Difference between acceleration due to gravity (g)and free fall.
Answers
Answer:
don't think there's any systematic difference you can count on in general.
However what's going on in this particular case (which you should remember till the exam and then ignore) is that between 13-13 and 13-14 the author has implicitly and rather confusingly switched from a measurement frame that's anchored to the center of the earth and non-rotating, to a different measurement frame rotating with the earth in which the scale is stationary.
In the new frame, the mass is stationary but has a force FN on it from the scale, so presumably (and in fact), if you took that force away by removing the scale suddenly, the object would fall with an acceleration of FN/m relative to the original position of the scale. And sure enough, that's the free-fall acceleration of objects as we normally measure them.
But that's not free-fall acceleration of the mass in the frame that the whole lead up to this point was in. What 13-13 really says if you just divide through by m is that there are two different objects accelerating independently: the scale by ω2R and the object by ag , and that, by a one line proof, the second derivative of the distance between them (not an actual acceleration for the purposes of Newton's laws) is g.