Lesson 22
Orbits, Centrifugal Energy and Effective Potential

Name:faculty Section:M2 Start Time:13:59:29 Instructor:pate Course:355

1) If we graph the central force potential energy U(r), the "centrifugal potential energy" Uc, and the effective potential energy Ueff = Uc + U(r) all as functions of r, we get a graph like the one shown to the right. [Note that the Marion and Thornton text represents the effective potential energy Ueff as "V(r)".]

In your own words, please explain this graph and its significance. In particular, consider three different cases for the total energy E of the particle undergoing central force motion -- say (i) lots of energy (how much is lots?), (ii) a medium amount of energy (how much is a medium amount?), and (iii) a tiny amount of energy (how much is a tiny amount?). (It may help to refer to the numbers included on the graph.) What kind of motion would result in each of the cases? Try to think of and include an example of each type of motion in your description. (Hint: You might consider thinking of planets, moons, and comets...)



2) Let Uc represent the "centrifugal potential energy" of a particle undergoing central force motion. Let U(r) represent the potential energy due to the central force f(r) acting on the particle.

Estimate the size of Uc for the Earth in its orbit about the Sun, and compare your estimate to the size of U(r) for the Earth in its orbit about the Sun. What do the relative sizes of the two "energies" tell you about the Earth's orbit?

As always, please state any assumptions or estimates you make in order to answer this question. (Thank you!)



3) The expressions for the Lagrangian L, the total energy Etot, the kinetic energy T, etc. in Sections 8.3 through 8.7 of the Marion and Thornton text (4th edition) are valid

  1. for all inertial reference frames.
  2. for all reference frames corotating with the system.
  3. only when the mass of the central object is much greater than the mass of the orbiting object.
  4. only in the reference frame in which the center of mass (CM) is fixed.



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