Now since the satellite is circling the earth it will be in one of two orbits. Geocentric or lower earth orbit. Geo orbit has the satellite circulating over one target, going to have to go to a lower orbit to aim at another target. To do this energy would have to be expended to get it into a lower or degrading orbit. It will not be the case of just dropping the rod. on its own it would take a long time to drop out of orbit. If it is held at a lower orbit it will pass over different parts of the Earth but it will have to be continuously 'topped up" with energy to remain in its orbit. That would mean topping up the gas tanks with fuel. Also the thing would be easy to track and sending up a rocket to disable it is a possibility. That would send debris all over making a big mess. Probably easiest to destroy it as it is being built might be a better idea. Or once one nation starts building one then another can do the same, arms race in space.
All great for the movies, not really practical. That is until we get to the point of mining space. Then just need the big ass engine and fuel to slow it down to drop it out of orbit.