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*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 g is 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

Gravitational potential energy Gravitation is a conservative force (work done by it is path-independent) For conservative forces (Ch. 8):

Gravitational potential energy To remove a particle from initial position to infinity Assuming U∞ = 0

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

Kepler’s laws Three Kepler’s laws Tycho Brahe/ Tyge Ottesen Brahe de Knudstrup (1546-1601) Johannes Kepler (1571-1630) 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 axis a and an eccentricity e 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 e but 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 4 2.13 × 10−8 N; (b) 60.6º

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

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

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