Orbital Motion and Angular Momentum

The motions of two bodies connected by an attractive force can also include rotation. In the simplest case (for example, rotation of the Earth around the Sun) one of the two bodies is much more massive than the other, and the heavier body hardly moves. The attractive force causes an acceleration through Newton s Second Law, but this does not necessarily imply that the speed changes—for a perfectly circular orbit the speed is constant. Velocity is a vector quantity, and so a change in direction is an acceleration as well. 

Bound orbits due to the attraction between unlike charges (or due to gravity, which has the same functional form) are all circles or ellipses. We will consider the circular case first. Suppose a particle of mass m is moving counterclockwise in a circular orbit of radius R in the. rv-plane. The particle s position r = (x,y,z) can be described by the equations 

The instantaneous velocity and acceleration are found by taking derivatives  

Notice that the acceleration vector is pointed directly toward the origin  

According to Newton s laws, this acceleration can only be sustained if there is an attractive force F = ma towards the origin  

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