Gravitation

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Chapter 13 Gravitation
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• Newtons law of gravitation
• Any two (or more) massive bodies attract each
  other
• Gravitational force (Newton's law of
  gravitation)
• Gravitational constant G 6.6710 11 Nm2/kg2
  6.6710 11 m3/(kgs2) universal constant

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• Gravitation and the superposition principle
• For a group of interacting particles, the net
  gravitational force on one of the particles is
• For a particle interacting with a continuous
  arrangement of masses (a massive finite object)
  the sum is replaced with an integral

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Chapter 13 Problem 9
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• Shell theorem
• For a particle interacting with a uniform
  spherical shell of matter
• Result of integration a uniform spherical shell
  of matter attracts a particle that is outside the
  shell as if all the shell's mass were
  concentrated at its center

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• Gravity force near the surface of Earth
• Earth can be though of as a nest of shells, one
  within another and each attracting a particle
  outside the Earths surface
• Thus Earth behaves like a particle located at
  the center of Earth with a mass equal to that of
  Earth
• g 9.8 m/s2
• This formula is derived for stationary Earth of
  ideal spherical shape and uniform density

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Gravity force near the surface of Earth In
reality g is not a constant because Earth is
rotating, Earth is approximately an ellipsoid
with a non-uniform density
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Gravity force near the surface of Earth Weight
of a crate measured at the equator
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• Gravitation inside Earth
• For a particle inside a uniform spherical shell
  of matter
• Result of integration a uniform spherical shell
  of matter exerts no net gravitational force on a
  particle located inside it

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• Gravitation inside Earth
• Earth can be though of as a nest of shells, one
  within another and each attracting a particle
  only outside its surface
• The density of Earth is non-uniform and
  increasing towards the center
• Result of integration the force reaches a
  maximum at a certain depth and then decreases to
  zero as the particle reaches the center

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Chapter 13 Problem 20
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• Gravitational potential energy
• Gravitation is a conservative force (work done
  by it is path-independent)
• For conservative forces (Ch. 8)

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• Gravitational potential energy
• To remove a particle from initial position to
  infinity
• Assuming U8 0

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• Escape speed
• Accounting for the shape of Earth, projectile
  motion (Ch. 4) has to be modified

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• Escape speed
• Escape speed speed required for a particle to
  escape from the planet into infinity (and stop
  there)

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• Escape speed
• If for some astronomical object
• Nothing (even light) can escape from the surface
  of this object a black hole

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Chapter 13 Problem 33
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• Keplers laws
• Three Keplers laws
• 1. The law of orbits All planets move in
  elliptical orbits, with the Sun at one focus
• 2. The law of areas A line that connects the
  planet to the Sun sweeps out equal areas in the
  plane of the planets orbit in equal time
  intervals
• 3. The law of periods The square of the period
  of any planet is proportional to the cube of the
  semimajor axis of its orbit

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• First Keplers law
• Elliptical orbits of planets are described by a
  semimajor axis a and an eccentricity e
• For most planets, the eccentricities are very
  small (Earth's e is 0.00167)

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• Second Keplers law
• For a star-planet system, the total angular
  momentum is constant (no external torques)
• For the elementary area swept by vector

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• Third Keplers law
• For a circular orbit and the Newtons Second law
• From the definition of a period
• For elliptic orbits

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• Satellites
• For a circular orbit and the Newtons Second law
• Kinetic energy of a satellite
• Total mechanical energy of a satellite

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• Satellites
• For an elliptic orbit it can be shown
• Orbits with different e but the same a have the
  same total mechanical energy

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Chapter 13 Problem 50
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Answers to the even-numbered problems Chapter
13 Problem 2 2.16
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• Answers to the even-numbered problems
• Chapter 13
• Problem 4
• 2.13 10-8 N
• (b) 60.6º

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• Answers to the even-numbered problems
• Chapter 13
• Problem 20
• G(M1 M2)m/a2
• (b) GM1m/b2
• (c) 0

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• Answers to the even-numbered problems
• Chapter 13
• Problem 32
• 2.2 107 J
• (b) 6.9 107 J

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Answers to the even-numbered problems Chapter
13 Problem 54 (a) 8.0 108 J (b) 36 N