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Physics for Scientists and
Engineers, 3rd edition
Fishbane
Gasiorowicz
Thornton
© 2005 Pearson Prentice Hall
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Chapter 12
Gravitation
Main Points of Chapter 12
• Early observations of planetary motion
• Kepler’s laws
• Universal gravitation
• Potential energy
• Planets and satellites
• Escape speed
• Orbits
• Gravitation and extended objects
Main Points of Chapter 12
• Tides
• Equality of inertial and gravitational mass
• Einstein’s theory of gravitation
• Equivalence principle
12-1 Early Observations of Planetary Motion
Just looking at the sky, one would assume that
the earth is the center of the universe, and that
everything we see is in orbit around us.
Early models of the solar system were
geocentric, but some of the planets were
observed to exhibit retrograde motion:
Models of Retrograde motion
• Ptolemaic Explanation(87 - 150 A.D.) The motions of the Sun,
Moon, and stars were based on perfect circles with the earth at
the center. The planets moved around small circular paths that
in turn moved around larger circular orbits around the Earth.
• Copernican Explanation (heliocentric)
• The planets further from the sun are moving more slowly in
their orbits than those closer to the sun. The retrograde motion
of Mars occurs when the Earth passes by the slower moving
Mars.
• http://www.lasalle.edu/~smithsc/Astronomy/retrograd.html
12-1 Early Observations of Planetary Motion
The geocentric model was
made more elaborate, with
epicycles upon epicycles,
but ultimately was unable
to explain retrograde
motion.
12-1 Early Observations of Planetary Motion
Kepler’s laws:
1. Planets move in planar elliptical
paths with the Sun at one focus of
the ellipse.
12-1 Early Observations of Planetary Motion
Kepler’s laws:
2. During equal time intervals the
radius vector from the Sun to a
planet sweeps out equal areas.
12-1 Early Observations of Planetary Motion
Kepler’s laws:
3. If T is the time that it takes for a planet to
make one full revolution around the Sun,
and if R is half the major axis of the ellipse
(R reduces to the radius of the planet’s orbit
if that orbit is circular), then:
(12-1)
where C is a constant whose value is
the same for all planets.
12-2 Newton’s Inverse-Square Law
Newton wanted to explain Kepler’s laws;
found that:
• Force must be central
• Inverse-square law
• Possible paths must be conic sections:
Newton’s explanation of Kepler’s 3rd law
• The elliptical paths described by Kepler could be
explained by the inverse square law k/r2 Where
k is a constant
• F = ma
• k/r2 = mv2/ R using T = 2pieR/v and squaring
• T2 = 4 pie 2 R2 / v2
• T2 = 4 pie 2 R2 / k/mR = (4 pie 2 m/k) R3
This is Kepler’s 3rd law with C= (4 pie 2 m/k)
12-2 Newton’s Inverse-Square Law
Law of universal gravitation includes all
requirements:
(12-4)
G is a constant that can be measured using
known masses; find:
(12-7)
Return to Kepler’s 3rd law
C= (4 pie 2 m/k) = (4 pie 2 m/GmM)
2
T=
(4 pie
2
3
R /GM)
12-2 Newton’s Inverse-Square Law
Potential energy can be derived from force:
(12-8)
12-3 Planets and Satellites
Newton realized that falling with a
sufficiently large initial horizontal
velocity is orbiting – that is, the
same force that causes the apple
to fall from the tree also keeps the
Moon in its orbit.
12-3 Planets and Satellites
Escape speed is outward speed needed to escape
from Earth’s gravitational potential well (that is, to
make total energy nonnegative):
(12-10)
Escape velocity of the earth
The total energy necessary for a rocket to escape the earths
gravitational attraction is 0
½ mvesc2 – Gmem/Re=0=E
Vesc = 11.2 km/s
The escape velocity is indepent of
direction
12-3 Planets and Satellites
Types of Orbits
Total energy:
(12-9)
If E is negative, orbits are
closed – elliptical or circular
If E is positive, orbits are
open – hyperbola
12-3 Planets and Satellites
Some facts and definitions:
• Orbits are all conical sections
• Ellipse has semimajor and semiminor axes
• Ellipse equation:
(12-11)
• Closest approach to sun is called
perihelion; farthest is aphelion
12-3 Planets and Satellites
• Semimajor axis:
(12-12)
• Eccentricity:
(12-13)
12-3 Planets and Satellites
Types of Orbits: circle, ellipses,
parabola, hyperbola
12-4 Gravitation and Extended Objects
The Gravitational Force Due to a Spherically
Symmetric Object
Within a hollow shell,
the gravitational force
is zero – forces from
opposite sides cancel
12-4 Gravitation and Extended Objects
The Gravitational Force Due to a Spherically
Symmetric Object
Within the object,
force at r is due to
mass inside r:
12-4 Gravitation and Extended Objects
Dark Matter:
Discovered within and around galaxies and
galaxy clusters
Within galaxies: needed to explain
rotational speed of outer areas
Within galaxy clusters: needed to make
clusters gravitationally bound
No matter of any kind is detectable where it
is needed – scientists still figuring it out
12-4 Gravitation and Extended Objects
Acceleration of gravity, g, varies with altitude
above Earth’s surface, due to changing distance
from Earth’s center:
g(h) = GM/(Re+h)2
= GM/Re(1+h/Re)2
For h/Re << 1 we can use 1/(1+x)2 = 1-2x
(12-19)
12-4 Gravitation and Extended Objects
Acceleration of gravity, g, varies with altitude
above Earth’s surface, due to changing distance
from Earth’s center:
(12-19)
This is a very small difference, even at the top
of Mt. Everest!
12-4 Gravitation and Extended Objects
Tidal Forces
Tidal forces occur because extended
objects are not points. Near side of Earth
is closer to Moon than far side, so the
magnitude of the gravitational force is not
the same on both sides.
This difference in gravitational forces on
different parts of an extended object is the
tidal force.
12-4 Gravitation and Extended Objects
Tidal Forces
Different accelerations at
A and B:
12-4 Gravitation and Extended Objects
Tidal Forces
The friction of the water provides the earth
a negative angular acceleration.
Conservation of momentum causes an
increase in distance from the earth, .5m a
year. Ultimately the earth will slow its
rotation until one face will always the
moon.
12-5 A Closer Look at Gravitation
Two massive bodies will exert equal and
opposite forces on each other – does that mean
they orbit around each other?
Yes – actually, they both orbit around their
common center of mass.
For a small planet and a large star, the difference
between this and orbiting the star’s center is
tiny.
For a large planet or a double star, it can be a
large effect.
12-5 A Closer Look at Gravitation
If several objects are exerting gravitational forces on
each other, net force is vector sum of forces, and net
potential energy is sum of potential energies.
12-5 A Closer Look at Gravitation
Equality of inertial and gravitational masses:
The m in F = ma is the inertial mass, a measure of
how resistant an object is to an external force.
The m in the gravitational force equation is the
gravitational mass, a measure of the gravitational
force the object exerts on another mass.
Apparently the inertial and gravitational masses of
an object are equal, but why?
12-6 Einstein’s Theory of Gravitation
Equivalence of gravitational and inertial
mass led Einstein to formulate the
equivalence principle:
No experiment can distinguish between the
following two situations:
(1) a physical system at rest that is subject to a
uniform gravitational force; and
(2) a physical system that is uniformly accelerating
in the absence of gravity.
12-6 Einstein’s Theory of Gravitation
Predictions of equivalence principle:
• Light falls under the influence of gravity
• Gravitational lenses
• Black holes
• Precession of planetary orbits
Summary of Chapter 12
• Kepler’s three laws:
1. Planets move in planar elliptical paths with the Sun at
one focus of the ellipse.
2. During equal time intervals, the radius vector from the
Sun to a planet sweeps out equal areas.
3. If T is the time that it takes for a planet to make one
full revolution around the Sun, and if R is half the major
axis of the ellipse (R reduces to the radius of the orbit of
the planet if that orbit is circular), then
(12-1)
where C is a constant whose value is the
same for all planets.
Summary of Chapter 12, cont.
• Newton showed that Kepler’s laws result from
universal law of gravitation:
(12-4)
• Not all masses are pointlike; if spherically
symmetric, can treat all mass within radius
of measurement to be concentrated at center
• Differences in gravitational force across an
extended object are tidal forces
Summary of Chapter 12, cont.
• Einstein’s general theory of relativity gives a
much more complete description of gravity;
starts with equivalence principle for inertial
and gravitational mass