We study the motion of the coupled system, S, constituted by a physical pendulum, B, with an interior cavity entirely filled with a viscous, compressible fluid, F. The system is constrained to rotate about a horizontal axis. The presence of the fluid may strongly affect the motion of B. In fact, we prove that, under appropriate assumptions, the fluid acts as a damper, namely, S must eventually reach a rest-state. Such a state is characterized by a suitable time-independent density distribution of F and a corresponding equilibrium position of the center of mass of S. These results are proved in the very general class of weak solutions and do not require any restriction on the initial data, other than having a finite energy. We complement our findings with some numerical tests. The latter show, among other things, the interesting property that “large” compressibility favors the damping effect, since it drastically reduces the time that S takes to go to rest.