Turn an axle with diameter 1 inch at 1 mile per hour, and an attached wheel with diameter 5 inches will spin at 5 miles per hour.
Why can't we apply this principle on a much bigger scale? Say you put an axle 1 foot in diameter at the peak of a mountain, and attached a cable to it 1 mile long (with another cable for counterweight), and then spun the axle 1000 miles per hour. The cable could have a satellite attached to it set to launch when top speed is reached.
That's already about 5,000,000 miles per hour (1 mile/1 foot =~ 5000, times 1000mph). It would take a lot of energy and very gradual acceleration, but I'm sure it's possible to get a 1-foot axle to spin at 1000 mph. Why doesn't this work in practice?