Two satellites are in orbit around the Earth. The first is in a geostationary orbit, the second satellite at radius half that of the first. What is the (approximate) period, in hours, of the second satellite?

A geostationary satellite is one that remains above the same location on Earth. This means it has to be above the equator, and its orbit period is exactly 24 hours! The period of a circular orbit is distance/speed, which is 2piradius/(tangential velocity). If the radius is halved, to remain in orbit the velocity has to go up. Now by equating Newton's gravitational equation with F=mv²/r, we have F=mv²/r=GM_Earthm/r² So v²=GM_Earth/r Halving r means v must go up by sqrt(2). So our new period is T_new=24*(1/2)*(1/sqrt(2))=8.5 hrs.