  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.
