Physics 2211, Spring 2002

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Physics 2211 Lecture 35Todays Agenda

• Newtons Law of Gravitation
• Shell theorems
• Kepler laws
• Satellite/Shuttle examples

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Newtons Law of Gravitation
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Newtons Law of Gravitation
m
z
M
y
x
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Shell Theorems

• A shell (or sphere) of uniform matter attracts a
  particle outside the shell (or sphere) as if all
  the shells (spheres) mass were located at its
  center.
• A uniform shell of matter exerts no net
  gravitational force on a particle located inside
  it.
• See Tipler 11-5 for proof. (Good example of
  problem solving in integral calculus.)

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Shell Theorems

• A shell (or sphere) of uniform matter attracts a
  particle outside the shell (or sphere) as if all
  the shells (spheres) mass were located at the
  center.

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Shell Theorems

• A uniform shell of matter with inner radius R
  exerts no gravitational force on a particle
  located at r lt R.

Force on m is zero.
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Application of Shell Theorems

• Find the gravitational force on particle m within
  a uniform spherical mass distribution M.

Only has an effect on m.
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RememberTransport Tunnel?
From Shell Theorems
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Keplers Laws

• Determined empirically by Kepler before law of
  gravitation was known
• 1st All planets move in elliptical orbits with
  the sun at one focus.
• 2nd A line joining any planet to the sun sweeps
  equal areas in equal times.
• 3rd The square of the period of any planet is
  proportional to the cube of the semi-major axis
  of it orbit.
• Keplers Laws can be derived from Newtons Law of
  Gravitation (see Tipler 11-2).

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Keplers Laws
planet
focus
sun (at focus)
a semi-major axis
elliptical orbit
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Application of Newtons Law of GravitationSatell
ite in circular orbit

• For a satellite of mass m in circular orbit
  around the earth at a distance r from the center
  of the earth

• The period, T, of the satellite is thus

• Notice
  (Keplers 3rd Law)

• For satellites in low Earth orbit,

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ExampleSatellite in Geo-synchronous circular
orbit

• For a satellite in geo-synchronous orbit,

• Thus, from

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Energy ofSatellite in circular orbit

• Consider the mechanical energy of a satellite of
  mass m in circular orbit around the earth at a
  distance r from the center of the earth

• But

• Therefore,
  

• Therefore, total orbital energy depends only on
  position.
  

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ExampleSpace Shuttle in circular orbit

• You are orbiting the earth in the space shuttle
  Columbia and you want to orbit the earth in less
  time. Do you
• increase your speed, or
• decrease your speed?

• Increasing your speed will increase E and thus
  increase r

• And since
  

• Orbital period, T, will increase. Therefore, you
  should decrease your speed.
  

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Recap of todays lecture

• Newtons Law of Gravitation
• Shell theorems
• Kepler laws
• Satellite/Shuttle examples
• Study for Final Exam