  We study the nonequilibrium steady state of a mechanical resonator in the
quantum regime realized by a suspended carbon nanotube quantum dot contacted by
two ferromagnets. Because of the spin-orbit interaction and/or an external
magnetic field gradient, the spin on the dot couples directly to the flexural
eigenmodes. Accordingly, the nanomechanical motion induces inelastic spin flips
of the tunneling electrons. A spin-polarized current at finite bias voltage
causes either heating or active cooling of the mechanical modes. We show that
maximal cooling is achieved at resonant transport when the energy splitting
between two dot levels of opposite spin equals the vibrational frequency. Even
for weak electron-resonator coupling and moderate polarizations we can achieve
ground-state cooling with a temperature of the leads, for instance, of
$T=10\omega$.
