Physics 1710 Chapter 14 The Theory of Gravity

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Physics 1710Chapter 14 The Theory of Gravity

• Quiz
• Consider a spool of thread. If one pulls on
  the thread, which way will the spool roll, if the
  thread comes over the top or under the bottom of
  the spool?

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Physics 1710Chapter 14 The Theory of Gravity

• Quiz
• What is the mass of the earth?
• How do you know?

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Physics 1710Chapter 14 The Theory of Gravity

• 1' Lesson
• The force of attraction between two bodies with
  mass M and m respectively is proportional to the
  product of their masses and inversely
  proportional to the distance between their
  centers squared.
• The proportionality constant in the Universal
  Law of Gravitation G is equal to 6.673 x 10 11 N
  m2 /kg2 .
• The gravitational force constant g is equal to
• G M/(Rh) 2, R is the radius of the planet.

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Physics 1710Chapter 14 The Theory of Gravity

• 1' Lesson
• Keplers Laws
• The orbits of the planets are ellipses.
• The areal velocity of a planet is constant.
• The cube of the radius of a planets orbit
• is proportional to the square of the period.
• The gravitation field is the force divided by
  the mass.
• The gravitation potential energy for a point mass
  is proportional to the product of the masses and
  inversely proportional to the distance between
  their centers.

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Physics 1710Chapter 14 The Theory of Gravity

• 1' Lesson
• The total energy of a gravitationally bound
  system is equal to the sum of the kinetic energy
  and (negative) gravitational potential energy.
• The escape velocity of an object projected
  upwards from the surface of the Earth is the
  velocity with a kinetic energy just equal to the
  gravitational kinetic energy of the particle at
  the radius of the Earth.

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Physics 1710Chapter 14 The Theory of Gravity

• Isaac Newtons
• Universal Law of Gravitation
• F - G M m/ r 2

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Physics 1710Chapter 14 The Theory of Gravity

• How did he discover this?
• Fact a moon circling a planet has an
  acceleration of a v 2 /r
• Fact a F/m.
• Fact Kepler had found that the square of the
  period was proportional to the cube of the
  radius
• T 2 k r 3 .
• Thus
• v 2? r / T

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Physics 1710Chapter 14 The Theory of Gravity

• And
• T 2 (2? r) 2 /(F r /m) k r 3
• Thus
• F (2?) 2 m/(k r 2 )
• An inverse square law. k 1/ (2?) 2 (G M)
• F G Mm/ r 2 ,
• But what value is G?

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Physics 1710Chapter 14 The Theory of Gravity

• Henry Cavendish
• And the
• Cavendish Experiment

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Physics 1710Chapter 14 The Theory of Gravity

• G 6.673 x 10-11 N ?m 2 /kg 2
• G 2/3 x 10-10 N ?m 2 /kg 2
• (to an accuracy of 0.1)

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Physics 1710Chapter 14 The Theory of Gravity

• Keplers Laws
• The orbits are ellipses. (Contrary to Aristotle
  and Ptolemy.)
• A central force F ? 1/ r 2 or r 2
• The areal velocity is a constant.
• Angular momentum is conserved
• ½ v r ?t constant implies that
• rmv L constant.
• T 2 ? r 3 implies F ? 1/ r 2, only.

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Physics 1710Chapter 14 The Theory of Gravity

• Gravitation Field
• g Fg / m
• At the surface of the Earth g 9.80 N/ kg.

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Physics 1710Chapter 14 The Theory of Gravity

• Gravitational Potential Energy
• U -? Fd r
• U -? G Mm / r 2 d r
• U GMm / r
• U 0 as r ?8

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Physics 1710Chapter 14 The Theory of Gravity

• Total Energy
• for Gravitationally Bound Mass
• E K U
• E ½ m v 2 - GMm/ r
• E L2 /2mr 2 GMm/ r
• One dimensional plot.
• Bound orbit if E 0.
• E - GMm/2r

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Physics 1710Chapter 14 The Theory of Gravity

• Escape Velocity
• If K is such that E gt 0, then
• K ½ m v 2 GMm/ r
• Thus
• v v 2GM/R

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Physics 1710Chapter 14 The Theory of Gravity

• Summary
• The force of attraction between two bodies with
  mass M and m respectively is proportional to the
  product of their masses and inversely
  proportional to the distance between their
  centers squared.
• F - G M m/ r 2
• The proportionality constant in the Universal
  Law of Gravitation G is equal to 6.673 x 10 11 N
  m2 /kg2 .

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Physics 1710Chapter 14 The Theory of Gravity

• Summary
• The gravitational force constant g is equal to
• G M/(Rh) 2, R is the radius of the planet.
• Keplers Laws
• The orbits of the planets are ellipses.
• The areal velocity of a planet is constant.
• The cube of the radius of a planets orbit
• is proportional to the square of the period.
• The gravitation field is the force divided by
  the mass.
• g Fg / m

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Physics 1710Chapter 14 The Theory of Gravity

• Summary
• The gravitation potential energy for a point
  mass is proportional to the product of the masses
  and inversely proportional to the distance
  between their centers
• U GMm / r
• The escape velocity is the minimum speed a
  projectile must have at the surface of a planet
  to escape the gravitational field.
• v v 2GM/R