Motion close to the surface of the earth

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on page 3 of the following
http://www.ph.ed.ac.uk/teaching/course-notes/documents/62/764-dr5.pdf
it discusses an example of motion close to the surface of the Earth and then says that the only force felt in S is [itex]m \vec{g}[/itex] and then rearranges for the force in S'.

Surely seeing as we exist in the frame S' i.e. we rotate round the axis as the Earth does and the only force we see is [itex]m \vec{g}[/itex] then we should be able to rearrange and find the force in S in terms of Coriolis andcentrifugal contributions. Clearly I'm going wrong here but I can't see why.

how do we know the force in S? nobody can be in the frame S as it requires us to be inertial wrt the rotation of the Earth?
 

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