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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


  • 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


  • 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


Chapter 13

Problem 9


  • 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


  • 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


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


Gravity force near the surface of Earth

Weight of a crate measured at the equator:


  • 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


  • 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


Chapter 13

Problem 20




  • Escape speed

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


  • Escape speed

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


  • Escape speed

  • If for some astronomical object

  • Nothing (even light) can escape from the surface of this object – a black hole


Chapter 13

Problem 33


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


  • 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)


  • 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


  • Third Kepler’s law

  • For a circular orbit and the Newton’s Second law

  • From the definition of a period

  • For elliptic orbits


  • Satellites

  • For a circular orbit and the Newton’s Second law

  • Kinetic energy of a satellite

  • Total mechanical energy of a satellite


  • Satellites

  • For an elliptic orbit it can be shown

  • Orbits with different ebut the same a have the same total mechanical energy


Chapter 13

Problem 50


Answers to the even-numbered problems

Chapter 13:

Problem 2

2.16





Answers to the even-numbered problems

Chapter 13:

Problem 54

(a) 8.0 × 108 J;

(b) 36 N


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