A solid conducting body of initial temperature $\backslash ($ T$\_\{1,0\}\backslash )$ is immersed suddenly in an amount of incompressible liquid of initial temperature $\backslash (\backslash$boldsymbol$\{$T$\}\_\{2,0\}\backslash ).$ The respective heat capacities of the immersed body and the initial liquid are $\backslash (($m c$)\_\{1\}\backslash )$ and $\backslash (($m c$)\_\{2\}\backslash ),$ where $\backslash ($ m $\backslash )$ and $\backslash ($ c $\backslash )$ denote the respective masses and specific heats of the two entities. The external $($wetted$)$ surface of the immersed body is $\backslash ($ A $\backslash ),$ and the convective heat transfer coefficient between the body and the liquid is $\backslash ($ h $\backslash )($constant$).$ Treating the immersed body and the surrounding liquid as two lumped capacitance systems, derive the expressions for the temperature of the immersed body and its surrounding liquid over time $\backslash (\backslash$left$($T$\_\{1\}($t$)\backslash$right$.\backslash )$ and $\backslash (\backslash$left$.$T$\_\{2\}($t$)\backslash$right$)\backslash ).$

$($Note that what distinguishes this problem form the system treated in class is the finiteness of the liquid pool. Therefore, the lumped temperature of the liquid $\backslash ($ T $2\backslash )$ varies under the influence of the heat transfer from the immersed body$).$