Formula Sheet: Buoyancy and Floatation | Fluid Mechanics for Mechanical Engineering PDF Download

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F ormula Sheet: Buo yancy & Floatation
Introduction to Buo yancy & Floatation
• Buo yancy : The upward force exerted b y a fluid on a body partially or fully
submerged in it, du e to pressure differences.
• Floatation : The ability of a body to float in a fluid when the buo yant force
equals the we ight of the body .
• Assumptions : Fluid is incompressible, at rest (h ydrostatic conditions), and
gr avity is uniform .
Archimedes’ Principle
• Buo yant F orce :
F
b
=?
f
gV
sub
whereF
b
= buo yant force,?
f
= fluid density ( 1000 kg m
-3
for water),g = gr av-
itational acceler ation (9.81 m s
-2
), V
sub
= volume of the submerged portion
of the body .
• Condition f or Floatation :
F
b
=W =?
b
gV
total
whereW = weight of the body ,?
b
= density of the body ,V
total
= total volume
of the body .
• Fr action Sub merged :
V
sub
V
total
=
?
b
?
f
Center of Buo yancy
• Definition : The center of buo yancy (B) is the centroid of the submerged
volume of the bo dy .
• Location : F or a partially submerged body , located at the centroid of the
displaced fl uid volume.
Stability of Floating Bodies
• Metacentric Height (GM) :
GM =BM +BG
where:
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F ormula Sheet: Buo yancy & Floatation
Introduction to Buo yancy & Floatation
• Buo yancy : The upward force exerted b y a fluid on a body partially or fully
submerged in it, du e to pressure differences.
• Floatation : The ability of a body to float in a fluid when the buo yant force
equals the we ight of the body .
• Assumptions : Fluid is incompressible, at rest (h ydrostatic conditions), and
gr avity is uniform .
Archimedes’ Principle
• Buo yant F orce :
F
b
=?
f
gV
sub
whereF
b
= buo yant force,?
f
= fluid density ( 1000 kg m
-3
for water),g = gr av-
itational acceler ation (9.81 m s
-2
), V
sub
= volume of the submerged portion
of the body .
• Condition f or Floatation :
F
b
=W =?
b
gV
total
whereW = weight of the body ,?
b
= density of the body ,V
total
= total volume
of the body .
• Fr action Sub merged :
V
sub
V
total
=
?
b
?
f
Center of Buo yancy
• Definition : The center of buo yancy (B) is the centroid of the submerged
volume of the bo dy .
• Location : F or a partially submerged body , located at the centroid of the
displaced fl uid volume.
Stability of Floating Bodies
• Metacentric Height (GM) :
GM =BM +BG
where:
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– BM = distance from center of buo yancy (B) to metacenter (M).
– BG = distance from center of buo yancy (B) to center of gr avity (G).
– GM >0 : Stable (M above G).
– GM =0 : Neutr ally stable.
– GM <0 : Unstable (M below G).
• Metacentric Radius (BM ) :
BM =
I
V
sub
whereI = second moment of area of the waterline plane about the axis of
tilt,V
sub
= submerged volume.
• Second Mom ent of Area (Examples) :
– Rectangular waterline: I =
LB
3
12
, whereL = length,B = breadth.
– Circular waterline: I =
pR
4
4
, whereR = r adius.
Restoring Moment
• Restoring Moment for Small Angular Displacement :
M
r
=?
f
gV
sub
GM sin?
where? = angle of tilt (small angles).
• Stability Condition : PositiveM
r
indicates restoring moment, ensuring sta-
bility .
Period of Oscillation
• Time Perio d of Rolling/Pitching :
T =2p
v
k
2
gGM
wherek = r adius of gyr ation of the body about the axis of rotation, GM =
metacentric height .
Applications
• Used in designing ships, submarines, and floating platforms to ensure sta-
bility and floa tation.
• Essential for GA TE problems on buo yant force, stability analysis, and meta-
centric height calculations.
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