We show theoretically that it is possible to trap and cool the rotational motion of a macroscopic mirror made of a perfectly reflecting spiral phase element using orbital angular momentum transfer from a Laguerre-Gaussian optical field. This technique offers a promising route to the placement of the rotor in its quantum mechanical ground state in the presence of thermal noise. It also opens up the possibility of simultaneously cooling a vibrational mode of the same mirror. Lastly, the proposed design may serve as a sensitive torsional balance in the quantum regime.PACS numbers: 42.50. Pq, 42.65.Sf, 85.85.+j, 04.80.Nn The optical control of the quantum mechanical centerof-mass motion of ions, neutral atoms, molecules and microscopic-scale objects has formed a dominant theme in atomic, molecular and optical physics in recent years [1,2]. These techniques are now being transferred to the macroscopic regime, where laser light has been found to be effective in cooling and trapping, for example, gramscale mirrors [3]. Such efforts are part of the rapidly emerging field of 'quantum optomechanics', one of the ambitious aims of which is to place a macroscopic object in its quantum mechanical ground state [4,5,6,7]. The accomplishment of this task would open up a host of fascinating scenarios ranging from the fundamental [8,9,10] to the applied [11,12,13], all following from the manifestation of quantum mechanical behavior in classical objects.So far most efforts have been aimed at quantizing linearly vibrating oscillators, which are prototypical mechanical objects [4,5,6]. Further, oscillators with breathing modes have been considered, driven by the centrifugal effects of radiation pressure [7,14,15]. A torsional mode has also been examined experimentally [16], but its effective dynamics was essentially vibrational in that the interaction mechanism was the mirror's exchange of linear momentum with radiation, a fact true of all the examples cited above.In this Letter we point out the possibility of quantizing instead a rotational mode of a classical torsional oscillator using the exchange of angular momentum with a radiation field. Combined with the use of the linear momentum of light, this results in the potential to simultaneously quantize a vibrational as well as a rotational mode of motion of an oscillator using the same radiation field. Also, (torque-induced) rotation of the mirror manifests itself as a change in cavity length, which may be measurable to a sensitivity better than the quantum noise limit, based on recent analyses [17,18]. We therefore expect our proposed design to possibly find use as a sensitive torque sensor, (i.e. a torsion balance) operating in the quantum regime.Specifically we consider the mechanical effect of a Laguerre-Gaussian beam interacting with a macroscopic mirror. In addition to their intrinsic spin angular momentum ≤ , photons in such beams also carry an integral orbital angular momentum l , see [19] and references therein. It has been experimentally demonstrated that Laguerre-Gauss...