Here is my theoretical development:

F = -kx

x(t) = Acos(ωt+θ) where ω = √(k/m)

of course, ω is the angular frequency and can be written as 2π/T

so let's isolate m.

m = k*T^2/4π^2

adding this into hooke's law,

g = -kx/(k*T^2/4π^2)

and therefor g = 4π^2*x/T^2

That's a very nice formula until you end up testing it. With a little work, we determined the measurement of g was always off by a factor of 1/T and therefor the formula should be g = 4π^2*x/T^3.

Where does this extra T come from? We've been trying to figure this out for a long time.