Actually, the Moon’s orbital plane is neither the ecliptic nor the equatorial planes. It is pretty close to the ecliptic – much closer than to the equatorial – but if it was spot-on, we’d have a solar and a lunar eclipse each month.
I’ve fired up my planetarium program and it is showing a not-quite full moon about 2.4 degrees south of the ecliptic. That’s about five moon diameters.
Unless the cable is significantly elastic (which is difficult to imagine alongside it being unbreakable), then bbig, bad things will happen straight away. There isn’t any suitable anchor point on the earth from which any anchor point on the moon is at a constant distance.
Ok, thanks. 
Has Commander Koenig signed off on this crazy scheme? Sounds like trouble.
I think “scrape along the ice” understates what would happen. The chord described by the cable would probably be significantly below the surface of the Earth. So “pare the surface like an apple” might be more appropriate.
Comparing the effects of the (presumably) inelastic and quite solid cable to that of gravity is a poor analogy. Gravity is OK with elliptical orbits. Unobtanium tethers not so much. The distance from the Earth to the Moon ranges from 220,000 miles to 250,000 miles. Exactly what’s going to happen depends on whether you attach the cable at apogee, perigee or somewhere in between. Best case is you have 30,000 miles of slack cable (of unknown diameter and mass) plunging to Earth on a monthly basis in a godawful tangle of kinetic energy.
Worst case is that you attach the cable at perigee and once the cable starts to tighten – well, “seismic activity” probably understates the result. I think we’d end up with a new axial tilt, and whatever one-celled organisms somehow survive the cataclysm will have to adjust to some new and interesting seasons.
Back to the equatorial tether, though, and assuming that the Earth and Moon are likewise made out of unobtanium (so the anchors don’t just pull free), you still wouldn’t get a collision. The cable would wind around the Earth some, reeling the Moon in, but eventually, the Moon would reach geostationary height, and the system would stabilize. I don’t know offhand precisely where the new geostationary height would be (though it could be calculated), but it would be higher than the current one, since the Earth’s rotation would be slower.
Will someone please think of the werewolves!
There’s this thing call conservation of momentum.
There’s this entity called the earth moon system.
83% of the momentum of this system resides with the moon.
That suggests to me that the earths rotation and equater will be most affected to maintain the total system momentum. Therefore I don’t believe the cable will coil.
Once the earth moon system is in equilibrium I see no reason for the cable to experience any tension.
I could be all wrong though.
Quick! Sell your beach property!
I’m looking forward to exploring the beach at the ultra-minus tides, though, aren’t you?