A uniform rod of length $\mathrm{L}$ and mass $\mathrm{M}$ has been placed on a rough horizontal surface as shown in the figure. The rod is pulled by applying a horizontal force. The coefficient of friction $\mathrm{\mu }$ between the surface and the rod is given by $\mathrm{\mu }=\left\{\begin{array}{ll}{\mathrm{\mu }}_0\mathrm{x},& 0\le \mathrm{x}\le \mathrm{L}\\ 0,& \mathrm{x}>\mathrm{L}\end{array}\right.$, where ${\mathrm{\mu }}_0$ is a positive constant. The heat generated due to friction as the rod moves by a distance $\mathrm{L}$ is