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Energy Changes in a Gravitational Field
A mass placed in a gravitational field experiences a force.  
If no other force acts on the mass, the total energy will remain constant but energy might be converted from g.p.e. to k.e.  
   
Energy Possessed by a Satellite in Orbit  
Let the mass of the planet be M, the mass of the satellite, m, and the radius of the orbit of the satellite r.  
The force of gravity must be just equal to the centripetal force needed to maintain the orbit .  
Therefore we can write  
 
This means that for a given radius of orbit, the satellite must have a speed given by  
 
which tells us that, if the radius of the orbit decreases, the speed must increase (to maintain the new orbit).  
   
Therefore, the kinetic energy, K possessed by the satellite is  
 
The potential energy possessed by the satellite is given by  
 
Looking at these last two equations, we see that  
1. if r decreases, K increases and P decreases (it becomes a larger negative value)    
2. the decrease in P is more than the increase in K (so the total energy decreases).    
   
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,   
   
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   
1. decrease the radius of the orbit and   
2. increase the speed of the satellite in its 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, as shown.   
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