Historically very interesting, Geocentric vs. heliocentric universe The main cast: Copernicus, Brahe, Galileo, Kepler, Newton Chapter 13: The Law of Gravity Reading assignment: Chapter 13.1 to 13.5 Homework :(due Monday, Nov. 12): Problems:3, 5, 9, 11, 17, 20

- Ptolemy (100 –170 A.D.) geocentric model: Sun revolves around earth (Wrong!) From astronomical observations: - Copernicus ( ) heliocentric model: Earth & planets revolve around sun - Galileo ( ) (1610) supports (loudly) the heliocentric model - Brahe ( ) Accurate observation of planetary motion - Kepler ( ), 1609: Laws I, II of planetary motion, - Kepler 1619: Law III of planetary motion - Aristotle ( B.C.) Heavier objects fall faster than light objects (Wrong!) - Galileo ( ) Neglecting air resistance, all objects fall at same acceleration Geocentric vs. heliocentric model of earth About falling objects

Newton’s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force of: G… Gravitational constant G = 6.673· N·m 2 /kg 2 m 1, m 2 …masses of particles 1 and 2 r… distance separating these particles … unit vector in r direction

Newton’s Law of Universal Gravitation - Particle 1 is attracted by particle 2 (and vice versa). - F 12 is force exerted by mass 1 on mass 2 (and vice versa). - r 12 goes from mass 1 to mass 2. - F 12 and F 21 form an action-reaction pair - Force drops off as 1/r 2 as distance r between particles increases - Can treat spherical, symmetric mass distributions as if the mass were concentrated in center of mass.

What is the attractive force you (m 1 = 100 kg) experience from the two people (m 2 = m 3 = 70 kg) sitting in front of you. Assume a distance r = 0.5 m and an angle  = 30° for both? Black board example 13.1 

Measuring the gravitational constant – Cavendish apparatus (1789)

Free-Fall Acceleration and the Gravitational Force Gravitational force: Thus : g is not constant as we move up from the surface of the earth!! G is a universal constant (does not change at all).

Earth’s gravitational field Close to the surface Far away from the surface Gravitational force acts from a distance through a “field”

a.What is the value of g in the ISS space station that is at an altitude of 400 km. Assume M E = 5.960·10 24 kg and R E = 6.370·10 6 m. b.Why does it feel like g = 0? Black board example 13.2 Variation of g with altitude The ISS photographed from shuttle Discovery in From

Kepler’s first two laws (1609): I.Planets move in elliptical paths around the sun. The sun is in one of the focal points (foci) of the ellipse II.The radius vector drawn from the sun to a planet sweeps out equal areas in equal time intervals (Law of equal areas). Kepler’s laws about planetary motion These laws hold true for any object in orbit Area S-A-B equals area S-D-C

Kepler’s third law (1619): III.The square of the orbital period, T, of any planet is proportional to the cube of the semimajor axis of the elliptical orbit, a. Kepler’s laws about planetary motion Thus, for any two planets:

Kepler’s laws about planetary motion Most planets, except Mercury and Pluto, are on almost a circular orbit Earth: Ratio of minor to major axis b/a = For planets around sun:

Black board example 13.3 The solar system If the Mars year is 1.88 earth years, what is Mars’ distance from the sun? Calculate the mass of the sun using the fact that the period of the earth’s orbit is 3.157·10 7 s and it’s distance from the sun is 1.496·10 11 m. Inner planets Further out: Saturn, Uranus, Neptun, Pluto

All nine eight planets of the solar system