Interesting way to measure weight. So the theory is all the different sides of the earth converge to the center and therefore if u have the center weighted all u need to do is then account for gravity. I even forgot about gravity lol. I am still studying and reading on this and cannot find anything that actually clicks or makes sense.
The strange thing is this it will be good if this can be demonstrated with a smaller sphere object or something close to the earth shape. Besides just weighing the center won't tell u much but a accurate figure of the center, there needs to some arimethic applied across the whole area of the earth once you have the center figure. Cause u can measure or something round and small like a ball but u do need to know the area of the ball and apply that figure and spread it across the whole radius of the ball. Well that's what I think, I could be wrong but I am not sure how you can get a center figure and assume it's accurate without taking that figure and applying it across the whole radius of the object.
This short video will help with deriving the formula
This video will explain how you find the mass of earth
Will find one that has lab work in it.