GRAVITY

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GRAVITY
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Chapter 13 Newton, Einstein, and Gravity
Isaac Newton 1642 - 1727
Albert Einstein 1872 - 1955
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Fundamentals of Physics

• Chapter 13 Gravitation
• Our Galaxy the Gravitational Force
• Newtons Law of Gravitation
• Gravitation the Principle of Superposition
• Gravitation Near Earths Surface
• Gravitation Inside Earth
• Gravitational Potential Energy
• Path Independence
• Potential Energy Force
• Escape Speed
• Planets Satellites Keplers 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
  Keplers Laws

But why??? Isaac Newton put it all together.
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Newtons Law of Gravitation
Every massive body in the universe attracts every
other massive body through the gravitational
force!
Newton (1665)
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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Newtons 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)
G 6.673 x 10-11 N m2 / kg2 (relatively
poorly known 1 part in 10,000)
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Gravitation Near Earths 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 Earths 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!
r is the distance between the centers of the two
bodies
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Newtons Law of Gravitation
Newton The gravitational force provides the
centripetal acceleration to hold the earth in its
orbit around the sun
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Orbital Motion
The gravitational attraction between the Earth
and the Moon causes the Moon to orbit around the
Earth rather than moving in a straight line.
Newton
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Gravitation Inside A Shell
Newton gravitational force inside a uniform
spherical shell of matter
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 shells matter were concentrated at
its center.
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 shells matter were concentrated at
its center.
Newton The gravitational force inside a
uniform spherical shell of matter is zero.
m oscillates back forth!
Hookes Law
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Gravitational Potential Energy of a 2-Particle
System
Gravity is a conservative force.
Gravity is a conservative force.
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
(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.
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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.

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.
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
Escape Speed
25,000 mi/hr
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http//ww2.unime.it/dipart/i_fismed/wbt/mirror/ntn
ujava/projectileOrbit/projectileOrbit.html
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Planets Satellites Keplers Laws
Tycho Brahe (1546-1601) Johannes Kepler
(1571-1630)
Sun
Path of Mars across the sky.
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Planets Satellites Keplers Laws
Sun
Keplers Laws

• Elliptical orbits with the Sun at one focus.

• Equal Areas in Equal Times by sun-planet line.

• T2 r3 (Period Mean Distance from sun)

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Keplers 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)
Sun is very near one focus.
perihelion - closest point to Sum aphelion -
farthest point
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Keplers 2nd Law
The Law of Areas A line that connects a planet
to the Sun sweeps out equal areas in the plane of
the planes 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/ntn
ujava/Kepler/Kepler.html
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Keplers 2nd Law
The Law of Areas A line that connects a planet
to the Sun sweeps out equal areas in the plane of
the planes orbit in equal times that is, the
rate dA/dt at which it sweeps out area A is
constant.
Conservation of Angular Momentum
Any Central Force
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Keplers 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.
Consider the special case of a circular orbit
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?
Keplers 3rd Law
acceleration of the moon
velocity of the moon
His own law!
A little algebra by Newton
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
• Newtons 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! (Newtons 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
• spacetime geometry ? mass (material bodies)

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Einstein Gravitation

• 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?
Einsteins Strong Equivalence Principle
No experiment can tell the difference!
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Gravity Light
Observe light in an accelerating elevator
Einsteins 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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Einsteins 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 space-time
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

• 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 space-time.
• 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 Einsteins 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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