  We analyze a quantum mechanical gyroscope which is modeled as a large spin
and used as a reference against which to measure the angular momenta of
spin-1/2 particles. These measurements induce a back-action on the reference
which is the central focus of our study. We begin by deriving explicit
expressions for the quantum channel representing the back-action. Then, we
analyze the dynamics incurred by the reference when it is used to sequentially
measure particles drawn from a fixed ensemble. We prove that the reference
thermalizes with the measured particles and find that generically, the thermal
state is reached in time which scales linearly with the size of the reference.
This contrasts a recent conclusion of Bartlett et al. that this takes a
quadratic amount of time when the particles are completely unpolarized. We now
understand their result in terms of a simple physical principle based on
symmetries and conservation laws. Finally, we initiate the study of the
non-equilibrium dynamics of the reference. Here we find that a reference in a
coherent state will essentially remain in one when measuring polarized
particles, while rotating itself to ultimately align with the polarization of
the particles.
