Theory

Theory 1

The pressure in a liquid is a function of the depth alone. This is a fundamental fact about liquids.

Pressure function

The fluid pressure in a liquid is a simple multiple of depth $h$:

$$p(h)=\rho gh$$

Constants:

• $\rho =\text{fluid density}$
• $g=\text{gravity constant}$

In SI units:

• $\rho =1000\mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
• $g=9.8\mathrm{m}/{\mathrm{s}}^2$

The pressure of a fluid acts upon any surface in the fluid by exerting a force perpendicular to the surface. Force is pressure times area. If the pressure varies across the surface, the total force must be calculated using an integral to add up differing contributions of force on each portion of the surface.

Fluid force on submerged plate

Total fluid force on plate:

$$F=\rho g{\int }_a^{b}h(x)w(x)dx$$

• $h(x)=\text{depth of horizontal slice}$
• $w(x)=\text{width of the slice}$
• $a,b=\text{vertical limits of surface}$

Use $x=a$ for top of plate (shallow edge) and $x=b$ for bottom of plate (deep edge).

Use $h(x)=x$ when $x=0$ at the water line, and $x$ increases with depth.

(Other $h(x)$ are possible. )

Vertical plate

This formula assumes the plate is oriented straight vertically, not slanting.

(Add the factor $\csc \theta$ for a plate tilted by angle $\theta$.)

Theory 2

What if the submerged surface is not oriented straight vertically?

The amount of surface for a horizontal strip at a given depth will be increased by a factor of $\csc \theta$ where $\theta$ is the angle of incline of the surface (with $\theta =0$ corresponding to horizontal and $\theta =\pi /2$ to vertical). Thus:

$$dz=\csc \theta dx$$

where $dz$ is the thickness of a strip.

So the total force formula becomes:

Fluid force for tilted surface

Total fluid force on tilted plate:

$$F=\rho g{\int }_a^{b}hwdz=\rho g{\int }_a^{b}h(x)w(x)\csc \theta dx$$

As before, $x$ measures depth with $x=0$ at the surface.