Derive expression for the interfacial position and density of two immiscible liquids in a centrifugal decanter.$($a$)$ A centrifuge bowl with $321$ mm internal diameter is rotating at a speed of $2000$ rpm$.$ It contains a layer of benzene $260$ mm thick. Calculate the gauge pressure exerted by benzene on the wall of the centrifuge bowl. Assume density of benzene is $832$ kg$/$m$3.($Note: Here, the centrifuge is consisting of only one liquid i$.$e$.$ benzene$)($b$)$ Suggest suitable option$($s$)$ to reduce the gauge pressure obtained in question $5($a$)$ by $50\%$