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Mechanics

Energy Changes in a Gravitational Field

A mass placed in a gravitational field experiences a force. If no other force acts, the total energy will remain constant but energy might be converted from g.p.e. to k.e.

Energy Possessed by a Satellite

If the mass of the planet is M and the radius of the orbit of the satellite is r, then it can easily be shown that the speed of the satellite, v, is given by

therefore, if r decreases, v must increase. If the mass of the satellite is m, then the kinetic energy, K, possessed by the satellite is given by

and the potential energy, P, possessed by the satellite is given by

These equations show

i) that if r decreases, K increases but P decreases (becomes a bigger negative number)
ii) the decrease in P is greater than the increase in K.

Therefore, to fall from one orbit to a lower orbit, the total energy must decrease. In other words, some work must be done to decrease the energy of the satellite if it is to fall to a lower orbit.

The work done, w, is equal to the change in the total energy of the satellite,
w = DELTA02K +
DELTA02P.

This work results in a conversion of energy from gravitational potential energy to internal energy of the satellite (it makes it hot!).

Air resistance can thus reduce the speed of the satellite along its orbit. This allows the satellite to fall towards the planet. As it falls, it gains speed.

So, if a viscous drag (air resistance) acts on a satellite, it will

i) decrease the radius of the orbit
ii) increase the speed of the satellite in itís new orbit.

In principle, the satellite could settle in a lower, faster orbit but in practice it will usually be falling to a region where the drag is greater. It will therefore continue to move towards the planet in a spiral path.

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