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Nothing would happen to the path of the satellite if the earth stopped rotating. The satellite is orbiting the earth at a constant speed which was supplied by the rocket which put it into orbit.
However, it would affect what some satellites observe or sense as they orbit the earth while it is rotating. Some satellites are in an orbit that is mainly north-south. In this case, as the satellite orbits the earth it keeps seeing different parts of the earth beneath it. If the earth stopped then the satellite would basically see the same surface of the earth all the time.
If the earth was not rotating, it would be more difficult to launch new satellites into space. Because the earth is rotating at about 1100 mph west to east, launching a rocket in an easterly direction gives it a higher starting speed so less energy is needed to get it moving fast enough to reach orbit.
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