System consisting of three stars: Alpha Centauri A, Alpha Centauri B, and Proxima Centauri

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Alpha Centauri

• System consisting of three stars Alpha Centauri
  A, Alpha Centauri B, and Proxima Centauri
• Alpha Centauri A and B (depicted at left) form a
  binary star system
• Binary star system two stars orbiting around
  their center of mass
• Video animation recorded at a speed 1,000,000x
  faster than real time

What force is responsible for the motion of Alpha
Centauri A and Alpha Centauri B?
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Universal Gravitation
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Gravity on Earth

• Neglecting air resistance, all objects near the
  surface of the Earth are in free-fall
• Know the acceleration due to gravity on the
  earths surface is 9.8 m/s2

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Objectives

• Discuss the historical development of the law of
  universal gravitation.
• Understand how Newtons law of universal
  gravitation explains both the motion of falling
  objects and the orbits of satellites and planets.
• Understand how the acceleration due to gravity
  acting upon a mass is affected by the location
  and mass of the other object in question.
• Quantitatively apply Newtons law of universal
  gravitation to solve problems.

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Newton and the apple

• Fiction Newton was sitting under an apple tree.
  Upon being struck upon the head by an apple,
  Newton realized gravity.
• Fact Newton observed an apple falling to the
  ground while sitting in his garden. He then
  reasoned that the same force that pulls an apple
  toward the ground is the same as the force that
  holds celestial objects in orbit.
• Gravitational force a force of mutual
  attraction between masses separated by a certain
  distance.

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Inverse Square Relationship

• Newton knew that the Moon was 60x farther from
  the center of Earth than it was from the Earths
  surface.
• If the force decreased at an inverse square rate,
  the gravitational force at the surface of the
  Moon would be 1/602 times the gravitational force
  on Earths surface.
• During Newtons time, the period of the Moon (?
  27 days) and the mass and radius of the Earth
  were known. Using these values, Newton was able
  to determine the acceleration of gravity on the
  Moon.
• Newtons value was not exactly correct since the
  known values were not known to great precision.
  Using values known today, Newton would have been
  correct.

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Philosophiae Naturalis Principia Mathematica

• Commonly referred to as the Principia
• Published July 5, 1687
• Newton discusses
• the Laws of Motion
• the Law of Universal Gravitation
• the derivation of Keplers Laws
• harmonic oscillation
• Detailed the law of universal gravitation in the
  third volume of the book - De mundi systemate (on
  the system of the world)
• Gravitas, Latin weight

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Universal Gravitation

• G universal gravitation constant
• G 6.67 x 10-11 N m2/ kg2
• m mass of an object
• r the distance between the center of mass of the
  two objects
• Fg is an attractive force that always exists
  between two masses, regardless of
• the medium separating them
• their size or composition

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The Gravitational Force Between a Point Mass and
a Sphere

• A satellite in orbit around a planet can be
  considered as a point mass and a sphere.
• Fg is the same as if all the mass of the sphere
  was concentrated at its center (the center of
  mass).

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The Gravitational Force

• The gravitational forces that any two masses
  exert on each other are always equal in magnitude
  and opposite in direction.
• The gravitational forces are an example of an
  action-reaction pair.

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Universal Gravitation Constant

• G 6.67 x 10-11 N m2/ kg2
• Since G is a very small number
• gravity has the lowest relative strength of the
  four fundamental forces
• force of gravity is negligible unless a very
  large mass involved

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The Gravitational Force and Newton's Second Law
of Motion, F ma

• Know that the gravitational forces acting on two
  masses are equal and opposite.
• The resulting acceleration of each mass is not
  necessarily equal and opposite.
• Consider the gravitational force that arises due
  to your interaction with the Earth using Newtons
  Second Law of Motion, F ma.

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The Gravitational Force and Newton's Second Law
of Motion, F ma
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The Gravitational Force

• The most important of the fundamental forces at
  long distances because of its infinite range
• Explains free-fall motion on Earth, planetary
  orbits, and large-scale order of galaxies.
• Can analyze the orbits of celestial objects to
  determine its distance from other celestial
  objects (the Sun)
• Allows researchers to detect the presence of
  matter that cannot be detected by telescopes
  dark matter
• Acts universally on all matter
• Unlike the electromagnetic force, the
  gravitational force acts universally on all
  matter since it does not depend on a mass
  electric charge.

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Gravitational Field

• Gravity is a field force
• Gravitational field strength g where g Fg/m
• Gravitational field is a vector with magnitude g
  pointing in the direction of Fg
• Gravitational field strength equals free-fall
  acceleration

The blue arrows correspond to the magnitude of
the gravitational field vectors of Earths
gravitational field at that point.
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Gravitational Field Strength

• The acceleration due to gravity decreases slowly
  with increasing height (altitude or distance
  between the center of mass of the Earth and the
  object in question)

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Gravitational Field Strength

• At distances comparable to or greater than the
  radius of the Earth, the acceleration due to
  gravity decreases at a faster rate.

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Weight

• An objects inertial mass is the same regardless
  of the acceleration due to gravity.
• Weight mass x gravitational field strength
• on earth, weight mass x 9.8 m/s2
• dependent upon gravitational field strength
  therefore weight changes with location
• m your mass M mass of planet r planet radius
• your weight on the surface of any planet will
  depend upon the planets mass and radius

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Summary

• The gravitational force is a field force that
  always exists between two masses, regardless of
  the medium that separates them.
• The same law of gravity applies everywhere in the
  universe.
• The magnitude of the gravitational force between
  two masses is given by the formula
• Fg between two masses is an action-reaction pair
  however, the resulting acceleration of each mass
  due to Fg will differ (if unequal masses).

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Assignment

• Holt, Rinehart, Winston Chapter 7
• Read pages 263 264
• Complete problems 39, 40
• Worksheet
• Complete problems 19, 25, 27, 29