Two identical spheres are connected by an elastic tether. The tether obeys Hooke's Law for ideal springs. At a particular moment in time, the tether is in a straight line, at its resting length, neither stretching nor contracting. This assembly is then placed into a circular orbit around the Earth, and oriented so that a line drawn from one sphere through the tether and the other sphere points directly at the Earth.
Give a qualitative description of the motion of the two spheres relative to each other over time.
(In reply to
Difficulty 5+++? by Hugo)
I think it is safe to asume that the earth is so masive that it remains fixed in place.
This is a two body problem.
That being said, I cannot find a simple "difficulty 3" answer for this question.
As people have already said, since the angular rotational velocity of
the ball closest to the earth is bigger than that of the other ball,
the two balls will begin to rotate and oscilate.
That being said, what happens to the center of mass?. Does it follow a
simple orbit, or does it bobble?. Is the bobble limited to a maximun
value (stable bobble) or does it go out of control (unstable bobble)?.
The same question can be asqued about the amplitudes of the oscilations, are they stable or do they get out of control.
I would quess that the system goes out of control over time and the
spring breaks, transforming the two initial orbits into two different
ones.
An orbit can be defined by it's angular momentum and energy. We know
that the total angular momentum of the two final orbits are equal to
the total angular momentum of the two initial orbits. Something similar
can be said about the total energy, but the "stolen" energy of the
broken spring can make the final energy smaller than the initial one
(if the spring has mass it can also steal some angular momentum).

Posted by ajosin
on 20050413 23:00:42 