UNIVERSAL LAW OF GRAVITATION

Geocentric vs. heliocentric model of earth - 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 Galileo (1564 - 1642) (1610) supports (loudly) the heliocentric model Brahe (1546 - 1601) Accurate observation of planetary motion 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

There is a popular story that Newton was sitting under an apple tree, an apple fell on his head, and he suddenly thought of the Universal Law of Gravitation.

What Really Happened with the Apple? Newton, upon observing an apple fall from a tree, began to think along the following lines: The apple is accelerated, since its velocity changes from zero as it is hanging on the tree and moves toward the ground. there must be a force that acts on the apple to cause this acceleration. Let's call this force "gravity", and the associated acceleration the "accleration due to gravity". Then imagine the apple tree is twice as high. Again, we expect the apple to be accelerated toward the ground, so this suggests that this force that we call gravity reaches to the top of the tallest apple tree.

Sir Isaac's Most Excellent Idea if the force of gravity reaches to the top of the highest tree, might it not reach even further; in particular, might it not reach all the way to the orbit of the Moon! the orbit of the Moon about the Earth could be a consequence of the gravitational force, because the acceleration due to gravity could change the velocity of the Moon in just such a way that it followed an orbit around the earth.

Newton concluded that the orbit of the Moon was of exactly the same nature: the Moon continuously "fell" in its path around the Earth because of the acceleration due to gravity, thus producing its orbit.

All nine eight planets of the solar system

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

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

Newton’s Law of Universal Gravitation - Particle 1 is attracted by particle 2 - Particle 2 is attracted by particle 1 - F12 and F21 form an action-reaction pair - Force drops off as 1/r2 as distance r between particles increases - Can treat spherical, symmetric mass distributions as if the mass were concentrated in center of mass.

Measuring the gravitational constant – Cavendish apparatus (1789)

Free-Fall Acceleration and the Gravitational Force Universal Gravitational force: Gravitational force near Earth Surface: Gravitational acceleration: g - is not constant as we move up from the surface of the earth - is not dependent on the mass of the falling object G is a universal constant (does not change at all).

Variation of g with altitude Black board example Variation of g with altitude Everest North Face and Rongbuk monastery (5030m), Tibet May, 1997 Photo credit: Philippe Noth What is the value of the acceleration due to gravity: 1. At the equator (R = 6.378 x 10 6 m) (Radius at the poles is 6. 356 x 10 6) 2. On top of mount Everest (h = 8848 m)? What is it in a space station that is at an altitude of 350 km Assume ME = 5.960·1024 kg and RE = 6.370·106 m.

q Board Example What is the attractive force you (m = 100 kg) experience from the person (m = 70 kg) sitting in front of you Assume a distance r = 0.5 m

q Black board example 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?

Gravity and Circular motion

Kepler’s laws about planetary motion These laws hold true for any object in orbit 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. Thus, for any two planets: