posted by Ray .
A orbiting satellite stays over a certain spot on the equator of (rotating) Mercury. What is the altitude of the orbit (called a "synchronous orbit")?
First you need to look up the period of rotation of Mercury. It is a rather slow rotator, locked to three rotations per two orbits about the sun. That make the sidereal rotation period 58.6 days. (That is NOT the length of a Mercury day).
Next you need the mass of Mercury to determine the geosynchronous orbital height. Set the gravitational pull equal to the centripetal force, and solve for R. The mass of the satellite will cancel out. The universal constant of gravity will appear in your formula.