Motion under a central force
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Energy and central force systems
Total energy is conserved when a particle is subject only to a central force. Gravitational potential energy is the integral of the gravitational force. [Equation goes here - download the original to see it.] The kinetic energy of a particle is [Equation goes here - download the original to see it.] Since total energy is conserved Kinetic Energy + Gravitational potential energy = Total energy Where the total energy, , is a constant. Thus [Equation goes here - download the original to see it.] If an object under a central force is in orbit on an elliptical path, then it is possible to deduce the values [Equation goes here - download the original to see it.], and [Equation goes here - download the original to see it.]of the distance of the ellipse along the major axis [Diagram goes here - download the original to see it.] To show this, total energy is [Equation goes here - download the original to see it.] Where [Equation goes here - download the original to see it.] represents the central force. When [Equation goes here - download the original to see it.] and [Equation goes here - download the original to see it.] [Equation goes here - download the original to see it.] the object cuts the x-axis, so the velocity is wholly transverse and [Equation goes here - download the original to see it.] , or [Equation goes here - download the original to see it.] . Substituting into the energy equation gives [Equation goes here - download the original to see it.] Hence [Equation goes here - download the original to see it.] or [Equation goes here - download the original to see it.] Which is a quadratic in [Equation goes here - download the original to see it.]giving roots [Equation goes here - download the original to see it.] which are the values of [Equation goes here - download the original to see it.] along the major axis of the elliptical orbit.
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Contents of
Motion under a central force
1 Motion under a central force
2 Transverse Velocity
3 Kepler's Second Law
4 Motion under gravity
5 Energy and central force systems
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