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General InfoEdit

WhatEdit

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

DefinitionEdit

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

Non Differential SolutionEdit

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

Pressure as a Function of DepthEdit

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 PrincipleEdit

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 HeightEdit

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