Actually, the space elevator IS in free orbit! The mass of the counterweight above the GSO station is equal to the mass of the ribbon below, so the system as a whole is in balance — a very long, thin structure with its axis pointed at the center of the Earth. Think of the elevator as a satellite 100,000 miles “tall” and you’ve got it.
I suppose that is probably true also, but still doesn’t answer the question of “How does something traveling at 25000 miles (the circumference of the earth) per day at the surface of the earth accelerate to 150000 miles (the circumference of a geosynchronous orbit) per day under the “Space Elevator” concept.
The only thing I can think of is that the elevator doesn’t go straight up as shown in the drawings, but rather is slightly angled to the west.