A satellite is in a geostationary orbit around the earth at some distant R from the earth's centre so that it's always over the same spot on the earth even though the earth is turning. How long does it take to orbit the earth if 1)it stays in its present position, 2)it moves out to an orbit twice as far from the earth's centre?

1) the same time of a full rotation of the earth. This is NOT 24 hours, but 24 hours multiplied by 365,26 and divided by 366,26 - the orbiting of the earth around the sun "steals" one day over the year.
2) Newton's gravity law and the centrifugal forces come in here. I'll check it and come back with this tomorrow.

The gravity force is G*m*M/(r*r)
The centripetal force is m*r*(w*w) - where w is the greek letter omega, standing for circular speed.
In orbit both forces are equal, or:
G*m*M/(r*r) = m*r*w*w
or w= squareroot of (G*M/r*r*r)
=> doubling the radius will divide the speed by squareroot of 8.