Problem $3.$The total hydrostatic force exerted on one face of a square plate $($the length of the side is $L)$ emersed vertically in a bath of an oil $($density ${}_o),$ with one edge on the free surface is given as $F($Figure P$2($a$)).$ Assume no air pressure. Give the answers for the following questions in the unit of $F.$

$($a$)$ If the plate is lowered vertically by a distance $L,$ what will be the total force $F_a$ on one face of the wall?

$($b$)$ The plate is now emersed vertically in a separated water$/$oil bath with one edge on the free surface. The $\frac{1}{3}($in length$)$ of the plate is in water and $\frac{2}{3}$ of the plate is in oil. $($Figure P$2($b$))$ What is the total force $F_h$ on one face of the wall? The density of water $({}_w)$ is the half of the oil.