Question $3(25$ points$)[$Problem $2\mathrm{.}73$ textbook$]$

A piston having a cross$-$sectional area $A$ is located in a cyclider containing water $({}_w)$ as shown in the

figure. An open U$-$tube manometer containing mercury $({}_{Hg})$ is connected to the cylider as shown.

The weight of the piston is neglegible.

a$)$ Derive an equation to determine the force, $F,$ acting on the piston. Give your answer as a function

of the given variables. $($Points $20)$

b$)$ For $A=0\mathrm{.}07m^2,h_l=60mm,$ and $h=100mm,$ what is the value of the applied force? $($Points $5)$

For part $($b$)$ the magnitude of the specific weights are:

Specific weight of water ${}_w=9,810\frac{N}{m^3}$

Specific weight of mercury ${}_{Hg}=133,000\frac{N}{m^3}$