|College of Science Physics Dept Tatsu Takeuchi Special Relativity Lecture Notes|
5. Laws of Physics in Non--Inertial Frames
For non-inertial frames, Newton's second law does not apply.
For instance, in a space shuttle in orbit, gravity is acting on everything but nothing is accelerated relative to the space shuttle. (This is known as weightlessness. Note that it is NOT gravitylessness. In fact, without gravity the space shuttle cannot stay in orbit. The space shuttle and everything inside it is always falling toward the Earth. It's just that it always "falls over the edge" of the Earth, so to speak, so that it never hits the ground!)
It is possible, however, to make it seem as if the relation F=ma is valid superficially even for non-inertial frames by introducing "fake" forces. For the space shuttle, we often say that the centrifugal force counteracts gravity and keeps objects from accelerating toward the ground. (Fake forces such as the centrifugal force are known as inertial forces.)
Just keep in mind that non-inertial frames have to be treated differently from inertial frames.