Hydrostatic Equilibrium

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 $$\rho _{water}$$.

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

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

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