$(23$ Marks$)$ A cubical block $($ m $)$ with a side length of $20\mathrm{.}0$ cm is hung from the ceiling via a massless rope. The figure to the right shows the block at rest when $\frac{3}{4}$ of its volume $V$ is submerged in a pool under the surface of a fluid with density $pr=681k\frac{g}{m^3}.$

a$)$ Find the force $($due to the fluid only$)$ that acts on one of the vertical faces of the block.

b$)$ What is the difference in pressure between the bottom and the top surfaces?

c$)$ Show that the buoyant force that acts on the block is $F{b}_b=40\mathrm{.}0N.$

d$)$ The tension in the rope is $T=58\mathrm{.}0N.$ Find the weight then deduce the density ${}_b$ of the block.

e$)$ The rope is now cut. Why does the block sink?

f$)$ Determine the normal force acting on the block when it is at rest at the bottom of the pool

g$)$ We now make the block hollow by removing $60\%$ of its volume $($or its mass$)$ from its center. We then completely fill the hollow part with water $($density $1000k\frac{g}{m^3}).$ Does it float or sink now?