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A Satellite with Drag

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

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 G 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

ANSWER:
increase
decrease
stay the same
vary in a more complex way than is listed here
Part B  
What is the potential energy U of the satellite?
Express your answer in terms of m, M, G, and r.
ANSWER:
  U =  -m*M*G/r 
Part C  
What is the kinetic energy K of the satellite?
Hint C.1 Centripetal acceleration
To keep an object moving in a circle at velocity v, there must be a centripetal acceleration with magnitude v^2/r 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 v of the satellite by setting the centripetal acceleration equal to the acceleration due to gravity.
Express your answer in terms of m, M, G, and r.
ANSWER:
  v =  sqrt(M*G/r) 
Express the kinetic energy in terms of m, M, G, and r.
ANSWER:
  K =  1/2*m*M*G/r 
Part D  
As the force of the air resistance acts on the satellite, the radius r of the satellite's orbit will __________.
Part D.1 Find the total energy
Find an expression for the total mechanical energy E of the satellite as a function of r, the radius of the satellite's orbit.
Express your answer in terms of m, M, G, and r.
ANSWER:
  E(r) =

Answer not displayed

ANSWER:
increase
decrease
stay the same
vary in a more complex way than is listed here
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

ANSWER:
increase
decrease
stay the same
vary in a more complex way than is listed here
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

ANSWER:
gravity alone
the drag force alone
both 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,

{\rm gravitional \; force = mass \times (centripetal\; acceleration + radial\; acceleration}).

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

ANSWER:
increase
decrease
stay the same
vary in a more complex way than is listed here
Part H  
Which force or forces will cause the magnitude of the satellite's angular momentum with respect to the center of the planet to change?
Part H.1 Consider the torques on the system

Part not displayed

ANSWER:
gravity alone
the drag force alone
both the drag force and gravity
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