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 Tethered satellites (Posted on 2005-04-13)
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.

 See The Solution Submitted by Larry Rating: 3.5000 (4 votes)

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The puzzle starts off with the entire assembly being in orbit and gives the initial orientation of the two spheres relative to one another and the earth.  But it does not specify the relative speeds of the two spheres.

The sphere closer to the earth could be traveling slower than the one farther one by an amount designed so that instantaneously the line between the two spheres will maintain its pointing toward the earth.  But in this case that situation won't last for long, as the bottom sphere does not have enough speed to remain in circular orbit, and will start to fall into a lower orbit, paradoxically speeding up in the process, were it not constrained by the elastic tether.

A different case would be if each sphere, disregarding the tether, is moving at a velocity that would keep it, by itself, in a circular orbit.  The lower one would be overtaking the upper one, as it must be going faster, even in an "absolute" sense (translational in addition to revolution sense). In this case the lower one is pulling ahead, but of course is also starting to be restrained by the tether attaching it to the upper sphere.

Where they go after the elasticity of the tether starts working is a more complicated matter.

 Posted by Charlie on 2005-04-13 19:29:35

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