  See http://youtu.be/Mf4IE8gWcJs for a YouTube video showing part of the
results in this paper. We consider helicoidal immersions in the Euclidean space
whose axis of symmetry is the z-axis that are solutions of the equation 2
H=\Lambda_0-a 1/2 R^2 where H is the mean curvature of the surface, R is the
distance form the point in the surface to the z-axis and a is a real number. We
refer to these surfaces as helicoidal rotating drops. We prove the existence of
properly immersed solutions that contain the z-axis. We also show the existence
of several families of embedded examples. We describe the set of possible
solutions and we show that most of these solutions are not properly immerse and
are dense in the region bounded by two concentric cylinders. We show that all
properly immersed solutions, besides being invariant under a one parameter
helicoidal group, they are invariant under a cyclic group of rotations of the
variables x and y.
  The second variation of energy for the volume constrained problem with
Dirichlet boundary conditions is also studied.
