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# Force Due to Hydrostatic Pressure4

## Force Due to Hydrostatic Pressure

(OP)
Hello all

I was hoping someone could clarify the difference between the equations both of which tells me the force due to water

The first equation, I think was to calculate the total force acting on a wall caused by a fluid in this case water:-

Where:-
p = density
g = gravity
w = width of wall
Y = height of the water on the wall

The second equation I have been told tells me the force acting on a submerged object:-

Where
F = Force
p = density
g = gravity
yc = Distance from water surface to the centroid of the submerged object

I am really struggling to understand the difference between the two equations and I was hoping you could or anyone else could shed some light - to me they should be the same.

Thanks

### RE: Force Due to Hydrostatic Pressure

I really hope you are a student.

OK, the common terms between both of your relations are rho (the 'p'-like symbol, greek 'r'), g and y.

rho x g = unit weight of water ~62.4 lb/ft3 (on earth); remember that rho = density and has units of mass per unit volume. Mass times acceleration, in this case 'g', = force (weight).

The water (hydrostatic) pressure on a wall vs. submerged depth = 62.4 x y, y = submerged depth. This is a linearly increasing distribution.

To get force per unit length of wall, integrate 62.4 x y with respect to y => 62.4 x (y^2) / 2
To get force per length of wall w, multiply the above result by w and you get your first equation.

For your second equation, assume the submerged object is relatively small compared to the distance below the water surface. Also assume a unit area for A.

The hydrostatic pressure acting at a depth of yc (or just y) is ... 62.4 x y per unit area.

### RE: Force Due to Hydrostatic Pressure

The first equation is for a wall the reaches above the surface of the water.

The second equation is for any area below the surface of the water. Do no try to memorize these specific equations, you can easily derive them yourself if you understand how hydrostatic pressure creates a force. The fact that you asked your question shows that you do not yet understand the basic principle.

### RE: Force Due to Hydrostatic Pressure

In both cases, ρ * g * y = pressure at depth y.

The first equation integrates pressure from the bottom of the wall to the top of the water, hence y^2/2, which you should recognize, assuming you took calculus. The limits of the integration are 0 to y, and the resultant is the total force on the wall, since the incremental area was w * dy

The second equation assumes that the height of the object is small, relative to the depth of the water, so no integration is done, and the pressure is assumed to be uniform across the object and is simply multiplied by the object's area to get the total force on the object.

TTFN (ta ta for now)
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### RE: Force Due to Hydrostatic Pressure

The second equation uses the water depth at the centroid of the area. The "integration" is done in calculating the centroid. It is not necessary to bring calculus into this, but you can if you want to.

### RE: Force Due to Hydrostatic Pressure

Matthew,

A square cup full of water, equation 1 calculates the horizontal force acting on each of the wall; equation 2 calculates the weight of the water above the base, also the force you need to hold the cup.

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