And the Moon orbit is not a perfect circle, the Earth-Moon distance varies quite a bit (about 40000km between perigee and apogee), so to actually keep the cable

And the Moon orbit is not a perfect circle, the Earth-Moon distance varies quite a bit (about 40000km between perigee and apogee), so to actually keep the cable at the same earth altitude you will need to change its length quite a bit (seems like around 125km/hour).