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Gravitation

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Gravitational force (Newton's law of gravitation) ... Tycho Brahe/ Tyge Ottesen. Brahe de Knudstrup (1546-1601) First Kepler's law ... – PowerPoint PPT presentation

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Title: Gravitation


1
Chapter 13 Gravitation
2
  • 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

3
  • 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

4
Chapter 13 Problem 9
5
  • 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

6
  • 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

7
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
8
Gravity force near the surface of Earth Weight
of a crate measured at the equator
9
  • 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

10
  • 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

11
Chapter 13 Problem 20
12
  • Gravitational potential energy
  • Gravitation is a conservative force (work done
    by it is path-independent)
  • For conservative forces (Ch. 8)

13
  • Gravitational potential energy
  • To remove a particle from initial position to
    infinity
  • Assuming U8 0

14
  • Escape speed
  • Accounting for the shape of Earth, projectile
    motion (Ch. 4) has to be modified

15
  • Escape speed
  • Escape speed speed required for a particle to
    escape from the planet into infinity (and stop
    there)

16
  • Escape speed
  • If for some astronomical object
  • Nothing (even light) can escape from the surface
    of this object a black hole

17
Chapter 13 Problem 33
18
  • 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

19
  • 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)

20
  • Second Keplers law
  • For a star-planet system, the total angular
    momentum is constant (no external torques)
  • For the elementary area swept by vector

21
  • Third Keplers law
  • For a circular orbit and the Newtons Second law
  • From the definition of a period
  • For elliptic orbits

22
  • Satellites
  • For a circular orbit and the Newtons Second law
  • Kinetic energy of a satellite
  • Total mechanical energy of a satellite

23
  • Satellites
  • For an elliptic orbit it can be shown
  • Orbits with different e but the same a have the
    same total mechanical energy

24
Chapter 13 Problem 50
25
Answers to the even-numbered problems Chapter
13 Problem 2 2.16
26
  • Answers to the even-numbered problems
  • Chapter 13
  • Problem 4
  • 2.13 10-8 N
  • (b) 60.6º

27
  • Answers to the even-numbered problems
  • Chapter 13
  • Problem 20
  • G(M1 M2)m/a2
  • (b) GM1m/b2
  • (c) 0

28
  • Answers to the even-numbered problems
  • Chapter 13
  • Problem 32
  • 2.2 107 J
  • (b) 6.9 107 J

29
Answers to the even-numbered problems Chapter
13 Problem 54 (a) 8.0 108 J (b) 36 N
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