Hydrostatics: Buoyancy & Stability

Hydrostatics: Buoyancy & Stability

Archimedes' principle, buoyant force calculations, and stability criteria for floating and submerged bodies.

Concept Overview

Buoyancy is the upward force exerted by a fluid that opposes the weight of an immersed object.

Archimedes' Principle

Archimedes' Principle

A body immersed in a fluid experiences a vertical upward buoyant force equal to the weight of the fluid it displaces.

Archimedes' Principle

Archimedes' principle, buoyant force calculations, and stability criteria for floating and submerged bodies.

F_B = \gamma V_{displaced} = \rho_{fluid} g V_{displaced}

Floating Body: Weight of body = Buoyant Force ( $W = F_B$ )
Submerged Body: Apparent Weight = True Weight - Buoyant Force ( $W_{app} = W - F_B$ )

Buoyancy Simulation: Adjust the density of the object and the fluid to see if it floats or sinks. Notice how the submerged volume changes.

Buoyancy & Stability Simulator

Fluid Density ($\rho_f$): 1000 kg/m$^3$

Object Density ($\rho_o$): 600 kg/m$^3$

Weight ($W$):0.00 kN

Buoyant Force ($F_B$):0.00 kN

Status:FLOATING

Submerged:0.0%

Fluid Surface

Object

SG=0.60

F_B

Stability of Floating Bodies

Concept Overview

Stability refers to the ability of a body to return to its original position after a small disturbance (tilt).

Metacenter ( $M$ )

The point of intersection between the vertical line through the center of buoyancy (

B

) in the upright position and the vertical line through the new center of buoyancy (

B'

) after a small angle of tilt.

Metacentric Height ( $GM$ )

The distance between the Center of Gravity (

G

) and the Metacenter (

M

). It is a key measure of stability.

Stability Criteria:

Stable Equilibrium: $M$ is above $G$ ( $GM > 0$ ). The body returns to upright.
Unstable Equilibrium: $M$ is below $G$ ( $GM < 0$ ). The body overturns.
Neutral Equilibrium: $M$ coincides with $G$ ( $GM = 0$ ).

Calculating $GM$ :

Metacentric Height ($GM$)

<strong>Buoyancy Simulation:</strong> Adjust the density of the object and the fluid to see if it floats or sinks. Notice how the submerged volume ...

GM = MB \pm GB

Use $+$ if $M$ is above $G$ , $-$ if below. Typically, we find $MB$ and compare locations.

Righting Moment: When a stable body is tilted by a small angle

\theta

, the buoyant force and weight create a restoring couple (Righting Moment).

M_R = W \cdot GM \sin\theta

Distance $MB$ :

MB = \frac{I}{V_{sub}}

Variables

Symbol	Description	Unit
$I$	Moment of inertia of the waterline area about the tilt axis.	-
$V_{sub}$	Volume of the submerged portion of the body.	-

Period of Oscillation

The time taken for a floating body to complete one full roll.

Rolling of Floating Bodies

When a stable floating body is disturbed, it will oscillate (roll) around its metacentric axis. The time period of this oscillation is given by:

Rolling of Floating Bodies

The time taken for a floating body to complete one full roll.

T = 2\pi \sqrt{\frac{k^2}{g \cdot GM}}

Variables

Symbol	Description	Unit
$T$	Time period of oscillation (seconds).	-
$k$	Radius of gyration of the body about its longitudinal axis.	-
$GM$	Metacentric height.	-
$g$	Acceleration due to gravity.	-

This shows that a larger

GM

(more stable) results in a shorter period of oscillation, meaning the ship snaps back quickly (which can be uncomfortable for passengers). A smaller

GM

gives a longer, more comfortable roll, but with less stability.

Stability of Submerged Bodies

Concept Overview

For fully submerged bodies (like submarines or balloons), the Center of Buoyancy (

B

) is fixed at the centroid of the displaced volume.

Stable: Center of Gravity ( $G$ ) is below Center of Buoyancy ( $B$ ).
Unstable: $G$ is above $B$ .

Key Takeaways

Floating Condition: A body floats if its average density is less than the fluid density. $W = F_B$ .
Period of Oscillation: The time period $T$ of rolling is inversely proportional to the square root of $GM$ .
Stability: Depends on the relative positions of $G$ , $B$ , and $M$ .
Metacenter ( $M$ ): Must be above $G$ for stability ( $GM > 0$ ).
Righting Moment: $M_{righting} = W \cdot GM \sin\theta$ .