"where along the length "

exactly- think of a rope whose CENTER is in orbit over a specific spot on the Earth- ( Dragging the two ends along, since they are balanced against each other)

If there is a break, it will change the balance ( center of mass)

the new center of mass will be either above or below geosync, leading to either

1. a closed conic section intersecting the Earth
2. an open conic section taht flies off into space

The CM at geosync gives the desired solution of an orbit around the Earth that does not intersect the surface ( at least, the CM does not.)

As stated before, drag is small (on 10 - 100 miles, out of 22,000) so can be ignored for the first order solution.


But not a great idea to build high drag structures on the lower ( high atmospheric density) section- that first 10 miles.