tough math question about spheres

It’s a frictionless environment, but gravity is the same as the earth’s. All the spheres described are of the same material and thus have the same specific gravity. The hollow sphere is not open; the solid metal spheres are teleported into it.

There is a hollow metal sphere of diameter x. A solid metal sphere of .618~(golden ratio)x is dropped into the hollow sphere. A second solid metal sphere of diameter .618~[sup]2[/sup] is then dropped into the hollow sphere. Imagine that the drop is not dead on; hence, the second sphere pushes the first solid metal sphere off of plumb, so to speak.

Since we are working with the golden ratio, the sum of the diameters of the two solid spheres are precisely equal to that of the hollow metal sphere.

The question is this: After all the spheres are in place, what is the measurement of the angle formed by A) the line going through the center of the hollow sphere and perpendicular to the plane of the earth (OK, it’s not flat, but) and B) the line going through the center of the first solid sphere and the point where this sphere is tangent to the hollow sphere?

Put simply, how far off plumb does the second solid sphere push the first?

Thanks for your help. I am actually going to use this information.

The volume of the larger solid sphere is 0.9888 x[sup]3[/sup], and that of the smaller is 0.2334 x[sup]3[/sup].

To make things easy, let’s take x = 1 m, and density = 1,000 kgm[sup]-1[/sup]. Their masses are then 988.8 kg and 233.4 kg respectively.

If the small sphere ended up on the bottom, the potential energy of the system would be:

m[sub]1[/sub] g h[sub]1[/sub] + m[sub]2[/sub] g h[sub]2[/sub]

= 988.8 x 9.8 x (0.618/2 + 0.382) + 233.4 x 9.8 x 0.382/2

= 7133 J

If the small sphere ended up on the top, the potential energy of the system would be:

988.8 x 9.8 x 0.618/2 + 233.4 x 9.8 x (0.618 + 0.382/2)

= 4845 J

If they ended up side by side, it’d be:

988.8 x 9.8 x 0.5 + 233.4 x 9.8 x 0.5

= 5989 J

The configuration where the small sphere is on top has the lowest potential energy, so the angle is -90[sup]o[/sup].

(note that you have to have some small amount of friction for the things to come to rest, otherwise they’ll just keep swinging around)

Sigh. Corrected version:

Another engineer just chiming in to say that Des is correct.