Archimedes' Principle

Archimedes’ principle defines a buoyant force inside fluids, and describes how bodies can float in a fluid.

Definition §

Formula §

Buoyant Force

$${\vec{F}}_b=-\rho \vec{g}V$$

Terms §

• $F_b$ = Buoyant force (in $\text{N}$)
• $\rho$ = Density of fluid (in ${\text{kg/m}}^3$)
• $\vec{g}$ = Acceleration due to Gravity (in ${\text{m/s}}^2$ )
• $V$ = Volume of fluid displaced (in ${\text{m}}^3$)

Implications §

Since the buoyant force is directly proportional to the density of the fluid, an object immersed in a fluid will have a Gravitational Force acting based on it’s density: ${\vec{F}}_w={\rho }_{obj}V\vec{g}$

Therefore the balance of forces depends on the density of the object, compared to the density of the fluid. For the object to float.

$$\Vert {\vec{F}}_b\Vert =\Vert {\vec{F}}_w\Vert$$

$${\rho }_{fluid}\cdot V_{displaced}\cdot \cancel{g}={\rho }_{object}\cdot V_{object}\cdot \cancel{g}$$

$${\rho }_{fluid}\cdot V_{disp}={\rho }_{obj}\cdot V_{obj}$$

$$\frac{{\rho }_{obj}}{{\rho }_{fluid}}=\frac{V_{fluiddisplaced}}{V_{object}}$$

900

An object sinks in a fluid if it’s mass is greater than that of an equivalent volume of fluid. More generally:

$$\{\begin{array}{ll}{\rho }_{object}<{\rho }_{fluid},& \text{Object floats to the top}\\ {\rho }_{object}={\rho }_{fluid}& \text{Neutral buoyancy, object stays at it’s height}\\ {\rho }_{object}>{\rho }_{fluid}& \text{Object sinks}\end{array}$$

Apparent Weight §

Since the buoyant force opposes gravity, it results in a reduction of the apparent weight:

$$\text{Apparent Weight}={\vec{F}}_w-{\vec{F}}_b$$

Derivation §

Let’s go back to our example for Hydrostatic Pressure:

Hydrostatic Pressure _2.excalidraw

The diagram was actually incomplete. Why? Because we defined the fluid to be at rest, and, but you can see that there is an imbalance of forces on the (hypothetical) cross sectional area.

What actually happens, is that we now have a buoyant force that arises in order to keep the fluid at rest.

Archimedes’ Principle .excalidraw

#todo explain semifloating and a floating object.