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If you (or e.g. a skater) spin on the spot with your arms outstretched you spin at a given speed, but when you retract your arms you spin much faster, extending your arms again will slow the spin down. Your arms are small in relation to your body but has a big effect in this case. Would a space-elevator have the same effect on earth's rotational speed, and by how much?

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Yes, but it will be of order $\frac {m_{elevator}}{m_{earth}}$ (quite small compared to your skater's arms example) and also depend on the latitude of where it is tethered.

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You should read the following information: http://users.wpi.edu/~paravind/Publications/PKASpace%20Elevators.pdf

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