PAP111_Lecture15 - PAP111 Lecture15 KeplersLaws KeplersLaws...

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PAP 111  Mechanics and Relativity  Lecture 15 Universal Gravitation Kepler’s Laws Newton’s Law of Universal Gravitation
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Kepler’s Laws Kepler’s detailed analysis on the data of planetary motion leads  to the following three mathematicbal laws known as the  Kepler’s  laws : 1. All planets move in elliptical orbits with the Sun at  one focus. 2. The radius vector drawn from the Sun to a planet  sweeps out equal areas in equal time intervals. 3. The square of the orbital period of any planet is  proportional to the cube of the semi-major axis of  the elliptical orbit. 
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A Note on Ellipses The  focus   F 1  and  F 2  of the ellipse  are located at a distance  c  from  the center. The longest distance through the  center is the major axis.  a  is the  semi-major axis . The shortest distance through the  center is the minor axis.  b  is the  semi-minor axis. The  eccentricity  of the ellipse is  defined as  e=c/a . The range of  eccentricities for ellipses is  0<e<1 For a circle,  e=0 .
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Active Figure Ellipse 13.04
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Ellipse and Planetary Motion The Sun is located at one focus of the ellipse. Nothing  is located at the other focus. The point furthest away from the sun is called the  aphelion . Its distance from the sun is  a  +  c . For the orbit of the moon around Earth (with the Earth at the  focus), this point is called the  apogee . The point nearest to the sun is called the  perihelion Its distance from the sun is  a  –  c  . For an orbit of the moon around Earth, this point is called the  perigee .
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Newton’s Law of Universal  Gravitation Newton’s  law of universal gravitation  states that: Every particle in the Universe attracts every other particle  with a force that is directly proportional to the product of  their masses and inversely proportional to the square of  the distance between them 2 2 1 r m m G F g = (25.1) G=6.673 x 10 -11  N.m 2 /kg 2  is called the  universal gravitational  constant .
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Remarks on Newton’s Law of  Universal Gravitation The magnitude of the force varies as the inverse square of the  separation of the particles –  inverse square law Gravitational force is attractive. Denoting  F 12  to be the force  exerted by particle 1 on particle 2, the vector form of the law is  as follows: 12 2 2 1 12 ˆ r F r m m G - = Newton’s third law implies that where  F 21  is the force exerted by  particle 2 on particle 1.
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