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GRAVITY
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Fundamentals of Physics
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Chapter 13: Newton, Einstein, and Gravity
Albert Einstein 1872 - 1955
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Isaac Newton 1642 - 1727
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Fundamentals of Physics
Chapter 13 Gravitation
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Our Galaxy & the Gravitational Force
Newton’s Law of Gravitation
Gravitation & the Principle of Superposition
Gravitation Near Earth’s Surface
Gravitation Inside Earth
Gravitational Potential Energy
Path Independence
Potential Energy & Force
Escape Speed
Planets &Satellites: Kepler’s Laws
Satellites: Orbits & Energy
Einstein & Gravitation
Principle of Equivalence
Curvature of Space
Review & Summary
Exercises & Problems
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The World & the Gravitational Force
Gravity is a very weak force, but essential for us to exist!
– Allowed matter to spread out after the Big Bang.
– Eventually pulled large amounts of mass together:
– Clouds of gas
– Brown dwarfs
– Stars
– Galaxies (The Milky Way, Andromeda)
– Clusters of Galaxies (“The Local Group”)
– Super Clusters
• “billions & billions” of galaxies and stars
– Big Bang - hydrogen, helium, lithium, & very little else
– Big Stars - burned out & exploded creating heavier elements carbon, oxygen, iron, gold, . . .
– Gravity then pulled the sun & earth together
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Physics ~1680
By now the world knew:
•
•
•
•
Bodies of different weights fall at the same speed
Bodies in motion did not necessarily come to rest
Moons could orbit different planets
Planets moved around the Sun in ellipses with the Sun at
one focus
• The orbital speeds of the planets obeyed “Kepler’s Laws”
But why???
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Isaac Newton put it all together.
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Newton’s Law of Gravitation
Every massive body in the universe attracts every other massive body
through the gravitational force!
Newton (1665):
F ~
m1 m2
r2
F  G
m1 m2
r2
G = 6.67 x 10-11 N m2 / kg2
Gravity Force is weak!
Principle of superposition: net effect is the sum of the individual forces.
(added vectorially).
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Newton’s Law of Gravitation
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Measurement of the Gravitational Force
If two masses are brought very close together in the laboratory,
the gravitational attraction between them can be detected.
Cavendish Experiment(~1760):
F  G
m1 m2
r2
G = 6.673 x 10-11 N m2 / kg2
(relatively poorly known: 1 part in 10,000)
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Gravitation Near Earth’s Surface
•
The Earth is not uniform.
•
The Earth is not a sphere.
– An ellipsoid ~0.3%
•
Earth is rotating.
– Centripetal acceleration
FN-mag = m(-w2R)
mg = mag-m(w2R)
(measured wt.) = (mag. of gravitational force) – (mass x centripetal acceleration)
g = ag –w2R
on equator difference ~ 0.034 m/s2
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Gravitation Near Earth’s Surface
Newton: The force exerted by any spherically symmetric object on a point
mass is the same as if all the mass were concentrated at its center.
r
Newton had to invent calculus to prove it!
F  G
m1 m2
r2
r is the distance between the centers of the two bodies
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Newton’s Law of Gravitation
Newton: The gravitational force provides the centripetal acceleration to hold
the earth in its orbit around the sun:
M m
v2
F  m
 G S2 E
r
r
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Orbital Motion
Newton:
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The gravitational attraction between the Earth and the
Moon causes the Moon to orbit around the Earth rather
than moving in a straight line.
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Gravitation Inside A Shell
Newton: gravitational force inside a uniform spherical shell of matter:
F0  G
m1 m0
m2 m0

G
r12
r2 2
But:
m1
r12
 2
m2
r2
F0  0
The forces on m0 due to m1 and m2
vary as ~1/r2, but the masses of m1 and
m2 grow as r2, hence the two forces
cancel out!
A uniform spherical shell of matter exerts no net gravitational
force on a particle located anywhere inside it.
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Gravitation Outside A Shell
Newton: A uniform spherical shell of matter attracts a particle that is outside the
shell as if all the shell’s matter were concentrated at its center.

gr 

dg
 r
G dM

dg r   2 cos
s
2  R sin  R d 
dA
dM  M
 M
A
4 R2
s 2  r 2  R 2  2 r R cos
2 s ds  2 r R sin  d
R 2  s 2  r 2  2 s r cos
Integrating from s = r - R to s = r + R:
GM
gr   2
r
Gravitational acceleration at a distance r from a point particle!
Newton invents calculus!!!
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Gravitation Inside & Outside a Uniform Sphere
Newton: Consider the sphere as a set of concentric shells:
The gravitational force inside a uniform spherical shell of matter is zero.
Only the matter inside radius r attracts an object at that radius.
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Gravitation Inside Earth
Newton: A uniform spherical shell of matter attracts a particle that is outside the shell as if all
the shell’s matter were concentrated at its center.
Newton: The gravitational force inside a uniform spherical shell of matter is zero.
gr  
GM
r
3
R
Fr  
GM m
r
3
R
m oscillates back & forth!


F ~ r
Hooke’s Law
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Gravitational Potential Energy of a 2-Particle System
Gravity is a conservative force.
m
The work done by the gravitational force on a particle moving
from an initial point A to a final point G is independent of the
path taken between the points.
Potential Energy:
U   U r  W  
 F  r  dr
M m
r2
GM m
U Ur 
r
F r   G

r
(attractive force)
Only DU is important; the location of U = 0 is arbitrary.
Choose U = 0 to be the point at which the masses are far apart.
U  0
U r   
GM m
r
M
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Escape Velocity
Consider a projectile of mass m, leaving the surface of a planet. The loses kinetic energy
and gains gravitational potential energy.
K  21 mv 2  0
U  G
E  K  U  constant
Mm
 0
R
Escape Speed: When the projectile reaches infinity, it has no
potential energy (U = 0) and no kinetic energy (stopped: v =
0): the total mechanical energy is zero.
K U  0
1
2
mv 2  G
Mm
 0
R
v 
2G M
R
Does not depend on the mass of the projectile nor on its
initial direction (i.e. escape speed not escape velocity).
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Escape Velocity
U r   
G ME m
r
Choosing
U   0
E  K  U  constant
U r 
E
ve  2 gr  2(9.81m / s 2 )(6.38 106 m)  11.2km / s
Escapes
Escape Speed
~ 25,000 mi/hr
Bound Orbit
No hydrogen & helium in the earth’s atmosphere:
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v  vesc
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http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/projectileOrbit/projectileOrbit.html
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Planets & Satellites: Kepler’s Laws
Tycho Brahe (1546-1601)
Johannes Kepler (1571-1630)
Sun
Path of Mars across the sky.
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Planets & Satellites: Kepler’s Laws
Sun
Kepler’s Laws:
– Elliptical orbits with the Sun at one focus.
– Equal Areas in Equal Times by sun-planet line.
– T2 ~ r3
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(Period & Mean Distance from sun)
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Kepler’s 1st Law
The Law of Orbits: All planets move in elliptical orbits, with the Sun at one focus.
ellipse - sum of distances from 2 foci is constant (2a).
eccentricity of earth = 0.0167
e = 0  circle (1 focus)
M  m
Sun is very near one focus.
perihelion - closest point to Sum
aphelion - farthest point
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Kepler’s 2nd Law
The Law of Areas: A line that connects a planet to the Sun sweeps out equal
areas in the plane of the plane’s orbit in equal times; that is, the rate dA/dt at
which it sweeps out area A is constant.
slower
faster
http://ww2.unime.it/dipart/i_fismed/wbt/mirror/ntnujava/Kepler/Kepler.html
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Kepler’s 2nd Law
The Law of Areas: A line that connects a planet to the Sun sweeps out equal areas
in the plane of the plane’s orbit in equal times; that is, the rate dA/dt at which it
sweeps out area A is constant.
Conservation of Angular Momentum:
dA 
1
2
 
r  v dt

1 
r  mv dt
2m
1 
dA 
L dt
2m
dA 
dA
1 

L  constant
dt
2m

L  constant
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Fundamentals of Physics
Any Central Force
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Kepler’s 3rd Law
The Law of Periods: The square of the period of any planet is proportional to the
cube of the semimajor axis of its orbit.
See Table 13-3.
2
3
T
~ r
Consider the special case of a circular orbit:
v2
Mm
F  m
 G 2
r
r
GM
v2 
r
2 r
Period : T 
v

T
2
4 2 3

r
GM
Determine the mass of a planet by measuring period
and mean orbital distance of a moon orbiting it.
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What did Newton know?
Kepler’s 3rd Law:
acceleration of the moon:
velocity of the moon:
His own law!
T 2 ~ r3
v2
a
r
v
2 r
T
F  ma
A little algebra by Newton:
F ~
1
r2
The force holding the moon in orbit depends on the
square of its distance from earth.
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Einstein & Gravitation
“Science is a perfectionist” - Narlikar
– Newton’s Law of Gravity works great!
– But, how & why does it work?
– A small problem with the planet Mercury.
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Einstein & Gravitation
A “Small” problem with the planet Mercury:
• Mercury takes 88 Earth days to orbit the sun.
• But, measurements show the perihelion “advancing”:
– 0.159o per 100 years! (Newton’s Law says zero!)
• Attractions by the other planets almost explain it.
– 0.147o per 100 years!
• 43 arc-seconds per 100 years unexplained.
Einstein explains it!
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Einstein & Gravitation
•
Special Theory of Relativity
– Space & Time are intimately connected.
c = constant everywhere
– Mass & Energy are equivalent.
E = m c2
•
General Theory of Relativity
– A theory of gravitation
space–time geometry
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
mass (material bodies)
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Einstein & Gravitation
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•
Galileo / Newton Universe
– No interdependence of time, space & mass.
• Time flows uniformly.
• Space is immutable.
– Euclidean geometry - “straight lines”
• Mass of a body is constant.
(shape & dimension of rigid bodies is also constant)
•
Einstein Universe
• Gravitational forces reach out to infinity.
– All bodies are moving in the gravitational field of other bodies.
Not moving in a straight line.
• Space is curved!
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Einstein & Gravitation
The General Theory of Relativity:
The Equivalence Principle:
Is it gravity or rockets that causes a?
Einstein’s Strong Equivalence Principle:
No experiment can tell the difference!
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Gravity & Light:
Observe light in an accelerating elevator:
Einstein’s Strong Equivalence Principle:
If light appears to follow a curved path in the elevator,
gravity must also cause it to curve.
Einstein: Light does not travel in a straight line!
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Einstein’s View of Gravitation
In his General Theory of Relativity, Einstein explained the force
of attraction between massive objects in this way:
“Mass tells space-time how to curve, and the curvature of spacetime tells masses how to accelerate.”
“space-time” refers to 4-dimensional space: x, y, z, ct
Einstein: Space-Time is curved by the presence of mass!
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Bending of Space Time
•
•
•
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Newton said the forces due to the mass of the Earth and the mass of
the Moon kept the Moon in orbit.
Einstein said the Moon was trapped in the funnel of the Earths gravity
well.
– A gravity well is formed when a mass bends the fabric of spacetime.
General Theory of Relativity
– Orbit is not caused by forces but by the curvature of space-time
(funnel curve)
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Einstein & Gravitation
Einstein: Space-Time is curved by the presence of mass!
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Einstein proposed a radical experiment to test his theory:
1915
1919: Einstein’s prediction verified by Eddington during a solar
eclipse by the moon.
Einstein becomes the most famous scientist of the 20th century!
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Gravitational Lensing
“An Einstein Ring”
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Gravitational Lensing
These usually involve light paths from quasars & galaxies being bent by intervening
galaxies & clusters.
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“Einstein Ring”
“Einstein Cross”
a galaxy behind a galaxy
multiple images
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