Express the kinetic energy in terms of the potential energy .

=

-1/2*U

Find the orbital period .

Express your answer in terms of , , , and .

=

2*pi*R^(3/2)*(G*M)^(-1/2)

Find an expression for the square of the orbital period.

Express your answer in terms of , , , and .

=

(2*pi)^2/(G*M)* R^3

Find , the magnitude of the angular momentum of the satellite with respect to the center of the planet.

Express your answer in terms of , , , and .

=

m*(G*M*R)^0.5

The quantities , , , and all represent physical quantities characterizing the orbit that depend on radius . Indicate the exponent (power) of the radial dependence of the absolute value of each.

Express your answer as a comma-separated list of exponents corresponding to , , , and , in that order. For example, -1,-1/2,-0.5,-3/2 would mean , , and so forth.

-1/2

,

-1

,

-1

,

1/2

Geosynchronous Satellite

A satellite that goes around the earth once every 24 hours is called a geosynchronous satellite. If a geosynchronous satellite is in an equatorial orbit, its position appears stationary with respect to a ground station, and it is known as a geostationary satellite.

Find the radius of the orbit of a geosynchronous satellite that circles the earth. (Note that is measured from the center of the earth, not the surface.)

You may use the following constants:

The universal gravitational constant is .

The mass of the earth is .

The radius of the earth is .

Give the orbital radius in meters to three significant digits.

=

4.225*10^7 (+/- 0.2%)
or (5.98*10^24*6.67*10^-11 /(2*pi/(24*60*60))^2)^(1/3)

m

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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.

The total mechanical energy of the satellite will __________.

Express the kinetic energy in terms of , , , and .

=

1/2*m*M*G/r

As the force of the air resistance acts on the satellite, the radius of the satellite's orbit will __________.

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

As the force of the air resistance acts on the satellite, the kinetic energy of the satellite will __________.

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

Which force or forces lead to a change in kinetic energy? That is, which force or forces do work on the satellite?

gravity alone
the drag force alone
both the drag force and gravity

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 __________.

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

Which force or forces will cause the magnitude of the satellite's angular momentum with respect to the center of the planet to change?

gravity alone
the drag force alone
both the drag force and gravity

Post-Collision Orbit

A small asteroid is moving in a circular orbit of radius about the sun. This asteroid is suddenly struck by another asteroid. (We won't worry about what happens to the second asteroid, and we'll assume that the first asteroid does not acquire a high enough velocity to escape from the sun's gravity). Immediately after the collision, the speed of the original asteroid is , and it is moving at an angle relative to the radial direction, as shown in the figure. Assume that the mass of the asteroid is and that the mass of the sun is , and use for the universal gravitation constant.

Since the asteroid does not reach escape velocity, it must remain in a bound orbit around the sun, which will be an ellipse. Take the following steps to find and , the aphelion and perihelion distance of the asteroid after the collision. (The aphelion is the point in the orbit farthest from the sun, and the perihelion is the point in the orbit closest to the sun.)

As in most orbit problems, the most fundamental principles involved are energy and angular momentum conservation.

What is the total energy of the asteroid immediately after the collision?

Express the total energy in terms of , , , , , and .

=

0.5*m*V_0^2-G*M*m/R_0

What is , the magnitude of the angular momentum of the asteroid immediately after the collision, as measured about the center of the sun?

Express the magnitude of the angular momentum in terms of , , , , , and .

=

m*V_0*R_0*sin(theta)

, , and are given initial conditions, and the values of , , and are known constants. The total energy and angular momentum may be expressed entirely in terms of these known, fixed quantities. This means that we should be able to express any of the fixed quantities related to the new orbit in terms of , , and known constants, instead of using the initial conditions.

Now focus on the new orbit of the asteroid. The final goal is to find and , the perhelion and aphelion distances of the new orbit, in terms of , , and known constants. Let and be the orbital speeds at aphelion and perihelion respectively.

Write an expression for the total energy at aphelion in terms of the variables characterizing the motion there (i.e., and ).