Buoyancy
What is Buoyant Force?
When a body is immersed in a fluid, the fluid exerts pressure on it from all sides.
Since pressure increases with depth, the upward force on the bottom is greater than the downward force on the top.
This results in a net upward force called:
Buoyant Force (Upthrust)
2. Archimedes’ Principle
When a body is wholly or partially immersed in a fluid, it experiences an upward force equal to the weight of the fluid displaced by the body.
Mathematically:
𝐹
𝐵
=
𝜌
𝑓
𝑔
𝑉
𝑑
F
B
=ρ
f
gV
d
Where:
Symbol Meaning
𝐹
𝐵
F
B
Buoyant force (N)
𝜌
𝑓
ρ
f
Density of fluid
𝑔
g Acceleration due to gravity
𝑉
𝑑
V
d
Volume of fluid displaced
3. Apparent Loss of Weight
When a body is immersed in fluid:
Apparent weight
=
𝑊
−
𝐹
𝐵
Apparent weight=W−F
B
Where:
𝑊
W = True weight of body
𝐹
𝐵
F
B
= Buoyant force
This is why objects feel lighter in water.
4. Conditions of a Body in a Fluid
Let:
𝑊
W = Weight of body
𝐹
𝐵
F
B
= Buoyant force
Condition Result
𝑊
>
𝐹
𝐵
W>F
B
Body sinks
𝑊
=
𝐹
𝐵
W=F
B
Body floats in equilibrium
𝑊
<
𝐹
𝐵
W<F
B
Body rises
5. Floatation
A body is said to float when it remains on the surface of a fluid.
Law of Floatation
A floating body displaces a volume of fluid whose weight is equal to the weight of the body.
𝑊
=
𝜌
𝑓
𝑔
𝑉
𝑑
W=ρ
f
gV
d
6. Fraction of Volume Submerged
For a floating body:
𝑉
𝑑
𝑉
=
𝜌
𝑏
𝜌
𝑓
V
V
d
=
ρ
f
ρ
b
Where:
Symbol Meaning
𝑉
𝑑
V
d
Volume submerged
𝑉
V Total volume
𝜌
𝑏
ρ
b
Density of body
𝜌
𝑓
ρ
f
Density of fluid
7. Stability of Floating Bodies
A floating body must not only float but also remain stable.
Important Points
Term Meaning
C.G. Centre of gravity of body
C.B. Centre of buoyancy (centroid of displaced fluid)
M Metacentre
8. Metacentre
When a floating body is slightly tilted:
Centre of buoyancy shifts from B to B₁
The vertical line through B₁ intersects original vertical line at point M
This point is called:
Metacentre
9. Metacentric Height (GM)
𝐺
𝑀
=
𝐵
𝑀
−
𝐵
𝐺
GM=BM−BG
Where:
𝐺
𝑀
GM = Metacentric height
𝐵
𝑀
BM = Distance between centre of buoyancy and metacentre
𝐵
𝐺
BG = Distance between centre of buoyancy and centre of gravity
For stability:
Condition Stability
𝐺
𝑀
>
0
GM>0 Stable equilibrium
𝐺
𝑀
=
0
GM=0 Neutral equilibrium
𝐺
𝑀
<
0
GM<0 Unstable equilibrium
10. Expression for BM
𝐵
𝑀
=
𝐼
𝑉
𝑑
BM=
V
d
I
Where:
Symbol Meaning
𝐼
I Moment of inertia of water-line area
𝑉
𝑑
V
d
Volume of displaced fluid
11. Types of Equilibrium
(A) Stable Equilibrium
Body returns to original position
Metacentre above C.G.
Example: Loaded ship
(B) Unstable Equilibrium
Body overturns
Metacentre below C.G.
Example: Top-heavy boat
(C) Neutral Equilibrium
Body stays in new position
Metacentre coincides with C.G.
Example: Floating sphere
12. Numerical Example
A wooden block of density 600 kg/m³ floats in water.
𝑉
𝑑
𝑉
=
600
1000
=
0.6
V
V
d
=
1000
600
=0.6
So:
60% volume is submerged
13. Applications in Civil Engineering
Design of ships and boats
Floating caissons
Pontoon bridges
Floating docks
Offshore platforms
Tags:
Fluid Mechanics