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How we know what we know An introduction into orbital mechanics Matt Hamill

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History Isaac Newton (1643 – 1727) Did he discover gravity? –NO! What did he do? –He used physics to connect the force that causes an apple to fall to the force that causes the moon to orbit the Earth

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Newton’s Thought Experiment If you launch a cannonball it will follow a curved path due to gravitational influences If the cannonball is launched at a high enough velocity it will “fall” around the Earth.

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Newton’s Hypothesis Every particle in the Universe attracts every other particle with a force (gravity) that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

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The legendary apple. Newton knew a falling apple accelerated toward the Earth around 32 ft/s 2 or 10 m/s 2

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The mysterious Moon Newton also knew –the Moon orbited the Earth in approximately 28 days –The moon’s distance from the Earth was about 60 times the Earth’s radius

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In the 17 th Century how can you prove Newton’s hypothesis? Newton used his inverse- square prediction to reason that the Moon’s acceleration toward the Earth (centripetal acceleration) should be proportional to:

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Inverse square continued… Using the same logic, the acceleration of the apple toward the Earth should be proportional to:

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Newton’s hypothesis predicts Newton predicted the ratio of the Moon’s acceleration (a M ) to the apple’s acceleration (g) would be Therefore the centripetal acceleration of the Moon should be around

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What is the Moon’s actual centripetal acceleration? We know centripetal acceleration can be calculated with the following formula If we assume the orbit of the Moon is circular it travels a distance that is equal to the circumference of a circle.

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What is the Moon’s actual centripetal acceleration? The Moon completes its orbital period in a time interval T = 27.32 days or 2.36 x 10 6 s.

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Predicted vs. Actual Predicted centripetal acceleration of the Moon Actual centripetal acceleration of the Moon Less than 1% difference

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Newton’s Law of Universal Gravitation G—gravitational constant G = 6.67 x 10 -11 N·m 2 /kg 2 m 1, m 2 —mass r—distance from their centers of mass

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Three ways to calculate gravitational force Equation1: Equation 2: Equation 3

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Resources Serway, Raymond, and John Jewett. Physics for Scientists and Engineers. 6th ed.. USA: Brooks/Cole, 2004. Print. http://en.wikipedia.org/wiki/File:GodfreyKneller- IsaacNewton-1689.jpghttp://en.wikipedia.org/wiki/File:GodfreyKneller- IsaacNewton-1689.jpg http://scienceiq.com/Images/FactsImages/apple _falling.gifhttp://scienceiq.com/Images/FactsImages/apple _falling.gif http://www.chemheritage.org/women_chemistry/ univ/images/clark_moon.jpghttp://www.chemheritage.org/women_chemistry/ univ/images/clark_moon.jpg http://www.astronautix.com/lvs/newannon.htm

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