Remember, with an orbital tower, you have two forces: Gravity working inwards and centrifugal working outwards. Gravity gets stronger further in, while centrifugal gets stronger further out. They balance exactly at the geostationary orbit point, which is at 36000km.
So the Gundam writers got it right: 10000km is closer to the Earth than the balance point, and thus the sum of the forces is still pointed towards Earth there. Further out from geostationary, for instance at 50000km, the centrifugal force is stronger, and things you let go of would be flung outwards.