Orbits of a Simple Harmonic Oscillator

Consider a body, of mass m, attached to one end of a spring with spring constant . The other end of the spring is fixed and the body is set into motion about the fixed point with angular momentum per unit mass J and total mechanical energy per unit mass E.

1. Using as a guide the planetary Kepler problem from lectures, show that the body describes an orbit given by

   

   in the plane.

2. What shape is this locus? Draw a labelled diagram in the (x,y) plane.
3. What condition must be satisfied by , J, and E in order for the orbit to be bound?
4. Again using lectures as a guide, derive an effective potential per unit mass, , for the body satisfying

   

   Interpret your answer to the previous question in terms of this effective potential.

Now imagine a planetary system in which the force of gravity is spring-like, i.e. it is attractive and directly (not inversely!) proportional to the square of the separation between two bodies. (Needless to say, this is not a real planetary system, but it is very important in other contexts, as you will see later in the course.)

5. Derive the equivalent of Kepler's Third Law for this planetary system. What is special about your result?
6. What would be the motion of Mars in the sky as seen from Earth if the Earth and Mars were in this planetary system?