|MadSci Network: Engineering|
Regardless of how it is done, there is a certain amount of energy required to lift a mass from the surface of the Earth to orbit and impart sufficient velocity (approximately 8 kilometers per second or 5 miles per second) to keep it in orbit.
Regarding your question about leaving the Earth end unattached, if Earth end is not anchored the cable will fly away much like a string will fly away if you attach it to a weight, twirl it quickly and let go.
Space elevators have been proposed as one replacements for rockets. In general, one end of the cable is anchored to the Earth's surface, and the other end is anchored to a mass (artificial satellite, small asteroid, etc.) that is approximately twice as far above the Earth's surface as the orbit you wish to achieve. A cargo is lifted along the elevator to the desired orbit and released or delivered to a waiting station. The time to get to orbit is longer than a rocket (hours or days versus minutes), but the advantages are potentially larger cargos and safety as the energy source is external to the vehicle.
The problem with space elevators is they simply cannot work using current conventional materials. The cables must run continuously at least 360 kilometers or 200 miles and more likely more than 500 kilometers to be used for a space elevator. A high-strength steel can achieve yield strengths of over 1000 megapascal or 150,000 pounds per square inch easily. However, its density is around 7.87 grams per cubic centimeter or 0.284 pounds per cubic foot. For a cable one meter in diameter and 360 kilometers in length capable of supporting 813 megaNewtons or 182,605,438 pounds force of load, the mass would be 2,225,195,280 kilograms or 4,895,429,616 pounds mass. With that strength and that mass, the cable will snap from centrifugal forces or collapse under its own weight.
Newer materials such as single-wall carbon nanotubes offer the potential for producing a cable with sufficient strength and low density to use for a space elevator. The density of carbon as diamond is only 2.62 grams per cubic centimeter or about one third the density of steel. The density of carbon nanotubes will likely be closer to 1.85 grams per cubic centimeter, a typical density of carbon fibers and pyrolytic graphite. The strength of the carbon nanotubes in long lengths is unknown since the longest ones produced to date are on the order of micrometers rather than meters. They are expected to be much stronger than carbon fibers which are commercially available with strengths around 4830 MPa or 750,000 pounds pr square inch in kilometer lengths. The high strength and low density may be sufficient to allow construction of the cable if sufficient lengths of nanotubes can be made.
So a space elevator has potential, but until materials are developed for the cable the potential cannot be realized, and rockets remain the most viable method of putting cargo into orbit.
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