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CIRCULAR MOTION
Energy Considerations in Planetary and
Satellite Motion
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Consider the system involving an object in the
vicinity of an even more massive object: perhaps the Earth-Sun system or
a satellite-Earth system.
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If we assume that the bigger mass is at rest
(v = 0), then the total energy of the system consists of the kinetic energy
of the smaller mass as it rotates around the bigger object, plus the gravitational
potential energy:
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Based on the equation above, the energy of the
system may be positive, negative or zero. For the systems we have been
studying such as planet-sun or satellite-planet system, the total Energy
is negative as we shall see.
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In any planetary system (Earth-Sun) or any satellite
system (Satellite-Earth), the gravitational force supplies the centripetal
force
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Therefore, the total energy of the system is
negative.
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This shows us that in the case of circular orbits,
the total energy of the system must be negative.
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Also, the kinetic energy is positive and is equal
to 1/2 the magnitude of the potential energy. Energy is CONSERVED in a
planet-sun system or satellite-planet system.
Gravitational Potential Energy
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Let us look at gravitational potential energy
in a bit more detail
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Let us begin with the gravitational force. From
this force, it is possible to calculate the change in the potential energy
of a system between 2 points.
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The choice of reference point is totally arbitrary,
therefore it is customary in systems like this to choose ri
such that U = 0 at that point. This occurs when r = infinity.
Escape Velocity
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Escape velocity is the minimum velocity needed
by an object in order to escape the gravitational pull of the Earth (or
any planet for that matter).
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We take rmax as our reference point
so is at infinity, and we assume that once it escapes to infinity, it has
zero velocity. So we proceed by conservation of energy of the system.
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vi = vesc when we set rmax
to infinity, therefore:
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The second equation is the escape velocity for
any arbitrary planet P.