Problem $5(40$ points$)$

Objective: The objective of this experiment is to determine the hydrostatic force $($and its location, also called center of pressure$)$ acting on a plane surface immersed in water when the surface is partially or fully submerged.

Apparatus: The apparatus $($shown below$)$ is a water vessel designed as a ring segment $($quarter toroid$).$ The top and bottom faces are concentric circular arcs centered on the pivot. The radii of the external and internal arcs are $200$ mm and $100$ mm $,$ respectively and the width of the vessel is $76\mathrm{.}2$ mm $.$

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I. vveignts

$4.$ Stop pin

$8.$ Handles

Theory: When the quadrant is filled with water to a certain height, $s,$ it is possible to analyze the forces acting on the surfaces of the quadrant as: $1)$ the hydrostatic force at any point on the curved surfaces is normal to the surface and therefore passes through the pivot point because the pivot point is located at the origin of the radii. Hydrostatic forces on the upper and lower curved surfaces therefore have no net torque effect; $2)$ the forces on the sides of the quadrant are equal and opposite horizontal forces; $3)$ the hydrostatic force on the vertical submerged face is counteracted by the balance weight $(F_G),$ shown as $7$ in the figure above. At equilibrium, the sum of the moments about the pivot point is zero.

What you have to do:

Calculate the magnitude $(F_p)$ and location of center of pressure $(l_d)$ of the hydrostatic force acting on the vertical plane surface $($see below$)$ for the following two cases $($a$)$ and $($b$)$:

Calculate the moment due to the appended weight for each case $(F_G^{**}l)$ and compare it to the moment due to the hydrostatic force calculated above $(F_p^{**}{l}_d).$ Briefly comment on the discrepancy $($if any$).$

$($D$)S>10U$

Water in the quadrant is filled up to a height s $($Case$-$a: $s100mm$ and Case$-$b: $s>100$ mm $).$ Note the pressure distribution profile for each case. Points C and D are the center of gravity and center of pressure respectively for the submerged plane area.

Use the following table to summarize your answers for the $3$ cases provided. Please show your work for each calculation.

$\backslash$table$[[\backslash$table$[[$Lever arm,$],[l,[mm]$