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

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.	$k g / m^{3}$
$g$	Acceleration due to gravity.	$m / s^{2}$
$V_{displaced}$	Volume of the displaced fluid (or submerged volume of the body).	$m^{3}$

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/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.

Stability of Floating Bodies

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 ( $G M$ )

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 $G$ ( $GM = 0$ ).

Metacentric Height ( $G M$ )

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
$G M$	Metacentric height.	m
$M B$	Distance from the center of buoyancy to the metacenter.	m
$G B$	Distance between the center of gravity and the center of buoyancy.	m

Note

Sign Convention: 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
$G M$	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 ( $M B$ )

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

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

Variables

Symbol	Description	Unit
$M B$	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
$G M$	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 ( $B$ ).

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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Concept Overview

Archimedes' Principle

Buoyant Force (Archimedes' Principle)

Buoyancy Simulation

Buoyancy & Stability Simulator

What this teaches

Try this

Stability of Floating Bodies

Metacenter (MMM)

Metacentric Height (GMGMGM)

Metacentric Height (GMGMGM)

Note

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

Metacenter ( $M$ )

Metacentric Height ( $G M$ )

Metacentric Height ( $G M$ )

Righting Moment ( $M_{R}$ )

Metacentric Radius ( $M B$ )

Period of Oscillation ( $T$ )