An orbiting tether tends to straighten out along a radial line because the force of gravity varies slightly along its length. The pull of gravity is stronger nearest the Earth and weakest furthest away. That means there is a net force on a tether which stretches it and keeps the line taut. This isn't just an exercise in physics, though, these tethers have lots of useful applications. Momentum exchange tethers allow momentum and energy to be transferred between objects in space.
Electrodynamic tethers interact with the Earth's magnetosphere to generate power or propulsion.
Electrodynamic tethers have already been flight tested, and the concept has proven to be a viable technology. In fact, it is being considered as a means to counteract the slight aerodynamic drag on the International Space Station , reducing the need for reboosts that rely on conventional chemical propellants. Such a system could possibly save the program about a billion dollars in operating costs over the life of the station. Both types of tethers promise to reduce the cost of getting satellites into orbit and keeping them there or removing them. After orbiting around the Earth once together, the rotating tether system tosses the satellite forward into a higher orbit, somewhat like a roller derby skater grabbing a teammate and slinging them forward.