lecture-ch13 - Contents 13 Gravitation 1 13.1 Newton’s...

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Unformatted text preview: Contents 13 Gravitation 1 13.1 Newton’s Law of Gravitation . . . . . . . . . . . . . . . . . . . 1 13.2 Gravitation and the Principle of Superposition . . . . . . . . . 3 13.3 Gravitation near the earth’s Surface . . . . . . . . . . . . . . . 5 13.3.1 1. The earth’s mass is not uniformly distributed . . . . 6 13.4 Gravitation inside the earth. . . . . . . . . . . . . . . . . . . . 7 13.5 Gravitational Potential Energy . . . . . . . . . . . . . . . . . 7 13.5.1 Spherically Symmetric Central Force . . . . . . . . . . 7 13.5.2 Gravitational Force . . . . . . . . . . . . . . . . . . . . 9 13.5.3 Potential Due to a Spherical Shell . . . . . . . . . . . . 10 13.6 Escape Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 13.7 Planets and Satellites: Kepler’s Laws . . . . . . . . . . . . . . 12 13.7.1 Kepler’s First Law. . . . . . . . . . . . . . . . . . . . . 12 13.7.2 Kepler’s Second Law . . . . . . . . . . . . . . . . . . . 17 13.7.3 Kepler’s Third law . . . . . . . . . . . . . . . . . . . . 18 13.8 Satellites: Orbits and Energy . . . . . . . . . . . . . . . . . . 19 13 Gravitation In this chapter we will explore the following topics:-Newton’s law of gravitation that describes the attractive force between two point masses and its application to extended objects.-The acceleration of gravity on the surface of the earth, above it, as well as below it.-Gravitational potential energy.-Kepler’s three laws on planetary motion.-Satellites (orbits, energy , escape velocity) 13.1 Newton’s Law of Gravitation Newton realized that the force which holds the moon it its orbit is of the same nature with the force that makes an apple drop near the surface of the earth. Newton concluded that the earth attracts both apples as well as the moon but also that every object in the universe attracts every other object. The tendency of objects to move towards each other is known as gravitation. 1 Newton formulated a force law known as Newton’s law of gravitation. Every particle attracts any other particle with a gravitational force that has the following characteristics: 1. The force acts along the line that connects the two particles. 2. Its magnitude is given by the equation: F = G m 1 m 2 r 2 Here m 1 and m 2 are the masses of the two particles, r is their separation and G is the gravitational constant. Its value is: G = 6 . 67 × 10 − 11 N.m 2 kg 2 . Let vectorx 1 be the position vector of m 1 and vectorx 2 be the position vector of m 2 . vectorx 2 ,m 2 vectorx 1 ,m 1 O The separation between the two masses r = | vectorx 1 − vectorx 2 | and the unit vector ˆ r 12 along the direction from vectorx 2 to vectorx 1 is ˆ r 12 = vectorx 1 − vectorx 2 | vectorx 1 − vectorx 2 | = vectorx 1 − vectorx 2 r Let vector F 12 denote the gravitational force exerted on m 1 due to m 2 . Then vector F 12 = − G m 1 m 2 r 2 ˆ r 12 = − G m 1 m 2 r 3 ( vectorx 1 − vectorx 2 ) Likewise vector F 21 = − G m 1 m 2 r 2 ˆ r 21 = − G m 1 m 2 r 3 (...
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This note was uploaded on 05/14/2011 for the course ECON 101 taught by Professor Asdaf during the Spring '11 term at Universidad de San Buenaventura Bogota.

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lecture-ch13 - Contents 13 Gravitation 1 13.1 Newton’s...

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