A Satellite with Drag

This problem concerns the properties of circular orbits for a satellite of mass orbiting a planet of mass in an almost circular orbit of radius .

In doing this problem, you are to assume that the planet has an atmosphere that causes a small drag due to air resistance. "Small" means that there is little change during each orbit so that the orbit remains nearly circular, but the radius can change slowly with time.

The following questions will ask about the net effects of drag and gravity on the satellite's motion, under the assumption that the satellite's orbit stays nearly circular. Use if necessary for the universal gravitational constant.

Part A
The total mechanical energy of the satellite will __________.
Part A.1 Consider the work done on the system Part not displayed
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Part B
What is the potential energy of the satellite?
 = -m*M*G/r
Part C
What is the kinetic energy of the satellite?
Hint C.1 Centripetal acceleration To keep an object moving in a circle at velocity , there must be a centripetal acceleration with magnitude directed toward the center of the circle.
Part C.2 Find the velocity
For the satellite, the centripetal acceleration is provided by gravity. Find the velocity of the satellite by setting the centripetal acceleration equal to the acceleration due to gravity.
 = sqrt(M*G/r)
Express the kinetic energy in terms of , , , and .
 = 1/2*m*M*G/r
Part D
As the force of the air resistance acts on the satellite, the radius of the satellite's orbit will __________.
Part D.1 Find the total energy
Find an expression for the total mechanical energy of the satellite as a function of , the radius of the satellite's orbit.
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Part E
As the force of the air resistance acts on the satellite, the kinetic energy of the satellite will __________.
Hint E.1 Kinetic energy as a function of radius Hint not displayed
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This may seem puzzling, but you will see why this happens in the next part.
Part F
Which force or forces lead to a change in kinetic energy? That is, which force or forces do work on the satellite?
Hint F.1 Definition of work Hint not displayed
Hint F.2 Direction of the velocity Hint not displayed
 gravity alonethe drag force aloneboth the drag force and gravity
Even though the drag force decreases the (nearly) tangential velocity of the satellite, this is more than offset by the increase in the radial velocity due to the action of the gravitational force, which is able to "reel the satellite in" slowly, as the tangential velocity decreases. Mathematically speaking, from Newton's 2nd law,

.

As the tangential velocity decreases, the magnitude of the centripetal acceleration required decreases. Also, as you saw earlier, the radius decreases, which means that the magnitude of the gravitational force increases. This means that the magnitude of the radial acceleration must increase, and so the radial velocity increases also. It is somewhat harder to show that this increase is greater than the decrease in the tangential velocity, but it must be so if the kinetic energy increases.
Part G
As the force of the air resistance acts on the satellite, the magnitude of the angular momentum of the satellite with respect to the center of the planet will __________.
Part G.1 Find the angular momentum Part not displayed