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Chapter 13 Gravitation

Chapter 13 Gravitation. Newton’s law of gravitation Any two (or more) massive bodies attract each other Gravitational force (Newton's law of gravitation) Gravitational constant G = 6.67*10 –11 N*m 2 /kg 2 = 6.67*10 –11 m 3 /(kg*s 2 ) – universal constant.

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Chapter 13 Gravitation

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  1. Chapter 13 Gravitation

  2. Newton’s law of gravitation • Any two (or more) massive bodies attract each other • Gravitational force (Newton's law of gravitation) • Gravitational constantG= 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – 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 Earth’s 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 gis 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 U∞ = 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. Johannes Kepler (1571-1630) Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup (1546-1601) • Kepler’s laws • Three Kepler’s 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 planet’s 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 Kepler’s law • Elliptical orbits of planets are described by a semimajor axisa and an eccentricitye • For most planets, the eccentricities are very small (Earth's e is 0.00167)

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

  21. Third Kepler’s law • For a circular orbit and the Newton’s Second law • From the definition of a period • For elliptic orbits

  22. Satellites • For a circular orbit and the Newton’s 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 ebut 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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