(Solved): Exercise 3 - Dam [12 points] a) Calculate the force exerted on the vertical wall of a dam by a wate ...

Exercise 3 - Dam [12 points] a) Calculate the force exerted on the vertical wall of a dam by a water impoundment, knowing that the height of the water is $H$ and the width of the dam $L$. First calculate the force exerted on an element of width $dz$ at a depth $z$, then integrate over the entire height of the dam. b) A gate $B$ at the lowest point has the shape of a $2m$ square. What is the total force supported by this gate? Data: density of water ${?}_{\text{water }}=10^3\mathrm{kg}/{\mathrm{m}}^3,H=30\mathrm{m},\mathrm{L}=100\mathrm{m},\mathrm{g}=10\mathrm{m}/{\mathrm{s}}^2$ r 1g. 2. vall. Exercise 4 - Pascal's barrel [12 points] Blaise Pascal (1623-1662) had a narrow tube about $10\mathrm{m}$ high attached to the lid of a barrel. This tube communicated in $A$ with the inside of the barrel, the whole being watertight. He filled the barrel with water, and once the barrel was full, he continued the operation by filling the thin tube above it. When the water reached a certain level, the barrel began to leak and exploded. To understand this spectacular phenomenon, let us assume that the lid, with surface area $S=0.25{\mathrm{m}}^2$, is weighted so that its mass is $m=500\mathrm{kg}$. a) At what critical height $h_c$ does the lid begin to lift? b) Knowing that the cross-sectional area of the thin tube is $s=1{\mathrm{cm}}^2$, what is the minimum volume $V$ of water required to move the lid? Exercise 5-Eiffel Tour's pistons [15 points] Consider the device below, where the liquid surfaces $S$ and $s$ support 2 pistons. If a force $F$ is exerted on the large piston, what force $f$ must be exerted on the small piston to maintain equilibrium? The Eiffel Tower has a mass of 8,000 tons and rests on the large pistons of 16 hydraulic presses. The diameter of the pistons is $6.2\mathrm{m}$ for the large ones and $17.3\mathrm{cm}$ for the small ones. What force must be exerted on each small piston to balance the tower? If one leg of the tower sinks by $1\mathrm{mm}$, what new force must be exerted? Use water as the liquid in the presses. Fig. 4. A two-piston system when the pistons are at the same height. Fig. 5. A two-piston system when they are not at the same height.