It depends on a few things. If the cylinder is rotating around a horizontal axis (like a baton), then the water certainly contributes to the moment of inertia.
If we assume that the water is an "inviscid" fluid, meaning that it flows perfectly smoothly through the object with no friction, then in the case of a sphere, when the sphere begins to rotate, the water will remain perfectly stationary and will not contribute to the moment of inertia of the sphere. It is similarly true if the cylinder is spinning on a vertical axis.
Hope that helps!
For a real viscous fluid, like water, the fluid will start to rotate gradually due to friction with the cylinder walls. So in the long run it does contribute to the moment of inertia. However, if you're turning the cylinder back and forth quickly, the water will hardly respond and thus will contribute little. What's the time scale to cross over from one type of behavior to another? It's approximately the square of the cylinder radius divided by the kinematic viscosity of the water. which is about 10-2 cm2/s at room temperature. Mike W.
(published on 03/29/2011)