Problem $3(20$ points$)$: Rectangular gate $AB$ is hinged at the free surface between fluid X and air, as shown in the figure below. The gate is $2$ m high and $3$ m wide. A block of wood floats such that $70\%$ of its volume is in fluid X $.$ The block of wood has a volume of $0\mathrm{.}15m^3$ and the specific weight of the wood, ${}_{\text{w}\text{o}\text{o}\text{d}},$ is $5900\frac{N}{m^3}.$

a$.$ What is the specific weight, ${}_x,$ of fluid X $?$

b$.$ Find the necessary force, $F,$ applied at the bottom of the gate just sufficient to keep it closed.