Why doesn't the moon leave earth's orbit?

Hey - why wait!
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But, yeah, we're living in an unstable set-up. Since the orbital period of the moon is different from the rotation period of the Earth, the tidal bulge due to gravity does not lie under the line joining the Earth and the moon. Because the Earth's rotation is obviously larger than the Moon's rate of revolution, the bulge gets ahead of the sub-lunar point on the Earth due to the delayed response, causing good old tidal lag. This has the net effect of imparting a continuous transverse acceleration on the moon, making it gain velocity and increase its orbital distance from the Earth. In reaction to providing this impetus to the moon and increasing its angular momentum, the Earth experiences a torque that tends to slow down its rotation. As the moon gets further away, the tidal bulge on the Earth and consequently, the impetus it imparts to the Moon will weaken. Also, the deformation of the Earth in response to Lunar tides causes energy to dissipate (which is what causes the tidal lag in the first place). Added to this, tidal friction causes the Earth to continually lose kinetic energy of rotation as heat, and as a result, its rotation rate slows down. Since the Moon is receding due to the tidal nudge it gets from Earth, its orbital period is also slowing down. It's expected that when the Moon is just far enough away and the Earth just slow enough that the tidal bulge always lies along the line joining the centres of the two, we'll reach real tidal lock (i.e. both bodies in the system), and the Moon-Earth system might then finally be considered to be in equilibrium. This is going to be really boring though, since one half of the world will have to make a significant journey if they ever want to see the Moon! The rate of this process is difficult to evaluate (my comments re tidal effects, tectonics and variable ocean response functions above cover this point), but if anyone wants to take a stab at estimating the point of equilibrium, feel free.

Now there's a provocative thought. I wonder what our world would be like without a Moon? We'd still have tides caused by the sun, I suppose, but I imagine there would be serious changes in climatic conditions (different airstreams etc), the Earth's axis would probably wobble like a top too, causing all sorts of variations in the seasons. The interesting(!) thing would be to observe the changes at the point of destruction (assuming we could avoid showering ourselves in Moon fragments).

This subject is quite a puzzle to me. The moon does all this goofy stuff and yet nothing else in the universe does anything similar. Do any of you know why you believe any of this theory? As to why the moon stays in orbit with Earth it is because they are both falling into the sun's gravity field at the same rate.

Believe in which bit? The Moon recession, the orbital mechanics or the tidal stuff? The first's covered by observation, the second's covered by Newton, and the third's a combination of both. There's another example, outside the Earth; similar stuff is happening in reverse with Phobos, which is below synchronous altitude with Mars, and being pulled into the planet through a combination of gravity and tidal force at a rate of about 1.8 metres per century.

How do you know the moon is receding at 4cm per year? The bits that make no sense to me are all the stuff you are saying is truth about the dynamics of the moon. Even the orbit is a puzzle since the period of the moon is not a simple matter to determine. Do you go by the sidereal month to fit the motion to the Kepler law? Then the tides are another puzzle. Does the ocean respond to the moon or does the whole Earth? And is there a tide on the moon? I don't think so.

I have inquired about the motion of the moon many times. NASA never says there is data indicating the motion and there must be so data. There are a lot of details in the motion of the moon that should be studied and are not studied because there are more important matters such as "zero and infinity" and stuff like that.