Equivalence Principle

Galileo famously demonstrated, or at any rate talked about demonstrating the equivalence principle by dropping heavy and light balls off the leaning tower of Pisa. Absent friction, the balls should fall at the same rate and reach the ground at the same time. WB has recently posted a puzzle on this subject. Since I found the puzzle hard, mainly because of what I considered miscelleaneous complications, and because Wolfgang didn't like my solution, I am posting my own more idealized, and hence slightly different, version here.

Consider a spherically symmetric matroyoshka Earth, with a very small inner sphere at its center, and a still smaller sphere at the center of it. The total mass of the Earth is M, the mass of the middle spherical shell (not counting the innermost sphere) is m, and the mass of the innermost sphere by itself is m1. Consider the following two experiments: (1) The spherical shell of mass m is separated from the Earth and raised to a distance between its center, and the center of the Earth, still occupied by the innermost sphere, is R. It is then dropped from rest, and the time until contact recorded. (2) The spherical shell, including its inner sphere (mass m + m1) is similarly raised to the same distance of separation and dropped, and its time recorded.

Note that the spheres dropped differ in mass, but not in diameter.

Will the two spheres have exactly the same time of fall? Assume no friction or external forces. For hints and answers consult WB's post and attached comments.