Gravity

1
Gravity

• Dont Let It Get You Down!

2
The Truth About Gravity

• Gravity is a phenomenon
• The phenomenon results in a force which can
  accelerate objects with mass
• g represents the acceleration due to the force
  caused by the phenomenon of gravity

3
Back to the Beginning

• Astronomy drove our understanding and discovery
• It did so without knowing it, however

4
The Importance of Being Gravity

• Gravity has basic properties that set it apart
  from the other forces (1) it is long-ranged and
  thus can act over cosmological distances (2) it
  always supplies an attractive force between any
  two pieces of matter in the Universe.
• Thus, although extremely weak, it always wins
  over cosmological distances and is the most
  important force for the understanding of the
  large scale structure and evolution of the
  Universe.

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So, let us deal with GRAVITY

• Well need a bit of a history lesson
• Brahe
• Kepler
• Newton
• Einstein

Pay close attention, gravity has many
implications!
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Tycho Brahe
1546 - 1601
A wild Dane, but he made and recorded large
quantities of accurate measurements of the
motions of the planets around the Sun.
Began working with Johannes Kepler in 1600.
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Keplers Laws of Planetary Motion
1) The planets move abort the sun in elliptical
orbits with the Sun at one focus. 2) The
radius vector joining a planet to the sun sweeps
over equal areas in equal intervals of time.
3) The square of the time of one compete
revolution of a planet about its orbit is
proportional to the cube of the orbit's
semi-major axis
The empirical discovery of these laws from Tycho
Brahe's mass of data constitutes one of the most
remarkable inductions ever made in science.
T12/ T22 R13/ R23 or T2k.R3
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Isaac Newton

• Born 1642, the year Galileo died
• Loner, tinkerer, paranoid
• 1665-1666 Plague was very good for him
• Suffered mental breakdown 1675
• Math, Chemistry, Theology, Parliament
• Died 1727
• Has his picture on the British pound note

He put the physics and mathematics to Keplers
Laws!
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Newtons Laws of Motion

• First Law - A body remains in its state of motion
  unless acted upon by an outside force
• Second Law - A body acted upon by an external
  force will change its momentum in the direction
  of the force such that the greater the force the
  greater the change in momentum (F ma).
• Third Law - Forces always occur in pairs, i.e.
  for every action there is an equal and opposite
  reaction

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Universal Law of Gravitation

• All objects in the Universe attract each other
  with a force that varies directly as the product
  of their masses and inversely as the square of
  their separation from each other.
• F G m m
• r

1
2
gravity
2
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General Relativity

• Einsteins Theory of Gravity, published 1915
• Principle of Equivalence Accelerations are
  indistinguishable from gravitational fields. They
  are equivalent.
• So, for example, when you are in freefall (like
  in an orbiting Shuttle), your downward
  acceleration is just enough to cancel the
  gravitational force.

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Einsteins View of Gravity

• Gravity is due to the curvature of spacetime.
• Spacetime is curved by mass.

13
GR Made Predictions

• Light would be bent by gravity
• Tested by Arthur Eddington during solar
  eclipse.
• Confirmed!
• Emission of gravitational radiation by
  accelerating objects
• Tested by observations of binary pulsars.
• Confirmed to 14 decimal places!

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Applications of Newtons 2nd Law and Einsteins GR

• Projectile Motion
• Pendulums
• Black Holes

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Projectile Motion
Projectile - any object given an initial velocity
which subsequently follows a path determined by
the gravitational force acting on it, and by the
frictional resistance of the atmosphere
Examples
bullet shot from a gun, rocket after the fuel is
exhausted, thrown or batted baseball
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Trajectory
Trajectory - the path followed by a projectile
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Our Assumptions

• We will consider only short length trajectories
  so that
• Gravitational force is considered constant in
  
• magnitude and direction
• Earth is an inertial system
• Air resistance is negligible

In other words, we will examine motion in a
vacuum on a flat, non-rotating Earth. In physics,
we call this creating an
Ideal Model
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First, the Forces

• Only force acting on the projectile is its
  weightremember we are in the Ideal Case.
• X-axis is horizontal y-axis is vertical
  origin is point where projectile starts its free
  flight
• So x-component of the force on the projectile
  is zero and y-component of the force is the
  weight, mg.

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Acceleration
Netwons 2nd law then tells us that the
x-component of the acceleration is zero and the
y-component is -g. In other words, trajectory is
a combination of a horizontal motion with
constant velocity and a vertical component with
constant acceleration.
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Result

• Under these conditions, projectiles travel in
  trajectories which are parabolas.
• You can derive the equation for the parabolic
  motion from Newtons Laws!

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Lets Play
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Pendulums
For small angles, sinQ Q Simple
Harmonic Motion
Q
L
Period 2p L/g
T
Measure period of oscillation and length of
pendulum, determine g!
mg sinQ
Q
x
mg
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Black Holes
A huge great enormous thing, like like
nothing. A huge big well, like a I dont
know like an enormous big nothing
Piglet describes the Heffalump,
in Winnie the Pooh by
A.A. Milne
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History

• Based on Newtons theory of gravity
• Proposed independently by
• 1783 Rev. John Mitchell
• 1796 Pierre Simon Laplace

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Structure of a Blackhole
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Artists View
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HST View
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Calculation of Critical Radius
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Calculation Continued
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Earth normal size
Earth Normal Size
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Earth half size
Earth Half Size
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Earth Quarter Size
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Earth Black Hole
Rbh 2GM/c2 R 9 mm
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Earth as Depression in Spacetime
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Blackhole is Bottomless Abyss in Spacetime
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Bending of Light
D
Q
d
D
photon
d is the approximate distance the photon falls
over the diameter D of an object with mass M
R
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From Newton, We Know...
F ma GMm/R2 Solving for a, a GM/R2
If you accelerate at rate a for time t, you move
a distance d of d 1/2 at2
Recall that t D/c. Putting everything together,
we see d 1/2 GMD2/(c2R2)
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Solving the Equations
D
Q
d
d 1/2 GMD2/(c2R2) but D2R and Q d/D
D
R
RESULT Q GM/c2R
Using GR, we would get 4 GM/c2R!
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Making the Prediction

• When you plug in the values for G, c, and the
  mass and radius of the Sun, you predict that
  light should be bent by 8.5 x 10-6 radians.
• This corresponds to 1.75 seconds of arc.
• Eddington measured 1.75probably.

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Light Benders, Mind Benders
41
Mass and Weight

• Weight is a force. It is the resultant
  gravitational force exerted on a body with mass m
  by all the other bodies on the Universe.
• Near Earths surface, the gravitational force
  from our planet dominates all others, so
  according to Newtons Laws, we can write
• Weight Fg G m Me / R2 mg
• where Me is the mass of the Earth and R is the
  radius of the Earth.

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But also...

• Since weight is a force, it is measured in
  units of force, namely Newtons.
• Remember what mass is measured in units of?

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Remember This!

• Mass is a fundamental, universal property. You
  have the same amount of mass no matter where you
  are in the Universe. The only thing that can
  change your mass is velocityaccording to General
  Relativity. As you approach the speed of light,
  you become infinitely heavy.

• Weight is not fundamental its value depends
  on what your circumstances are in the Universe.