I refer you to Archimedes: "Give me a fulcrum and I will move the World!".

You could mount a drive shaft on the poles, or put a gearwheel round the equator, or put ratchets and springs on tall towers. In all cases, the problem is that you never have anything to pull against that is not rotating with the Earth. So there is no force to act, and therefore no energy can be extracted.

Nature (or perhaps some benign aliens) solved this rather neatly about 4 billion years ago. By placing a large Moon quite close by (but not where it would be in geostationary orbit), and covering Earth with a fair amount of hydraulic fluid (oceans), we got a continuous source of energy driven by Earth's rotation and anchored externally by the Moon's inertia. It's called tidal energy. (There is a similar but smaller effect on the solid rock, too.)

It's not permanent, however. The pull of the tides on the moon is gradually boosting it into a higher orbit. The tides were much stronger earlier in the earth's life, although the changing disposition of the continents would have modified the tidal flow in odd ways.

You seem to be sharing a problem that Archimedes encountered: a lever long enough plus a fulcrum. Fortunately I can help you in that matter. The first thing to bear in mind is that we have long been storing the energy you describe in the kinetic and potential energy of the Earth-moon system. Our days are being lengthened by tidal forces that have been driving the moon into an ever-widening orbit at the expense of Earth’s angular momentum.

Now, all we need do, is to install rope from both poles to the moon (using just one pole would cause destructive precession, and we must be practical about the design of these systems). Then we use the momentum of the moon to drive dynamos large enough to supply the entire planet’s electricity from the tension in those ropes; there should be plenty. The energy we harvest would be sustainable for billions of years because it would reduce the moon’s orbit, increasing its orbital velocity and increasing its tidal force with the inverse FOURTH power of its orbital radius!

And it gets better! As the moon comes closer, our oceanic tides will increase drastically, in turn widening the orbit. We could harness the tides in their turn to drive our dynamos, for a long, long time. In fact we could continue till the orbit of the moon matched the planetary day length.

By that time we should be a billion or so years on, and it would be high time to abandon our planet anyway, because our sun would be getting into its red giant phase, ready to swallow our inner planets, probably including our planet. By then however, tectonic activity probably would have swallowed the great pyramid of Giza, Hollywood, and Graceland, so there would be nothing for us to stay back for. Exactly HOW to leave our planet, and probably our solar system as well is another matter; but if we couldn’t manage it, we would deserve to broil. A more relevant question would be: what would we be by then?

Would we even care?

Stick around and see.

Jon

Of course, if you want to think small instead, there are many options. You could deflect ocean currents north and south to extract the Coriolis energy. Or you could fly massive airships similarly north and south for similar effects. You could erect HIGH towers with weights from which you could extract piezoelectric tidal energy, possibly in combination with raising and lowering the weights at strategic points in the lunar cycle for best effect. You could send spacecraft up to slingshot round the moon in a figure-8 trajectory partly to generate current with on-board magnets to slow them down on passing earth.  This would slow down the moon as well and speed it up as its orbit contracts.

Archimedes woould go green with envy!

But not I!

All the best,

Jon