Chapter 13: The Law of Gravity Reading assignment: Chapter 13.1 to 13.5 Homework (due Wednesday, Nov. 14): OQ1, OQ2, OQ5, OQ8, 2, 3, 6, 10, 13, 14, 18, 28, 33 Historically very interesting, heliocentric vs. geocentric universe The main cast: Copernicus, Brahe, Galileo, Kepler, Newton Satellite motion Copernicus 1473 – 1543 Brahe 1546 – 1601 Galileo 1564 – 1642 Kepler 1571 – 1630 Newton 1643 – 161727

Geocentric vs. heliocentric model of earth Earth is round! Slowly accepted, starting in ancient Greece, Magellan + Elcano circumnavigation (1519 – 1522). Ptolemy (100 –170 A.D.) geocentric model: Sun revolves around earth (Wrong!) From astronomical observations: Copernicus (1473-1543) heliocentric model: Earth & planets revolve around sun Brahe (1546 - 1601) Accurate observation of planetary motion Galileo (1564 - 1642) (1610) supports the heliocentric model Kepler (1571-1630), 1609: Laws I, II of planetary motion, Kepler 1619: Law III of planetary motion About falling objects Aristotle (384-322 B.C.) Heavier objects fall faster than light objects (Wrong!) Galileo (1564 - 1642) Neglecting air resistance, all objects fall at same acceleration

Newton’s Law of Universal Gravitation Every particle in the Universe attracts every other particle with a force of: G… Gravitational constant G = 6.673·10-11 N·m2/kg2 m1, m2 …masses of particles 1 and 2 r… distance separating these particles … unit vector in r direction Experimentally confirmed. Measuring the gravitational constant – Cavendish apparatus (1789)

q Black board example 13.1 What is the attractive force you (m1 = 100 kg) experience from the two people (m2 = m3 = 70 kg) sitting in front of you. Assume a distance r = 0.5 m and an angle q = 30° for both?

Free-Fall Acceleration and the 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).

Variation of g with altitude Black board example 13.2 Variation of g with altitude The ISS photographed from shuttle Discovery in 2006. From http://www.daviddarling.info/encyclopedia/I/ISS.html What is the value of g in the ISS space station that is at an altitude of 400 km. Assume ME = 5.960·1024 kg and RE = 6.370·106 m. Why does it feel like g = 0?

U will get smaller (more negative) as r gets smaller. Gravitational potential energy (similar equation for charge-charge interaction) Notice the – sign U = 0 at infinity U will get smaller (more negative) as r gets smaller. “Falling down” means loosing gravit. potential energy. Use only when far away from earth; otherwise use approximation U = mgh.

A) ~10,000 m/s B) ~11,000 m/s C) ~20,000 m/s D) ~22,000 m/s Black board example 13.3 First Rocket Launch from Cape Canaveral (NASA); July 1950 What is the escape speed of a particle on earth (ignore air resistance)? A) ~10,000 m/s B) ~11,000 m/s C) ~20,000 m/s D) ~22,000 m/s

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

Kepler’s laws about planetary motion 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. G… Gravitational constant G = 6.673·10-11 N·m2/kg2 MS … central mass, (i.e. mass of sun for planet motion) Thus, for any two planets:

All nine eight planets of the solar system

Inner solar system: Outer solar system: Mercury Venus Earth Mars Jupiter Saturn Uranus Neptune (Pluto) My Very Educated Mother Just Served Us Nine (Plums)