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09/23/09 - Newton's Law of Gravitation
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 | Newton's Guess about the Moon based on the Apple
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 | Newton hypothesized that the moon was falling towards the Earth, as an apple does, but because of its sideways velocity, it didn't get appreciably closer. Instead it moved in a roughly circular orbit.
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| The force of gravity between two objects depends on the product of their two masses.
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| Newton reasoned that the force of gravitational attraction on the moon depended both on the mass of the moon and the mass of the Earth, in fact, on the product of the two masses. One unit of mass and one unit of mass produces one unit of force on each object. One unit of mass and two units of mass produces two units of force on each object. Two units of mass and two units of mass produces four units of force on each object.
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| The force of gravity between two objects depends on the inverse square of the distance between them.
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| Newton reasoned that things that spread out from a point, such as light from a lamp, get weaker as the square of the distance from the source.
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| Newton combined these two ideas and hypothesized that the force of gravity is proportional to the product of the two masses and inversely to the square of the distance between their centers.
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| Newton tested his hypothesis with the Moon. He said an apple falls towards the earth with an acceleration of 9.8 m/s2. Since he knew the Moon was about 60 times farther from the center of the Earth than the apple, it should have 1/602 = 1/3600 times the acceleration (9.8 m/s2/3600 = 0.0027 m/s2).
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| He calculated the centripetal acceleration of the Moon by noting that it took about 27.3 days (= 27.3*24*60*60 = 2.36 x 106 s) to go around the Earth, so its period T = 2.36 x 106 s. So the angular velocity was ω = 2π/T = 2π/(2.36 x 106) = 2.66 x 10-6 rad/s. Thus the centripetal acceleration of the Moon was ac = Rω2 = (60*6.37 x 106)(2.66 x 10-6)2 = 0.0027 m/s2, which Newton thought was pretty good support for his hypothesis.
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| Kepler's Laws of Planetary Motion
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| An astronomer named Kepler studied the motion of the planets before Newton was born and came up with three observations (called Kepler's Laws).
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| The planets move in elliptical orbits with the Sun at one focus.
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| The planets move around the sun in such a manner that a straight line from the sun to a planet sweeps out equal areas in equal times. That means a planet moves faster when it's closer to the sun and slower when it's farther from the sun.
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| The ratio of the cube of the average distance of a planet from the sun to the square of the time it takes to go around the sun is the same constant for all the planets: R3/T2 = k.
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| Newton showed that his law of gravitation was consistent with Kepler's Laws. His law of gravitation predicts that Kepler's Laws should be observed.
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| Example of Geosynchronous Satellite Orbit
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| At what velocity and radius will a satellite orbit above the Earth's equator in 24 hours?
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| Determining the mass of Jupiter from the period and radius of its moon Io's orbit.
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| T = 152854 s, R = 4.217 x 108 m
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| Read the remainder of Chapter 5, pp. 145-158.
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| Practice Exam on Chapters 1-5 (double-click on the page below to see the whole document -- 2 pages)
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| Practice Exam on Chapters 1-5 (answers)
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