Hydrostatics: Buoyancy & Stability

Hydrostatics: Buoyancy & Stability

Learning Objectives

Define buoyancy and apply Archimedes' Principle to floating and submerged bodies.
Determine the metacentric height ( $GM$ ) to evaluate the stability of floating bodies.
Calculate the righting moment and period of oscillation for floating bodies.
Differentiate between stable, unstable, and neutral equilibrium.

Hydrostatics involves the study of fluids at rest. Buoyancy and stability are core concepts governed by Archimedes' Principle, dictating whether bodies float, sink, or remain neutrally suspended. Understanding buoyant force calculations and stability criteria is critical in civil engineering. These principles are applied in the design of pontoons, floating breakwaters, submarines, and determining the stability of large concrete caissons during towing and installation phases in offshore construction.

Concept Overview

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

Archimedes' Principle

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

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

Buoyant Force (Archimedes' Principle)

Calculates the upward buoyant force acting on a submerged or floating body.

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

Variables

Symbol	Description	Unit
$F_B$	Buoyant force.	N
	Specific weight of the fluid.	$N/m^3$
	Mass density of the fluid.	$kg/m^3$
$g$	Acceleration due to gravity.	$m/s^2$
$V_{displaced}$	Volume of the displaced fluid (or submerged volume of the body).	$m^3$

Interactive 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/m³

Object Density (

\\rho_o

): 600 kg/m³

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

W

What this teaches

This explores Archimedes' principle: the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid that the body displaces. It shows the conditions for floating (object density < fluid density) and sinking (object density > fluid density).

Try this

Set the Fluid Density to 1000 kg/m³ (Water). Move the Object Density slider from 500 kg/m³ to 1500 kg/m³. Watch the object submerge further until it sinks.
Find the neutral buoyancy point by setting both Object Density and Fluid Density to exactly 1000 kg/m³. The object should be 100% submerged but not sinking.

Center of Buoyancy ( $B$ )

The centroid of the displaced volume of fluid. It is the point through which the resultant vertical buoyant force acts.

Center of Gravity ( $G$ )

The point through which the entire weight of the body acts.

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.

Stability of Floating Bodies

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

Metacentric Height ( $GM$ )

The distance between the Center of Gravity ( $G$ ) and the Metacenter ( $M$ ) is called the metacentric height ( $GM$ ). It is a key measure of stability.

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 $G$ ( $GM = 0$ ).

Small-Angle Approximation

The formulas for metacentric height ( $GM$ ) and righting moment assume a small angle of tilt (typically less than 10 to 15 degrees). For larger angles, the metacentric radius ( $MB$ ) is not constant, and more complex stability curves are required to determine true stability.

Metacentric Height ( $GM$ )

Determines the metacentric height based on relative positions of the metacenter, center of gravity, and center of buoyancy.

GM = MB \pm GB

Variables

Symbol	Description	Unit
$GM$	Metacentric height.	m
$MB$	Distance from the center of buoyancy to the metacenter.	m
$GB$	Distance between the center of gravity and the center of buoyancy.	m

Sign Convention for Metacentric Height

Typically, the sign in $GM = MB \pm GB$ depends on whether the center of gravity ( $G$ ) lies above or below the center of buoyancy ( $B$ ). Specifically, $GM = MB - GB$ if $G$ is above $B$ , and $GM = MB + GB$ if $G$ is below $B$ (assuming $M$ is above $B$ ).

Righting Moment

When a stable body is tilted by a small angle $\theta$ , the buoyant force and weight create a restoring couple (Righting Moment).

Righting Moment ( $M_R$ )

Calculates the restoring couple (righting moment) acting on a tilted floating body.

M_R = W \cdot GM \sin\theta

Variables

Symbol	Description	Unit
$M_R$	Righting moment.	N·m
$W$	Total weight of the floating body.	N
$GM$	Metacentric height.	m
	Angle of tilt.	rad or degrees

Distance to Metacenter

The distance $MB$ is a geometric property determined by the shape of the waterline area and the submerged volume of the body.

Metacentric Radius ( $MB$ )

Calculates the distance from the center of buoyancy to the metacenter.

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

Variables

Symbol	Description	Unit
$MB$	Distance from the center of buoyancy to the metacenter.	m
$I$	Moment of inertia of the waterline area about the tilt axis.	$m^4$
$V_{sub}$	Volume of the submerged portion of the body.	$m^3$

Rolling of Floating Bodies

When a stable floating body is disturbed, it will oscillate (roll) around its metacentric axis. The time taken for a floating body to complete one full roll is the period of oscillation.

Period of Oscillation ( $T$ )

Calculates the rolling period of a floating body.

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

Variables

Symbol	Description	Unit
$T$	Time period of oscillation.	s
$k$	Radius of gyration of the body about its longitudinal roll axis.	m
$GM$	Metacentric height.	m
$g$	Acceleration due to gravity.	$m/s^2$

Period and Stability Tradeoff

This physical relationship 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

For fully submerged bodies (like submarines or balloons), the Center of Buoyancy ( $B$ ) is fixed at the centroid of the displaced volume.

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

Civil Engineering Applications

Naval Architecture: Designing the hull shape to ensure a large enough metacentric height ( $GM$ ) for stability without causing an uncomfortable roll period.
Floating Structures: Design of floating breakwaters, pontoons, and platforms where weight distribution ( $G$ ) must be carefully balanced against buoyancy ( $B$ ).
Offshore Construction: Towing and installing concrete caissons or jacket structures, requiring precise ballasting to control stability.

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: $M_{righting} = W \cdot GM \sin\theta$ .

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Learning Objectives

Concept Overview

Archimedes' Principle

Buoyant Force (Archimedes' Principle)

Interactive Simulation

Buoyancy & Stability Simulator

What this teaches

Try this

Center of Buoyancy (BBB)

Center of Gravity (GGG)

Metacenter (MMM)

Stability of Floating Bodies

Metacentric Height (GMGMGM)

Small-Angle Approximation

Metacentric Height (GMGMGM)

Sign Convention for Metacentric Height

Righting Moment

Righting Moment (MRM_RMR​)

Distance to Metacenter

Metacentric Radius (MBMBMB)

Rolling of Floating Bodies

Period of Oscillation (TTT)

Period and Stability Tradeoff

Stability of Submerged Bodies

Civil Engineering Applications

Center of Buoyancy ( $B$ )

Center of Gravity ( $G$ )

Metacenter ( $M$ )

Metacentric Height ( $GM$ )

Metacentric Height ( $GM$ )

Righting Moment ( $M_R$ )

Metacentric Radius ( $MB$ )

Period of Oscillation ( $T$ )