Section 7'2 Using the Law of Universal Gravitation

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Section 7.2 Using the Law of Universal
Gravitation

•  Objectives
• Solve orbital motion problems.
• Relate weightlessness to objects in free fall.
• Describe gravitational fields.
• Compare views on gravitation.
•  

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ORBITS OF PLANETS AND SATELLITES

• Go over Cannonball example p. 179
•  
• The curvature of the projectile would continue to
  just match the curvature of Earth, so that the
  cannonball would never get any closer or farther
  away from Earths curved surface. The cannonball
  therefore would be in orbit. This is how a
  satellite works.
•  
• Thus a cannonball or any object or satellite at
  or above this altitude could orbit Earth for a
  long time.
•  
• A satellite in an orbit that is always the same
  height above Earth move in uniform circular
  motion.
•  

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ORBITS OF PLANETS AND SATELLITES

• ac v2 / r
•  
• Fc mac mv2 / r
•  
• Combining the above equation with Newtons Law of
  Universal Gravitation yields the following
  equation
• GmEm mv2
• r2 r

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ORBITS OF PLANETS AND SATELLITES

• Speed of a Satellite Orbiting Earth is equal to
  the square root of the universal gravitation
  constant times the mass of Earth divided by the
  radius of the orbit.
• v v (GmE) / r
• Period of a Satellite Orbiting Earth is equal
  to 2? times the square root of the radius or the
  orbit cubed divided by the product of the
  universal gravitation constant and the mass of
  Earth.
• T 2? v r3 / (GmE)

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ORBITS OF PLANETS AND SATELLITES

• The equations for the speed and period of a
  satellite can be used for any object in orbit
  about another object.
•  
• The mass of the central body will replace mE in
  the equations and r will be the distance between
  the centers of the orbiting body and central
  body.
•  
• Since the acceleration of any mass must follow
  Newtons 2nd Law (F ma) more force is needed to
  launch a more massive satellite into orbit. Thus
  the mass of a satellite is limited to the
  capability of the rockets used to launch it.

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ORBITS OF PLANETS AND SATELLITES

• Do Example Problem 2 p. 181
• v v (GmE) / r
• v v (6.67 10-11)(5.97 1024) / (6.38 106
  .225 106)
• v v (3.98199 1014) / (6.605 106)
• v v (6.029 107)
• v 7.765 103 m/s
• T 2? v r3 / (GmE)
• T 2? v (6.605 106)3 / (6.67 10-11)(5.97
  1024)
• T 2(3.14) v (2.8815 1020) / (3.98199 1014)
• T 6.28 v (7.236 105)
• T 6.28 (850.647)
• T 5342.062 s (If convert 89 min or 1.5
  hours
• Do Practice Problems p. 181 12-14

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ACCELERATION DUE TO GRAVITY

• F GmEm / r2 ma
•  
• a GmE / r2
•  
• a g and r rE so
•  
• g GmE / rE2
•  
• mE grE2 / G
•  
• a g (rE / r)2
• This last equation shows that as you move farther
  from Earths center (r becomes larger) the
  acceleration due to gravity is reduced according
  to the inverse square relationship.

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ACCELERATION DUE TO GRAVITY

• Weightlessness an objects apparent weight of
  zero that results when there are no contact
  forces pushing on the object. This is also
  called Zero g.
• There is gravity in space. Gravity is what causes
  the shuttle and satellites to orbit Earth.

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THE GRAVITATIONAL FIELD

• Gravity acts over a distance. It acts on objects
  that are not touching.
•  
• Michael Faraday invented the concept of the
  field to explain how a magnet attracts objects.
  Later the field concept was applied to gravity.
•  
• Gravitational Field is equal to the universal
  gravitational constant times the objects mass
  divided by the square of the distance from the
  objects center.
• g Gm / r2

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THE GRAVITATIONAL FIELD

• The Gravitational Field can be measured by
  placing an object with a small mass in the
  gravitational field and measuring the force on
  it. Then the gravitational field is the force
  divided by a mass. It is measured in Newtons per
  kilogram (N/kg) which m/s2. Thus we have
• g F / m
• The strength of the field varies inversely with
  the square of the distance from the center of
  Earth. The gravitational field depends on
  Earths mass but not on the mass of the object
  experiencing it.

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TWO KINDS OF MASS

• Inertial Mass is equal to the net force exerted
  on the object divided by the acceleration of the
  object. It is a measure of the objects
  resistance to any type of force.
• mInertial FNet / a
•  
• Gravitational Mass is equal to the distance
  between the objects squared times the
  gravitational force divided by the product of the
  universal gravitational constant times the mass
  of the other object.
• mGravitational r2FGravitational / Gm
•  
• Newton claimed that Inertial and Gravitational
  Mass are equal in magnitude. All experiments
  done since then show this is the case.

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EINSTEINS THEORY OF GRAVITY

• Einstein proposed that Gravity is not a force but
  rather an effect of space itself. According to
  Einstein mass changes the space around it. Mass
  causes space to be curved and other bodies are
  accelerated because of the way they follow this
  curved space.
•  
• Einsteins General Theory of Relativity makes
  many predictions about how massive objects affect
  one another. It predicts the deflection or
  bending of light by massive objects. In 1919, an
  eclipse of the sun proved Einsteins theory.
• Do 7.2 Section Review p. 185 15-21