Geostationary Orbit

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I have been running over this question in my mind now for quite some time and I just can not come up with an answer. Here it is.... Lets say we have come up with a way to manufacture something close to the strength and weight of spider silk and we put 35,800 kilometers of the stuff on a spaceship and flew the spaceship into geostationary orbit around the Earth. Once arriving in orbit we start to let out the spider silk in the direction of the Earth. Will the silk string ever reach the ground? What happens when the silk string starts to interface with the Earts atmosphere? Would the string extend in a straight line all the way to the Earth or would it become electrically charged and bend into the shape of the earths mangetic field? What would happen to the spaceship?