A particle of mass m is attached to a rigid support by a spring with force constant k. At equilibrium, the spring hangs vertically downward. To this mass-spring combination is attached an identical oscillator, the spring of the latter being connected to the mass of the former. Calculate the characteristic frequencies for one-dimensional vertical oscillations, and compare with the frequencies when one or the other of the particle is held fixed while the other oscillates. Describe the normal modes of motion for the system.