newtonian mechanics - An alternate approach to find variation of gravitational acceleration with respect to depth of an object in Earth

As we go down the Earth, assuming it to be a uniform sphere abiding only to Newtonian physics, the gravitational acceleration imparted to an object by the Earth goes on decreasing, and decreases to 0 m/s^2 at the center.

Usually, we use the shell law to do these calculations. They are made easy and concise, but I would like to have an alternate approach.

I am going to do some really interesting calculations regarding how the acceleration due to gravity changes as we go down the Earth. Suppose we are at a depth 'h' in the Earth. If we choose to split the Earth for calculation purposes into two halves: one below us and the other part above, we can say that the part below would attract us in one direction and the other in the direction opposite to it(because their center of masses lie in one line)

Now, if we figure out the respective attractions between these two sectors of the sphere and the body, we can determine the value of gravitational acceleration by subtracting one force by another and then dividing the resultant force by our mass to obtain the acceleration.

But I have a doubt. While I calculate the force, I take the the two sectors of the spheres to be point bodies at their respective center of masses and then calculate. Can this be done?

I have already used integration to find out the masses of these two sectors, but have yet to find the center of masses.

Can this assumption be made? If it cant, then how do you solve for gravity in objects with complex shapes?

Also, while finding the center of mass, i plan on cutting up the Earth into infinite infinitely thin plates, finding the mass of each plate and then defining the co-ordinates of each plate(with respect to a common origin), then integrate(co-ordinate*mass) with suitable limits.

Will this method yield an accurate result? I used a similar technique to find the masses, and they turned out to be accurate.

Answer

The point mass assumption is only good if you're sufficiently far away such that the angle formed between the center of mass and the farthest extent of the massive body is sufficiently small. (i.e. the level of error of using the approximation sin x = x is considered acceptable).

You can certainly divide the Earth into two pieces, but you'll have to integrate the effect of each piece.