If you are on the Moon, does the Earth move in the sky?
Generally, the Earth will not "move across the sky"; it pretty much "stays put" in one location. That is not to say, however, that the appearance of the Earth does not change. Read on!
Our Moon spins on its axis so that as it orbits the Earth, it always presents the same face to the Earth. As a result, when viewed from the Moon, the Earth will always remain in about the same spot in the sky all the time! (This may be easier to see if you set up two balls (using a light as the Sun) and make a model of the situation; place yourself on the Moon ball and you'll see what the Earth then looks like at any point in your orbit.)
I say that it is about the same because there are some differences. For example, there are slightly different apparent sizes of the Earth due to the fact that the orbit of the Moon is not a perfect circle; sometimes the Earth is closer (and appears larger) and sometimes it is farther away (and appears smaller) in each orbit. Also, because the orbit of the Moon is tilted about 5 degrees with respect to the Earth's equator, from the Moon there will be locations where the Earth will slowly rise and set during the lunar month as seen from the surface. The Moon undergoes a motion called "libration" which causes it to rock slightly back and forth relative to a line connecting the centers of the Earth and the Moon. This libration effect, as seen from the Moon, will cause the Earth to move slightly back and forth in the sky relative to a fixed point above the lunar horizon.
It is also important to note that the Earth will go through a complete set of phases each lunar month, with a "Full Earth" happening when it is "New Moon", and a "New Earth" happening during "Full Moon" (in other words, they will appear to be in exactly opposite phases). For additional information on the lunar phases, see StarChild's Question of the Month response for "What are the phases of the Moon?"
In summary, while it generally remains in the same location, the Earth does not remain perfectly stationary in the lunar sky from every point on the Moon, but moves in a rather complicated way depending on your location on the lunar sphere!
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