What is the total height of a system consisting of 4 spheres of radius $x$ on top of which is a sphere of radius $y$? (Such that the top sphere is in the 'crevice' of the others, as seen when packing spheres.)
I know a solution that works for spheres of the same radius, but not one where the radii are different.
Is there a generalised equation that works for varying numbers of spheres? For example, where the 9 bottom spheres have radius $x$, the 4 middle spheres have radius $y$, and the top sphere has radius $z$.
Let the bottom four spheres have radius $R_1$ , and the the top sphere have radius $R_2$, then using the distance formula between the center of one the four spheres and the center of the top sphere we can find the position of this top sphere.
Let the center of one of the bottom spheres be $( R_1, R_1, R_1 )$ and the top sphere $ (0, 0, z_2 )$ , then expressing the distance between the two centers as the sum of the two radii, we end up with
$ R_1^2 + R_1^2 + (z_2 - R_1)^2 = (R_1 + R_2)^2 $
Hence,
$ z_2 = R_1 + \sqrt{ (R_1 + R_2)^2 - 2 R_1^2 } $
For the three-layer set up with $9$ balls at the bottom layer, then $4$ in the second layer, then $1$ at the top, you will have similar to the above
$ z_2 = R_1 + \sqrt{ (R_1 + R_2)^2 - 2 R_1^2 } $
and
$ z_3 = z_2 + \sqrt{ (R_2 + R_3)^2 - 2 R_2^2 } $