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# HR13 - Chapter 13 Gravitation In this chapter we will...

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Chapter 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)

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m 1 m 2 Newton realized that the force which holds the moon it its orbit around the earth is of the same nature with the force that makes an apple drop near the surface of the earth. Newton's Law of Gravitation 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. Newton formulated a force law known as Every particle any other particle with a gravitational force that has the following characteristics: attracts 1. Newton's law of gravitation. 12 2 The force acts along the line that connects the two particles Its magnitude is given by the equation: Here and are the masses of the two particles, is their separation an i d mm r m F G m G r = 2. 11 2 2 s the gravitational constant. Its value is: 6.67 10 N.m / Gk g 2 FG r = (13-2)

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Henry Cavendish 1731-1810
(12-16) 2 2 2 24 In his experiment Cavendish measured the mass of the earth! Gravitational force on apple = Gravitational force on apple = Solve for M = 5.96 10 kg mg mMG R mMG mg R gR M G = How does one measure the radius of the earth? This was already done by a librarian in Alexandria called Eratosthenes (around 200 BC). Eratosthenes knew that on a particular day every year sunlight reached the bottom of a very deep well in Syene (modern Aswan). He also knew the distance between Alexandria and Syene. From this information he was able to determine s R

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(12-17) The distance between Alexandria and Syene is give by: . Here is the angle between the sun's rays and the vertical in Alexandria at the time when the sun's rays in Syene reach the bottom o s sR θθ = f the well. The angle was determined using Eratosthene's walking stick and the length of the shadow it cast in Alexandria. tan . Erathosthenes measured to be 7 h θ = ° A A C θ θ R R Sun’s rays s Alexandria Syene well θ θ h A Ground in Alexandria Sun’s rays Erathosthene’s walking stick
m 1 m 2 F 12 F 21 r 12 1 2 21 21 12 21 The gravitational force exerted on by is equal in magnitude to the force exerted on by but opposite in direction. The two forces obey Newton's third law: 0 Fm m F mm FF += GG 12 1 2 Newton proved that a uniform shell attarcts a particle that is outside the shell as if the shell's mass were concentrated at the shell center If the particle is inside the shell, the FG r = Note : net force is zero Consider the force the earth (radius , mass ) exerts on an apple of mass . The earth can be thought of as consisting of concentric shells. Thus from the apple's point of view the earth behaves li FR M m ke a point mass at the earth center The magnitude of the force is given by the equation: (13-3) m 1 m 2 r F 1 m 2 m 1 2 r = 2 mM R =

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HR13 - Chapter 13 Gravitation In this chapter we will...

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