So, we must add an extra force,
the buoyant force, to the formula which we obtained
previously.
Assuming that the total body of volume
V is submerged in the fluid
and that m=Vρs
we then get
ma
=
-mg + FArchimedes +
C1rv +
C2r2v2
ma
=
-Vρsg +
Vρfluidg +
C1rv +
C2r2v2
ma
=
-mg(ρs-ρfluid)/ρs +
C1rv +
C2r2v2
We seem to may account for the buoyant force by substituting
in the old expression g
by g(ρs-ρfluid)/ρs.
For solid objects with densities
equal or larger
than the density of water ρwater=1.0×103 kg/m3,
we are dealing with errors of one promille or less
if we do not take into account the density of air of about
1 kg/m3.
However, for solid metal spheres in water
the error would be some ten percent or more, hence non-negligible.