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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?

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    $\begingroup$ You can spin things roughly so fast that the rim speed approaches the speed of sound in the material, beyond that they break apart (catastrophically). The method you are proposing has been extensively analyzed, see "space tethers". $\endgroup$ – CuriousOne Feb 15 '16 at 22:17
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    $\begingroup$ And, just try to find a motor with enough torque to get the thing to spin in the first place... $\endgroup$ – Jon Custer Feb 16 '16 at 0:20
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This is done with gear works all the time - from watches to large scale machinery; as the comments have already pointed out, maximum speed is limited by the materials used and the torque available.

The total energy remains unchanged, minus losses, so a complete analysis is required to see what you gain vs what you lose.

This site discusses gear reduction, but illustrates the issues: http://www.teamdavinci.com/understanding_gear_reduction.htm

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1 mile long rigid wire! Can you imagine the material which moves rigidly if you move its one end with such length. calculate the young and shear modulus of the material. Consider this with such length and with normal material (Young's/Shear modulus in the possible range) and with your speed it will not move the pipe/wire/rod (with 1 mile length) from other side but produce waves only.

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