The grand-design spiral galaxy M~51 was observed at 40pc resolution in CO(1--0) by the PAWS project. A large number of molecular clouds were identified and we search for velocity gradients in two high signal-to-noise subsamples, containing 682 and 376 clouds. The velocity gradients are found to be systematically prograde oriented, as was previously found for the rather flocculent spiral M~33. This strongly supports the idea that the velocity gradients reflect cloud rotation, rather than more random dynamical forces, such as turbulence. Not only are the gradients prograde, but their $frac{partial v}{partial x}$ and $frac{partial v}{partial y}$ coefficients follow galactic shear in sign, although with a lower amplitude. No link is found between the orientation of the gradient and the orientation of the cloud. The values of the cloud angular momenta appear to be an extension of the values noted for galactic clouds despite the orders of magnitude difference in cloud mass. Roughly 30% of the clouds show retrograde velocity gradients. For a strictly rising rotation curve, as in M~51, gravitational contraction would be expected to yield strictly prograde rotators within an axisymmetric potential. In M~51, the fraction of retrograde rotators is found to be higher in the spiral arms than in the disk as a whole. Along the leading edge of the spiral arms, a majority of the clouds are retrograde rotators. While this work should be continued on other nearby galaxies, the M~33 and M~51 studies have shown that clouds rotate and that they rotate mostly prograde, although the amplitudes are not such that rotational energy is a significant support mechanism against gravitation. In this work, we show that retrograde rotation is linked to the presence of a spiral gravitational potential.
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