1 / 29

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.

carrie
Télécharger la présentation

Chapter 13 Gravitation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related