  We describe a simple mechanical system that operates as a ponderomotive
particle trap, consisting of a circular hoop and a frictionless bead, with the
hoop rotating about a horizontal axis lying in the plane of the hoop. The bead
in the frame of the hoop is thus exposed to an effective sinusoidally-varying
gravitational field. This field's component along the hoop is a zero at the top
and bottom. In the same frame, the bead experiences a time-independent
centrifugal force that is zero at the top and bottom as well. The system is
analyzed in the ideal case of small displacements from the minimum, and the
motion of the particle is shown to satisfy the Mathieu equation. In the
particular case that the axis of rotation is tangential to the hoop, the system
is an exact analog for the rf Paul ion trap. Various complicating factors such
as anharmonic terms, friction and noise are considered. A working model of the
proposed system has been constructed, using a ball-bearing rolling in a tube
along the outside of a section of a bicycle rim. The apparatus demonstrates in
detail the operation of an rf Paul trap by reproducing the dynamics of trapped
atomic ions and illustrating the manner in which the electric potential varies
with time.
