Ack! Damn' submit button!

To continue, the cable suddenly becomes a separate unit, with its centre of gravity 9,900 Km above the surface. I'll have to sleep on this, to work out how conservation of energy affects matters, but I suspect the cable will immediately acquire apparent horizontal velocity in the direction of the planet's rotation i.e. the cable will appear to fall forwards, rather than be dragged around by the planet. Once this forward motion is acquired, it will be combined with the downward acceleration due to gravity to indeed wrap the planet.

I suspect that even if I am wrong, the cable will have sufficient rigidy to prevent it simply falling vertically and creating a pan-scrubber which would undoubtedly make the "Guiness Book of Records".