Hydrostatic Equilibrium

General Info

What

Hydrostatic equilibrium is a condition that, if held, guarantees that pressure is sufficient to counteract the force by gravity

Definition

$\frac {\textrm{d}P}{\textrm{d}r} = - \frac {GM_r\rho}{r^2} = -\rho g$

Non Differential Solution

Note that $\frac {\textrm{d}P}{\textrm{d}r} \sim \frac {P_2 - P_1}{R_2 - R_1}$

Pressure as a Function of Depth

Assuming that density is constant

$\frac {\textrm{d}P}{\textrm{d}r} = -\rho g$

$\frac {\Delta P}{\Delta r} = -\rho g$

$\frac {P_z - 0}{0 - z} = - \rho g$

$P_z = \rho g z$

Archimedes Principle

This principle states that the mass of water displaced is equal to the mass of the floating object. Assuming that the density of water is constant with respect to depth:

$P(h) = \rho _{water}gh = \frac {m_{obj}g}{A}$

$\rho _{water} hA = m_{obj}$

$m_{water-disp} = m_{obj}$

Using Archimedes Principle to Find Submerged Height

Suppose we want to find how far a block of wood with density $\rho _{wood}$ is submerged in water with density $\rho_{water}$.

$\textrm{mass~wood} = \textrm{mass~water}$

$\rho_{wood} A h = \rho _{water} A d$

$d = \frac {\rho _{wood}}{\rho _{water}} h$