(Solved): 3. (a) A very long solid body (S) having a square cross section of side length a floats at an inter ...

3. (a) A very long solid body (S) having a square cross section of side length $a$ floats at an interface with the configuration sketched. Determine $h$ and $?$ at equilibrium in terms of $a$, densities ${?}_{\mathrm{A}}$ and ${?}_{\mathrm{S}}$, and interfacial tension ${?}_{\mathrm{AB}}$. (It is not necessary to solve explicitly for $h$ and $?$ from the final equations you obtain.) Hint: The differential equation of interfacial statics does not need to be solved if one is interested only in finding $h$ and $?$ (and not details of the interfacial profile). One can instead write a horizontal force balance for the entire fluid interface on one side of the solid. (b) A very long solid body (S) having a square cross section floats at an air-liquid interface with the configuration sketched. Please express the volume per unit length in Region 1 in terms of the liquid density ${?}_{\mathrm{A}}$, interfacial tension ${?}_{\mathrm{AB}}$, and angle $?$ only. If you need any more information (say ${}_2{?}_{\mathrm{S}},a,h,\dots$ ) to do this, please specify it.