Physics and Engineering: Fluid Pressure and Force

Pressure ($p$) is force per unit area applied in a direction perpendicular to the surface of an object. While pressure may be measured in any unit of force divided by any unit of area, the SI unit of pressure (the newton per square meter) is called the pascal (Pa).

Fluid Pressure and Force

Pressure as exerted by particle collisions inside a closed container.

Mathematically, $p = \frac{F}{A}$, where $p$ is the pressure, $\mathbf{F}$ is the normal force, and $A$ is the area of the surface on contact. 

Pressure is a scalar quantity. It relates the vector surface element (a vector normal to the surface) with the normal force acting on it. The pressure is the scalar proportionality constant that relates the two normal vectors: 

$d\mathbf{F}_n=-p\,d\mathbf{A} = -p\,\mathbf{n}\,dA$

The subtraction (–) sign comes from the fact that the force is considered towards the surface element while the normal vector points outward. The total force normal to the contact surface would be:

 $\displaystyle{\mathbf{F}_n = \int d\mathbf{F}_n=- \int p\,d\mathbf{A} = - \int p\,\mathbf{n}\,dA}$

Pressure is an important quantity in the studies of fluid (for example, in weather forecast). For fluids near the surface of the earth, the formula may be written as $p = \rho g h$, where $p$ is the pressure, $\rho$ is the density of the fluid, $g$ is the gravitational acceleration, and $h$ is the depth of the liquid in meters. Using this expression, we can calculate the total force that the fluid pressure gives rise to:

$\mathbf{F_n} = -(\int \rho g h \, dA) \, \mathbf{n}$

This equation, for example, can be used to calculate the total force on a submarine submerged in the sea.