Two solid plates with differing thermal properties are in contact with one another as shown in the figure below. The bottom of plate $1$ is at temperature T$\_$o$,$ and at the top of plate $2$ there is convective heat transfer to the air $($with ambient temperature T $\backslash$infty and heat transfer coefficient h $).$ The horizontal dimensions are large enough that, for positions of interest, T$=$T$($y$)$ only. If one parallel plate slides along the other, the rate of frictional heating at the plane of contact is: H$\_$s$=$c s U where U is the relative velocity, s is the force per unit area normal to the interface, and c is the dimensionless coefficient of dry friction for the materials involved. $/$r$/$n $($d$)$ Assume now that plate $2$ is moved horizontally at speed U$,$ with plate $1$ stationary and the two held in contact only by gravity. Derive expressions for T$\_1($y$)$ and T$\_2($y$)$ as a function of H$\_$s$.$