How can I measure the orbital period of a satellite around Earth?

In order to find the orbital period of a satellite around earth, you must understand the forces at play in this situation. You must equate the two radial forces acting on the satellite - these are Newton's law of gravitational attraction between the satellite and the earth, and the centripetal force acting on the satellite. (You can imagine the centripetal force as the tension in an invisible wire connecting the satellite to the earth as it attempts to move away and is pulled back in as it orbits). By equating these two forces you will be able to cancel down the like-coefficients and be left with the equation of orbital velocity for the Satellite. After this you must realise that the period can be related to the velocity by the speed-distance-time relation, where orbital period is the inverse of orbital time.