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MIT16_07F09_Lec15

# MIT16_07F09_Lec15 - J Peraire S Widnall 16.07 Dynamics Fall...

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Unformatted text preview: J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 1.2 Lecture L15- Central Force Motion: Kepler’s Laws When the only force acting on a particle is always directed to- wards a fixed point, the motion is called central force motion . This type of motion is particularly relevant when studying the orbital movement of planets and satellites. The laws which gov- ern this motion were first postulated by Kepler and deduced from observation. In this lecture, we will see that these laws are a con- sequence of Newton’s second law. An understanding of central force motion is necessary for the design of satellites and space vehicles. Kepler’s Problem We consider the motion of a particle of mass m , in an inertial reference frame, under the inﬂuence of a force, F , directed towards the origin. We will be particularly interested in the case when the force is inversely proportional to the square of the distance between the particle and the origin, such as the gravitational force. In this case, µ F = − r 2 m e r , where µ is the gravitational parameter, r is the modulus of the position vector, r , and e r = r /r . It can be shown that, in general, Kepler’s problem is equivalent to the two-body problem, in which two masses, M and m , move solely due to the inﬂuence of their mutual gravitational attraction. This equivalence is obvious when M m , since, in this case, the center of mass of the system can be taken to be at M . 1 However, even in the more general case when the two masses are of similar size, we shall show that the problem can be reduced to a ”Kepler” problem. Although most problems in celestial mechanics involve more than two bodies, many problems of practical interest can be accurately solved by just looking at two bodies at a time. When more than two bodies are involved, the problem is considerably more complicated, and, in this case, no general solutions are known. The two body problem was studied by Kepler (1571-1630) who lived before Newton was born. His interest was in describing the motion of planets around the sun. He postulated the following laws: 1.- The orbits of the planets are ellipses with the Sun at one focus 2.- The line joining a planet to the Sun sweeps out equal areas in equal intervals of time 3.- The square of the period of a planet is proportional to the cube of the major axis of its elliptical orbit In this lecture, we will start from Newton’s laws and verify that the above three laws can indeed be derived from Newtonian mechanics. Equivalence between the two-body problem and Kepler’s problem Here we consider the problem of two isolated bodies of masses M and m which interact though gravitational attraction. Let r M and r m denote the position vectors of the two bodies relative to a fixed origin O . Since the only force acting on the bodies is the force of mutual gravitational attraction, the motion is governed by Newton’s law with an equal and opposite force acting on each body....
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MIT16_07F09_Lec15 - J Peraire S Widnall 16.07 Dynamics Fall...

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