1. Universal Gravitation Ptolemy (150AD) theorized that since all objects fall towards Earth, the Earth must be the center of the universe. This is known as a “Geocentric” view In the early 1500’s Copernicus hypothesized that the Earth is actually revolving around the Sun. This is known as a “Heliocentric” view.

2. Universal Gravitation 1. All Planets move about the Sun in an elliptical orbit with the Sun at it’s focus. 2. If a line is drawn from the Sun to a planet it will pass through (over) equal areas in equal time intervals. 3. The square of the ratio of the revolving time period for any 2 planets around the Sun equals the cube of the ratio of their average distances from the Sun. In the late 1500’s Johannes Kepler argued 3 Laws: (T A ) 2 = (r A ) 3 (T B ) 2 (r B ) 3

3. Newton’s Universal Law of Gravitational Attraction In 1666, based on the research conducted by Johannes Kepler, Newton argued that all objects are surrounded by a gravitational field and the magnitude of the Force acting on a planet keeping it in orbit must be inversely proportional to the square of the distance between the centers of mass of the planet and the Sun. F = Gm 1 m 2 r 2 Where: F = the Gravitational Attractive Force G = Newton’s Universal Gravitational Constant between any 2 objects 6.7 x 10 -11 N * m 2 /Kg 2 m 1 = Mass of object 1 m 2 = Mass of object 2 r = Radius between m 1 & m 2 center of masses

4. Newton’s Universal Law of Gravitational Attraction Find the Gravitational Force (F G ) acting on your body, assuming 2.2 lb = 1 Kg. Estimating 4.45 N = 1 lb, convert the F G acting on your body into pounds. Calculate the Attractive Force pulling you towards the person sitting next to you. Convert that into pounds.

5. Satellite Motion Recall, a c = v 2 /r and F = ma so when we consider the Centripetal Forces acting on an object we can say: F c = m a c or F c = mv 2 /r Orbital (& Escape) Velocities: occur when:F c = F G so: m s v 2 /r = Gm s m e r 2 where: m s = mass of satellite in orbit m E = mass of the Earth

6. Satellite Motion Since: m s v 2 /r = Gm s m E r 2 m s cancels m s, r cancels r leaving: v 2 = Gm E /r Isolating “v” we solve for the orbital velocity or escape velocity, v orbit or v escape : _________ v escape = \ / Gm E /r Squaring both sides: (2  r) 2 = ( \ / Gm E /r) 2 4  2 r 2 = Gm E (T) 2 T 2 r 4, , G and m E are all constants, isolating them leaves:r 3 = K where K is a constant: T 2 K Sun = 3.35 x 10 18 m 3 /s 2 K Earth = 1.01 x 10 13 m 3 /s 2

7. Satellite Motion Calculate the orbital velocity of a 2500 Kg telecommunications satellite in geosynchronous orbit at 225Km above the surface of the Earth. M E = 6.0 x 10 24 Kg r E = 6.4 x 10 6 m r S = 2.25 x 10 5 m r net = 6.625 x 10 6 m v escape = \ / Gm E /r v escape = \ /(6.7 x 10 -11 N * m 2 /Kg 2 )(6.0 x 10 24 Kg) 6.625 x 10 6 m v escape = 7789.69 m/s

8. Gravitational Attraction Problems: 1. Calculate the speed of a satellite shot into orbit at a radius of 150 Km above the surface of the Earth. 2. Find the speed that the moon orbits the Earth. Find the time period for one revolution. 3. Two bowling balls have masses of 8 Kg and 6 Kg respectively. They are placed 2 m apart. What is the Gravitational Force acting between them? 4. The Gravitational Force between 2 electrons 1 m apart is 5.42 x 10 -71 N. Find the mass of an electron. 5. Find the attractive force between 2 people (m 1 = 60 Kg, m 2 = 100 Kg) when they are standing 10 m apart, 5m and 1m.