  In this Letter, we study the motion and wake-patterns of freely rising and
falling cylinders in quiescent fluid. We show that the amplitude of oscillation
and the overall system-dynamics are intricately linked to two parameters: the
particle's mass-density relative to the fluid $m^* \equiv \rho_p/\rho_f$ and
its relative moment-of-inertia $I^* \equiv {I}_p/{I}_f$. This supersedes the
current understanding that a critical mass density ($m^*\approx$ 0.54) alone
triggers the sudden onset of vigorous vibrations. Using over 144 combinations
of ${m}^*$ and $I^*$, we comprehensively map out the parameter space covering
very heavy ($m^* > 10$) to very buoyant ($m^* < 0.1$) particles. The entire
data collapses into two scaling regimes demarcated by a transitional Strouhal
number, $St_t \approx 0.17$. $St_t$ separates a mass-dominated regime from a
regime dominated by the particle's moment of inertia. A shift from one regime
to the other also marks a gradual transition in the wake-shedding pattern: from
the classical $2S$~(2-Single) vortex mode to a $2P$~(2-Pairs) vortex mode.
Thus, auto-rotation can have a significant influence on the trajectories and
wakes of freely rising isotropic bodies.
