If you tethered a space station to the earth by a long cable, you could get to space in an elevator that rides up the cable--much simpler and cheaper than riding to space on a rocket. There's one big problem, however: There is no way to create a cable that is long enough. The cable would need to reach 36,000 km upward, to the height where a satellite orbits at the same speed as the earth rotates; a cable this long made of ordinary materials couldn't even support its own weight. Consider a steel cable suspended from a point high above the earth. The stress in the cable is highest at the top; it must support the weight of cable below it.
What is the greatest length the cable could have without failing? For the purposes of this problem, you can ignore the variation in gravity near the surface of the earth. Hint: The mass of the cable is the volume of the cable multiplied by the density. The density of steel is 7900 kg/m3.