An object at the equator is moving about 1050 miles per hour due to the rotation of the Earth. An object at geosynchronous orbit altitude must move at 3430 mph to be "weightless." Therefore, as you pull an object from the surface to geosynchronous altitude, you must accelerate it by 2380 mph to avoid pulling the cable like a bowstring.
That's why a space elevator isn't going to save any money over a rocket. You still have to add the same amount of energy to an object to get it to stay in orbit. A space elevator doesn't provide any cost advantage.