITATION

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Chapter 14 GRAVITATION
14.1 The World and the Gravitational Force
Claudius Ptolemy (A. D. 2nd century) Nicolaus
Copernicus (1473 ? 1543) Johannes Kepler (1571 ?
1630) Isaac Newton (in 1665) Newtons
theory of celestial mechanics has stood up
remarkably well to the test of centuries and it
remains one of the most accurate and successful
theories in all of physics, that is to say, in
all of science.
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aM/aa (0.0027 m/s2)/(9.8 m/s2) 2.8
? 10?4
(6400 km)2/(380,000)2 2.8 ? 10?4
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14.2 Newtons Law of Universal Gravitation
14-2-1 Newtons law of universal gravitation
The law of universal gravitation formulated by
Newton states
Every particle attracts every other
particle with a force directly proportional to
the product of their masses and inversely
proportional to the square of the distance
between them.
The constant G is known as the gravitational
constant.
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At the surface of the Earth
At high altitude above the surface of the Earth
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14-2-2 The vector force The vector form
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14.3 Gravitation and the Principle of
Superposition 14-3-1 Principle of Superposition
When more than two bodies are
interacting, the net gravitational force on any
one of them follows the principle of
superposition
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14-3-2 The Gravitational constant G
The first laboratory determination of G
from the force between spherical masses at close
distance was done by Henry Cavendish in 1798.
The presently accepted value of G G 6.673
? 10?11 N?m2/kg2 with an uncertainty about ?
0.15. Thus, the mass of the Earth can be
determined.
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14.4 Gravitation Near Earths Surface
The magnitude of the gravitational force
acting on a particle of mass m, located at an
external point a distance r from the Earths
center, can then be written as
From Newtons second law, F mag, we obtain
See TABLE 14-1 Variation of ag with Altitude
(page 299).
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• The real Earth differs from our model Earth in
  three ways
• The Earths crust is not uniform. The density of
  Earth varies radially as shown in Fig. 14-6 (page
  299). There are also local density variations
  everywhere.
• The Earth is not a sphere. The Earth is
  approximately an ellipsoid, flattened at the
  poles and bulging at the equator.
• The Earth is rotating. The Earth is no longer a
  inertial reference frame.

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The variation of g with latitude
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At the equator
go ? g ?2RE 0.034 m/s2
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14.4 Gravitation Inside Earth
There two shell theorems are very useful
in simplifying the analysis of the gravitational
force. Shell Theorem 1 A uniformly
dense spherical shell attracts an external
particle as if all the mass of the shell were
concentrated at its center. Shell Theorem 2
A uniformly dense spherical shell exerts
no gravitational force on a particle located
anywhere inside it.
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Therefore, to calculate the gravitational
force on a particle when it is a distance r from
the center of Earth, we may use shell theorems
The gravitational force on the particle is
due only to that portion of the Earth that lies
inside the sphere of radius r, and from shell
theorem 1 it is concluded that we can consider
that mass to be concentrated at the center of the
Earth.
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14-6 Gravitational Potential Energy 14-6-1
Calculating the potential energy
The gravitational force is conservative
Thus, we can find the change in the potential
energy of the system as m moves between points a
and b
Take rb ? and Ub 0, we have
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The potential energy defined above leads
to the familiar mgy for a small difference in
evaluation y near the surface of the Earth
When y ltlt RE, we can use the binomial expansion
to approximate the last term as (1 x)?1 1 ? x
??? ? 1 ? x, which gives
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14-6-2 Escape speed
We can find the escape speed v for the
Earth (or any other bodies) by using conversation
of energy
and solving for v we find
See TABLE 14-2 Some Escape Speeds (page 305).
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14-6-3 Potential energy of many-particle systems
We now consider another interpretation
for U(r). Consider two objects, of masses m and
M, separated by an infinitely large distance and
at rest. We take one of the particles (m, for
example) and move it slowly and at constant
velocity toward the other, until the separation
of the two particles is r. Wnet Wext Wgrav
0, Thus the work done by our hand is Wext
? Wgrav ? GMm/r.
We can give this alternative view of the
potential energy The potential energy of
a system of particles is equal to the work done
by an external agent to assemble the system,
starting from the standard reference
configuration.
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Consider three bodies of masses m1, m2,
and m3. Let them initially be at rest infinitely
far from one another.
The total work done by the external agent in
assembling the system is
If we wanted to separate the system into three
isolated masses once again, we would have to
supply an amount of energy. This energy is called
binding energy.
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14-7 Planets and Satellites Keplers Laws
1. The law of orbits All planets move in
elliptical orbits having the Sun at one focus.
a the semimajor axis e the eccentricity
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Aphelion the maximum distance Ra, Ra a(1
e) Perihelion the closest distance Rp, Rp
a(1 ? e) For circular orbits, Ra Rp a.
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2. The law of Areas A line joining any planet to
the Sun sweeps out equal areas in equal times.
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The rate at which the area is swept out is
Assuming we can regard the more massive body M as
at rest, the angular momentum of the orbiting
body m relative to the origin at the central body
is
Thus
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3. The law of periods The square of the period
of any planet about the Sun is proportional to
the cube of the semimajor axis of its orbit. We
take an example from circular orbits
The rotational period T 2?r/v. Thus,
A similar result is obtained for elliptical
orbits, with the radius r replaced by the
semimajor axis a.
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Keplers laws also apply to stars orbiting about
each other.
The orbits of the two stars in the binary
system Krüger 60
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The orbital motion of Earth is perturbed by
inter-planetary forces, mainly due to Jupiter.
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14-8 Energy Considerations in planetary and
satellite motion
Consider again the motion of a body of
mass m about a massive body of mass M. We
consider M to be at rest in an inertial reference
frame, with the body m moving about it in a
circular orbit with tangential speed v and
angular speed ?. The potential energy of the
system is
The kinetic energy is
The total mechanical energy is
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For an elliptical orbit
For an elliptical orbit the total energy
is also negative.The quantity r must be taken
equal to the semimajor axis of the ellipse. The
total energy does not depend on the shape of the
ellipse, but only on its overall size.
Incidentally, the angular momenta of these orbits
differ.
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E lt 0
E 0
E gt 0
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The total energy conservation
The angular momentum conservation
Solving above equations
Focus parameter
Ellipticity
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14-9 The Gravitational Field
The point of view of action-at-a-distance
The interaction between the two particles is
direct. The point of view of gravitational field
A particle modifying the space around it in some
way and setting up a gravitational field, then
acts on any other particle, exerting the force of
gravitational attraction on it.
We must determine the gravitational field
established by a given distribution of particles.
Thus, the gravitational field strength at a point
is defined as the gravitational force per unit
mass at that point. In terms of a test mass mo,
And the force on a particle at any point in that
field
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14-9 Modern Developments in gravitation 14-9-1
Dark matter
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14-9-2 Inertial mass and gravitational mass
The inertial mass and the gravitational
mass of a body are the same.
14-9-3 The principle of equivalence
The similarity between the effects of
gravity and the effects of a suitable accelerated
motion of the frame of reference is called the
Principle of Equivalence.
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14-9-3 The principle of equivalence
The similarity between the effects of
gravity and the effects of a suitable accelerated
motion of the frame of reference is called the
Principle of Equivalence.
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14-9-4 The general theory of relativity
Gravitation is due to a curvature (or shape) of
space that is caused by the masses.

• Most significant experimental tests
• Precession of the perihelion of Mercury.
• Bending of light.
• Delay of radar echoes.

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Problems

1. 14-13 (on page 317),
2. 14-17,
3. 14-20,
4. 14-25,
5. 14-44,
6. 14-59.