  Prandtl-Tomlinson (PT) model has been very successful in explaining nanoscale
friction in a variety of situations. However, the simplistic PT model, on
account of having a point mass being dragged across a sinusoidal force field,
cannot be used for studying rolling friction at nanoscales. In this manuscript,
we generalize the PT model as a collection of point particles arranged in a
circle of radius $R$. The resulting ``rigid body'' is driven in a composite
force field by a moving spring (of stiffness $k$) connected to the center of
mass of the rigid body in presence of damping. The force field is a product of
the familiar sinusoidal function used in the PT model with a parametrically
controlled ($\lambda$) exponentially varying function that is dependent on the
vertical coordinates of the particles. Our generalized model degenerates to the
standard PT model if $R \ll 1$ and $\lambda \to 0$. With $R \sim 1$ and
$\lambda \to 0$, the model undergoes a transition from sticky dynamics to
smooth dynamics as $k$ is increased to a critical value. The analytical
expression agrees well with the simulation results. Similar analytical
expressions have been derived for $ \lambda \neq 0$ as well. In this scenario,
the sticky dynamics is experienced in both $x$ and $y$ directions, and our
numerical results agree with the analytical solution for $x$ direction. The
dynamics, investigated numerically for the general case of $R \sim 1$ and
$\lambda \neq 0$, reveals several interesting aspects of nanoscale tribology
including the regimes where energy dissipation due to friction is minimum.
Further the results from our proposed model are in qualitative agreement with
those from MD simulations as well. We believe that the simplicity of our model
along with its similarity to the PT model may make it a popular tool for
analyzing complicated nanotribological regimes.
