# Buoyancy

In science, buoyancy (pronunciation: /ˈbɔɪ.ənsi/ or /ˈbjənsi/; also known as upthrust) is an upward force exerted by a fluid that opposes the weight of an immersed object. In a column of fluid, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bottom of a column of fluid is greater than at the top of the column. Similarly, the pressure at the bottom of an object submerged in a fluid is greater than at the top of the object. This pressure difference results in a net upwards force on the object. The magnitude of that force exerted is proportional to that pressure difference, and (as explained by Archimedes' principle) is equivalent to the weight of the fluid that would otherwise occupy the volume of the object, i.e. the displaced fluid.

For this reason, an object whose density is greater than that of the fluid in which it is submerged tends to sink. If the object is either less dense than the liquid or is shaped appropriately (as in a boat), the force can keep the object afloat. This can occur only in a reference frame which either has a gravitational field or is accelerating due to a force other than gravity defining a "downward" direction (that is, a non-inertial reference frame). In a situation of fluid statics, the net upward buoyancy force is equal to the magnitude of the weight of fluid displaced by the body.

${\displaystyle {\text{apparent immersed weight}}={\text{weight}}-{\text{weight of displaced fluid}}\,}$
${\displaystyle {\frac {\text{density}}{\text{density of fluid}}}={\frac {\text{weight}}{\text{weight of displaced fluid}}},\,}$
${\displaystyle {\frac {\text{density of object}}{\text{density of fluid}}}={\frac {\text{weight}}{{\text{weight}}-{\text{apparent immersed weight}}}}\,}$
${\displaystyle \mathbf {f} +\operatorname {div} \,\sigma =0}$
${\displaystyle \sigma _{ij}=-p\delta _{ij}.\,}$
${\displaystyle \mathbf {f} =\nabla p.\,}$
${\displaystyle \mathbf {f} =-\nabla \Phi .\,}$
${\displaystyle \nabla (p+\Phi )=0\Longrightarrow p+\Phi ={\text{constant}}.\,}$
${\displaystyle p=\rho _{f}gz.\,}$
${\displaystyle \mathbf {B} =\oint \sigma \,d\mathbf {A} .}$
${\displaystyle \mathbf {B} =\int \operatorname {div} \sigma \,dV=-\int \mathbf {f} \,dV=-\rho _{f}\mathbf {g} \int \,dV=-\rho _{f}\mathbf {g} V}$
${\displaystyle B=\rho _{f}V_{\text{disp}}\,g,\,}$
${\displaystyle B=\rho _{f}Vg.\,}$
${\displaystyle F_{\text{net}}=0=mg-\rho _{f}V_{\text{disp}}g\,}$
${\displaystyle mg=\rho _{f}V_{\text{disp}}g,\,}$
${\displaystyle m=\rho _{f}V_{\text{disp}}.\,}$
(Note: If the fluid in question is seawater, it will not have the same density (ρ) at every location. For this reason, a ship may display a Plimsoll line.)
${\displaystyle T=\rho _{f}Vg-mg.\,}$
${\displaystyle N=mg-\rho _{f}Vg.\,}$
'Buoyancy force = weight of object in empty space − weight of object immersed in fluid'
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