Pressure

Pressure ($P$) is the force exerted per unit area:

$$ \begin{equation}\begin{aligned} P&=\frac{F}{A}\\ \end{aligned}\end{equation} $$

The SI unit is the Pascal($Pa$), equivalent to the pressure exerted by applying a one Newton ($1 N$) force over an area of one square-metre ($m^2$):

$$ \begin{equation}\begin{aligned} P&=\frac{F}{A}\\ &\rightarrow \frac{1N}{1 m^2}\\ &=1 Pa\\ \end{aligned}\end{equation} $$

Pressure in a fluid

In a fluid, pressure increases with depth. The pressure is also directly proportional to the density, $\rho$ of the fluid as well as acceleration due to gravity, $g$. For a closed vessel (one not exposed to the atmosphere), the pressure is given by

$$ \begin{equation}\begin{aligned} P_{fluid}=\rho gh\\ \end{aligned}\end{equation} $$

For an open vessel, we must consider the atmospheric pressure. The total pressure is $$ \begin{equation}\begin{aligned} P_{total}&=P_{fluid}+P_{atmos}\\ P_{total}&=\rho gh + P_{atmos}\\ \end{aligned}\end{equation} $$