UNIVERSAL LAW OF GRAVITATION

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UNIVERSAL LAW OF GRAVITATION
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Geocentric vs. heliocentric model of earth

• - Ptolemy (100 170 A.D.) geocentric model Sun
  revolves around earth (Wrong!)
• From astronomical observations
• Copernicus (1473-1543) heliocentric model
  Earth planets revolve around sun
• Galileo (1564 - 1642) (1610) supports (loudly)
  the heliocentric model
• Brahe (1546 - 1601) Accurate observation of
  planetary motion
• Kepler (1571-1630), 1609 Laws I, II of
  planetary motion,
• Kepler 1619 Law III of planetary motion

About falling objects

• Aristotle (384-322 B.C.) Heavier objects fall
  faster than light objects (Wrong!)
• Galileo (1564 - 1642) Neglecting air
  resistance, all objects fall at same acceleration

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• There is a popular story that Newton was sitting
  under an apple tree, an apple fell on his head,
  and he suddenly thought of the Universal Law of
  Gravitation.

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What Really Happened with the Apple?

• Newton, upon observing an apple fall from a tree,
  began to think along the following lines
• The apple is accelerated, since its velocity
  changes from zero as it is hanging on the tree
  and moves toward the ground.
• there must be a force that acts on the apple to
  cause this acceleration. Let's call this force
  "gravity", and the associated acceleration the
  "accleration due to gravity".
• Then imagine the apple tree is twice as high.
  Again, we expect the apple to be accelerated
  toward the ground, so this suggests that this
  force that we call gravity reaches to the top of
  the tallest apple tree.

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Sir Isaac's Most Excellent Idea

• if the force of gravity reaches to the top of the
  highest tree, might it not reach even further in
  particular, might it not reach all the way to the
  orbit of the Moon!
• the orbit of the Moon about the Earth could be a
  consequence of the gravitational force, because
  the acceleration due to gravity could change the
  velocity of the Moon in just such a way that it
  followed an orbit around the earth.

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• Newton concluded that the orbit of the Moon was
  of exactly the same nature the Moon continuously
  "fell" in its path around the Earth because of
  the acceleration due to gravity, thus producing
  its orbit.

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All nine eight planets of the solar system
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Earths gravitational field
Gravitational force acts from a distance through
a field
Close to the surface
Far away from the surface
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Newtons Law of Universal Gravitation
Every particle in the Universe attracts every
other particle with a force of
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Newtons Law of Universal Gravitation
- Particle 1 is attracted by particle 2 -
Particle 2 is attracted by particle 1 - F12 and
F21 form an action-reaction pair - Force drops
off as 1/r2 as distance r between particles
increases - Can treat spherical, symmetric mass
distributions as if the mass were concentrated in
center of mass.
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Measuring the gravitational constant Cavendish
apparatus (1789)
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Free-Fall Acceleration and the Gravitational Force
Universal Gravitational force Gravitational
force near Earth Surface
Gravitational acceleration
g - is not constant as we move up from the
surface of the earth - is not dependent on the
mass of the falling object G is a universal
constant (does not change at all).
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Black board example Variation of g with altitude
Everest North Face and Rongbuk monastery (5030m),
Tibet May, 1997 Photo credit Philippe Noth

• What is the value of the acceleration due to
  gravity
• 1. At the equator (R 6.378 x 10 6 m) (Radius
  at the poles is 6. 356 x 10 6)
• 2. On top of mount Everest (h 8848 m)?
• What is it in a space station that is at an
  altitude of 350 km
• Assume ME 5.9601024 kg and RE
  6.370106 m.

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• Board Example

• What is the attractive force you (m 100 kg)
  experience from the
• person (m 70 kg) sitting in front of you
• Assume a distance r 0.5 m

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Black board example
What is the attractive force you (m1 100 kg)
experience from the two people (m2 m3 70 kg)
sitting in front of you. Assume a distance r
0.5 m and an angle q 30 for both?
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Gravity and Circular motion
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Keplers laws about planetary motion These laws
hold true for any object in orbit

• Keplers first two laws (1609)
• Planets move in elliptical paths around the sun.
  The sun is in one of the focal points (foci) of
  the ellipse
• The radius vector drawn from the sun to a planet
  sweeps out equal areas in equal time intervals
  (Law of equal areas).

Area S-A-B equals area S-D-C
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Keplers laws about planetary motion
Keplers third law (1619) III. The square of
the orbital period, T, of any planet is
proportional to the cube of the semimajor axis of
the elliptical orbit, a.
Thus, for any two planets