Formula Sheets: Buoyancy & Floatation | Fluid Mechanics for Civil Engineering - Civil Engineering (CE) PDF Download

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Fluid Mec hanics: Buo y ancy & Floatation F orm ula
Sheet for Civ il GA TE
Buo y ancy
• Arc himedes’ Principle : Buo y an t force equals the w eigh t of the fluid displaced b y a
submerged or floating b o dy .
F
b
=?gV
displaced
where ? is fluid densit y ( kgm
-3
), g is gra vitational acceleration (981m/s
2
), V
displaced
is the v olume of fluid displaced ( m
3
).
• Sp ecific Gra vit y : Relates densit y of the ob ject or fluid to w ater.
S =
?
?
w ater
, ?
w ater
= 1000kgm
-3
• W eigh t of Bo dy :
W =?
b o dy
gV
b o dy
where ?
b o dy
is the densit y of the b o dy , V
b o dy
is the v olume of the b o dy .
Floatation
• Condition for Floatation : A b o dy floats when the buo y an t force equals its w eigh t.
F
b
=W or ?gV
displaced
=?
b o dy
gV
b o dy
• F raction of V olume Submerged : F or a floating b o dy .
V
displaced
V
b o dy
=
?
b o dy
?
or, using sp ecific gra vit y:
V
displaced
V
b o dy
=
S
b o dy
S
fluid
• Cen ter of Buo y ancy : The cen troid of the displaced fluid v olume, lo cated at depth
h
b
.
h
b
= Depth to cen troid of s ubmerged v olume
Stabilit y of Floating Bo dies
• Metacen ter : The p oin t where the line of action of the buo y an t force in tersects the
v ertical axis of the b o dy after a small tilt.
• Metacen tric Heigh t (GM) : Distance b et w een the cen ter of gra vit y (G ) and the
metacen ter (M ).
GM =
I
V
displaced
-BG
where I is the momen t of inertia of the w aterline area ab out the axis of tilt (m
4
), BG
is the distance b et w een the cen ter of buo y ancy (B ) and the cen ter of gra vit y (G ).
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Fluid Mec hanics: Buo y ancy & Floatation F orm ula
Sheet for Civ il GA TE
Buo y ancy
• Arc himedes’ Principle : Buo y an t force equals the w eigh t of the fluid displaced b y a
submerged or floating b o dy .
F
b
=?gV
displaced
where ? is fluid densit y ( kgm
-3
), g is gra vitational acceleration (981m/s
2
), V
displaced
is the v olume of fluid displaced ( m
3
).
• Sp ecific Gra vit y : Relates densit y of the ob ject or fluid to w ater.
S =
?
?
w ater
, ?
w ater
= 1000kgm
-3
• W eigh t of Bo dy :
W =?
b o dy
gV
b o dy
where ?
b o dy
is the densit y of the b o dy , V
b o dy
is the v olume of the b o dy .
Floatation
• Condition for Floatation : A b o dy floats when the buo y an t force equals its w eigh t.
F
b
=W or ?gV
displaced
=?
b o dy
gV
b o dy
• F raction of V olume Submerged : F or a floating b o dy .
V
displaced
V
b o dy
=
?
b o dy
?
or, using sp ecific gra vit y:
V
displaced
V
b o dy
=
S
b o dy
S
fluid
• Cen ter of Buo y ancy : The cen troid of the displaced fluid v olume, lo cated at depth
h
b
.
h
b
= Depth to cen troid of s ubmerged v olume
Stabilit y of Floating Bo dies
• Metacen ter : The p oin t where the line of action of the buo y an t force in tersects the
v ertical axis of the b o dy after a small tilt.
• Metacen tric Heigh t (GM) : Distance b et w een the cen ter of gra vit y (G ) and the
metacen ter (M ).
GM =
I
V
displaced
-BG
where I is the momen t of inertia of the w aterline area ab out the axis of tilt (m
4
), BG
is the distance b et w een the cen ter of buo y ancy (B ) and the cen ter of gra vit y (G ).
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• Stabilit y Conditions :
– Stable: M is ab o v e G (GM > 0 ).
– Unstable: M is b elo w G (GM < 0 ).
– Neutral: M coincides with G (GM = 0 ).
• Momen t of Inertia for Common W aterline Areas :
– Rectangle (length L , breadth B ):
I =
LB
3
12
(ab out longitudinal axis)
– Circle (radius r ):
I =
pr
4
4
Key Notes
• Standard v alues: ?
w ater
= 1000kgm
-3
, g = 9.81ms
-2
.
• Use SI units consisten tly for GA TE calculations.
• F or stabilit y , ensure the momen t of inertia is calculated ab out the correct axis of tilt.
• If sp ecific gra vit y is giv en, use ? =S ·?
w ater
for calculations.
• A ccoun t for atmospheric pressure (P
atm
= 101.3kPa ) if relev an t to the problem.
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