Probably you have noticed that the same part of the moon is always facing the earth (the other side of it being called the dark side of the moon - also a famous Pink Floyd album :-) ). It appears therefore to be standing still. What really happens is that the moon rotates around its axis at exactly the same speed as earth does. As a result when the surface of the earth turns a little bit, the moon has turned by the same bit and as a result the same people on the surface always see the same side of the moon.
Of course, the real question is why does the moon rotate at the same speed as earth. This is due to a procedure called mode locking, a nonlinear phenomenon common in many electrical and mechanical systems. The basic principle behind it is that when there are two systems oscillating (in the general sense) with certain frequencies, and these systems can be affecting one another, then they tend to alter their frequency of oscillation so that they finally match.
In the case of the moon and the earth, when the moon was first created it didn't have the same period as earth, and at that time different sides of it were visible from earth. However due to the gravitational attraction between them, the earth's much stronger attraction caused the moon to eventually alter its period so that it matched earth's. At this situation the systems locks, because it is the more convenient way of oscillating for both systems.
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