Re: Where is the center of mass for the Earth-Moon system?

The centre of mass of the Earth-Moon system is quite easy to calculate.  
You just have to find the point where is the two bodies were connected by a 
rigid rod, they would balance.  The Earth is about 1/80th the mass of the 
Earth so the point at which they would balance is about 1/80th of the 
distance from the Earth to the Moon.  Since this is about 250,000 miles, 
the balance point (centre of mass) is just below the surface of the Earth 
on the side of the Earth facing the Moon.
The Earth is indeed slowing down due to tidal friction.  We are all 
familiar with water tides - but the same effect is going on within the 
solid Earth too and indeed the Earth raises much larger solid tides on the 
Moon (which have already stopped the rotation of the Moon relative to the 
Earth).  As the Earth rotates, there is a friction which is slowing it 
down.  Similarly this tidal friction affects the Moon and due to a physical 
law called the conservation of angular momentum, the Moon is actually 
getting further away from us, it used to be much closer in the past. This 
means that tidal friction is gradually slowing too so that it is not a 
simple matter to calculate how long it will take the Earth to slow down 
completely.  In fact the best and most sensitive method of measuring how 
much the Earth is slowing down is to predict when and where total eclipses 
of the Sun (by the Moon) would have occurred in the past (over the past few 
thousand years).  Ancient records tell us where the eclipse was actually 
seen - on the right predicted day but in the wrong place.  That's because 
the Earth has slowed down just a little in that time and the cumlulative 
effects over so many rotations over a couple of thousand years throw off 
the calculated position by hundreds of miles.  From this a very accurate 
figure for the slowing down can be determined.