Someone commented with an interesting question on a YouTube video I posted last year:
“I wonder how far from the earth the end of the cable would have to be to allow for the equivalent of 1G due to centrifugal force.”
I didn’t know so I asked Ben Shelef, CEO of the Spaceward Foundation (host of the Space Elevator Games) – this was his reply:
At any point on the tether, in your reference frame, you have two forces (accelerations) acting on you. Gravity downwards, and the centrifugal acceleration outwards.
Gravity diminishes with distance square, so at a height of 6000 km, you’re at double the distance from the center of the earth as you were when you took off, and so the force of gravity is 1/4 what it was.
At ground level, the centrifugal acceleration is very small, but it increases linearly with the radius. (a=omega^2*r) [omega is the spin rate of the Earth].
At GEO, the two accelerations are equal. (and each is very small, basically 1/50g)
So as we move out, at some point, the outwards acceleration will equal 1g. how far? We can neglect gravity, since it diminishes even further. The Centrifugal acceleration has to increase a factor of 50! So 50 times as far as GEO – way beyond the end of a 100,000 km long tether.
- omega is 6.28/24/3600 = 7.3E-5 1/Sec
- omega-square is 5.3E-9 1/Sec2
- r = g/omega2 = 1.9E9 m, or 1.9E6 km – or ~20 times longer than the 100,000 km tether
The mean distance between the earth and the moon is ~384,400 km, so a tether long enough to generate 1g at its tip would need to be nearly 5 times LONGER than that mean distance between the earth and the moon! I don’t think we’ll be seeing it anytime soon ?