Actually, all stars are set in a fixed position in the sky relative to each other. It is the movement of the Earth, rotating on its axis, that makes the stars, including the sun, appear to move across the sky. (Planets, however, do move on their own through the sky and can thus be differentiated from stars.)
The North Star is situated directly above the northern axis of Earth. Thus, as Earth spins, Polaris, on the northern pivot point of rotation, does not appear to move from its polar position. Hence the name Polaris.
The stars appear to move because of the Earth’s rotation around its axis, and the North Star is called that because it aligns with the axis on the north end, and appears to remain in a constant position that marks where north is.
The Earth’s axis isn’t actually constant. It naturally shifts and wobbles, but over a period that is too long to be noticeable. So eventually, the axis of the earth with no longer point directly at Polaris (the north star). I recall that far in the future, the axis will actually align with another star that could be used to mark north at that point, but I don’t remember which star. So the north star does eventually move in relation to the earth, but not in a way that would be obvious within a normal human lifespan.
Reference: Astronomy class
Yes, it does. Everything in space is moving relative to everything else; there is no “absolute rest”. In terms of our view from Earth, the North Star, Polaris, does not appear to move, because its movement is due to what is called the procession of the equinoxes, which is a very slowly changing cycle in the night sky. About 14,000 years ago, we had a different North Star, which was Vega, in the constellation Lyra. It will become our north star again in about 12,000 years, but not before another star, Gamma Cephei, will fill the position for awhile, starting in a little less than 1000 years.
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