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Gravity as a central force

# Gravity as a central force - with itself is zero and since...

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Gravity as a central force Newton's Universal Law of Gravitation produces a central force. The force is in the radial direction and depends only on the distance between objects. If one of the masses is at the origin, then ( ) = F ( r ) . That is, the force is a function of the distance between the particles and completely in the direction of . Obviously, the force is also dependent on G and the masses, but these are just constant--the only coordinate on which the force depends is the radial one. It is easy to show that when a particle is in a central force, angular momentum is conserved, and motion takes place in a plane. First, let us consider the angular momentum: = ( × ) = × + × = ×( m ) + × = 0 The last equality follows because the cross product of
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Unformatted text preview: with itself is zero, and since is entirely in the direction of , the cross product of these two vectors is zero also. Since angular momentum does not change over time it is conserved. This is essentially a more general expression of Kepler's Second Law, which we saw ( here ) also asserted the conservation of angular momentum. At some time t , we have the position vector and velocity vector of the motion that define a plane P with a normal given by = × . In the previous proof we showed that × does not change in time. This means that = × does not change in time either. Therefore, × = for all t . Since must be orthogonal to , it must always lie in the plane P ....
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