Forces on surfaces. In a chemical processing plant, a large storage tank $($shown below$)$ stores a

hazardous liquid $($density $).$ The upper corner of the tank has corroded, so it is removed and

covered with a new panel shaped like an isosceles right triangle $($side length $$ as shown$).$ The panel

is screwed on with three bolts $($assume the bolts are exactly at the corners of the triangle$).$ The top

of the tank is open to the atmosphere and triangular plate$$s top edge is level with the liquid surface.

a$.$ What is the net force, $$ on the triangular plate in terms of $,,$ and $($acceleration of

gravity$)?$

b$.$ Using the coordinate system shown $($ positive left and

$$ positive down, origin at the upper right corner of the

plate$),$ where does this force act $($what is the center of

pressure, CoP$)?$ Your answer should be in terms of $.$

Hint: pressure does not vary horizontally in a static fluid, so the

x$-$coordinate of the CoP is the same as the x$-$coordinate of the

triangle$$s centroid.

c$.$ What are the three forces the three bolts must exert ON

the plate to keep it attached, in terms of $,,$ and $?$

Let$$s call them $($left bolt$),($right bolt$),$ and $($bottom bolt$).$ Hint $1$: the bolts must exert

a total force on the plate that is equal and opposite to $.$ Hint $2$: the bolts must also exert moments on

the plate that are equal and opposite to the moment being applied by $,$ no matter what axis you are

taking the moments around! A few moment calculations about different axes will give you the tools to solve

this problem.