# Calculatinf Force Due To Hydrostatic Pressure

#### jot_dain

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

#### benit13

Consider a rock thrown into a water tank. I drew a simple diagram of this situation and labelled some of the forces that are acting:

This isn't *all* of the forces, but the forces indicated are the ones which relate to your formulae.

If we consider the arrow on the rock first, this force is the weight of water pushing on the top of the rock. Water, like other masses, has weight. This weight increases the deeper you go into the water because there's more water above you pushing down on you. This is your second formula.

Now lets' consider the forces on the walls. Gravity is pulling down on all of the water in the tank and, like any other collection of masses, wants to spread out as much as possible. Consequently, the water is constantly pushing on the walls of the tank. The force pushing on the side walls of the tank is your first formula.

#### jot_dain

Consider a rock thrown into a water tank. I drew a simple diagram of this situation and labelled some of the forces that are acting:

View attachment 2897

This isn't *all* of the forces, but the forces indicated are the ones which relate to your formulae.

If we consider the arrow on the rock first, this force is the weight of water pushing on the top of the rock. Water, like other masses, has weight. This weight increases the deeper you go into the water because there's more water above you pushing down on you. This is your second formula.

Now lets' consider the forces on the walls. Gravity is pulling down on all of the water in the tank and, like any other collection of masses, wants to spread out as much as possible. Consequently, the water is constantly pushing on the walls of the tank. The force pushing on the side walls of the tank is your first formula.

Thank you benit13

#### jot_dain

Thank you benit13

Hi Benit13

After some help and more web searches, I have just realized that the two formulas are the same:-

Force = p * g * w * Y^2/2
Force = p * g * w * Y * Y * 1/2
(w * Y = Area)
Force = p * g * Area * Y /2
Y/2 = is the location of the centriod a rectangular section thus given the name Yc. Yc can vary depending on the submerged shape.
Force = p * g * Area * Yc

#### benit13

Right... I misunderstood what your second formula meant.

In general, for a submerged object, the pressure as a function of height is:

$$\displaystyle p(z) = \rho g z$$

Exploiting symmetry in the horizontal plane, we integrate over height only:

$$\displaystyle F = w \int_{0}^{h} \rho g z dz$$
$$\displaystyle = \rho g w \left[\frac{z^2}{2}\right]_{0}^{h}$$
$$\displaystyle = \rho g w \frac{h^2}{2}$$

where w is the width of the tank. Integration could be important if the tank doesn't have convenient dimensions (such as an unusual wall shape or angle).