Test: Submerged Bodies - Civil Engineering (CE) MCQ

Buoyancy - 
When a body is either wholly or partially immersed in a fluid, the hydrostatic lift due to the net vertical component of the hydrostatic pressure forces experienced by the body is called the “Buoyant Force” and the phenomenon is called “Buoyancy”.
The Center of Buoyancy is a point through which the force of buoyancy is supposed to act.
When a body is submerged in a liquid (or a fluid), the equilibrium requires that the weight of the body acting through its Center of Gravity should be co-linear with the Buoyancy Force acting through the Center of Buoyancy.
If the Body is Not Homogeneous in its distribution of mass over the entire volume, the location of the Center of Gravity (G) does not coincide with the Center of Volume (B).
Depending upon the relative locations of (G) and (B), the submerged body attains different states of equilibrium: Stable, Unstable, and Neutral.
Stable Equilibrium:
(G) is located below (B).
A body is given a small angular displacement and then released, returns to its original position by retaining the original vertical axis as vertical because of the restoring couple produced by the action of the Buoyant Force and the Weight. 
Unstable Equilibrium:
(G) is located above (B).
Any disturbance from the equilibrium position will create a destroying couple that will turn the body away from the original position.
Neutral Equilibrium:
(G) and (B) coincide.
The body will always assume the same position in which it is placed. A body having a small displacement and then released, neither returns to the original position nor increases its displacement- It will simply adapt to the new position. 

Now, 
∑M₀ = 0 for equilibrium.

∴ h = 1.596 m

• Stable Equilibrium: If the body returns to its original position by retaining the originally vertical axis as vertical.
• Unstable Equilibrium: If the body does not return to its original position but moves further from it.
• Neutral Equilibrium: If the body neither returns to its original position nor increases its displacement further, it will simply adapt its new position

Concept:
Buoyancy:
When a body is either wholly or partially immersed in a fluid, a lift is generated due to the net vertical component of hydrostatic pressure forces experienced by the body. This lift is called the buoyant force and the phenomenon is called buoyancy.
Centre of buoyancy:
The line of the buoyant force is vertical and passes through the center of gravity of the displaced fluid i.e. the centroid of the displaced volume which is known as the center of buoyancy. 

Concept:
A body in a liquid is said to be stable when given small displacement, it returns to its original position.
For stability of completely submerged Bodies, the center of gravity ‘G’ is below the center of Buoyancy ‘B’.

Concept:
Condition of stable equilibrium for a submerged body in terms of center of buoyancy and the center of gravity:

• Stable equilibrium: If its center of gravity is directly below the center of buoyancy
• Neutral equilibrium: If its center of gravity is coincident with the center of buoyancy
• Unstable equilibrium: If its center of gravity is directly above the center of buoyancy
  

Condition of stable equilibrium for a floating body in terms of metacentric height (GM) as follows:

• Stable equilibrium: GM > 0 (M is above G) i.e. Metacentric height is positive.
• Neutral equilibrium : GM = 0 (M coinciding with G)
• Unstable equilibrium : GM < 0 (M is below G)

Metacentric height (GM):
GM = I/V - BG
Note:
1) M.O.I. (I) of top view is calculated about water surface about y-y axis.
2) V = Volume of body submerged in water.
Now,

∴ I = 13.5 m⁴
Now,
V = 6 × 3 × 0.8 = 14.4 m³
BG = AG – AB – 0.6 – 0.4 = 0.2 m
Thus,
GM = 13.5/14.4 - 0.2
∴ GM = 0.7375 m

Stability of unconstrained Submerged Bodies in Fluid:
The equilibrium of a body submerged in a liquid requires that the weight of the body acting through its centre of gravity should be colinear with an equal hydrostatic lift acting through the centre of buoyancy.

• Stable Equilibrium: If the body returns to its original position by retaining the originally vertical axis as vertical
• Unstable Equilibrium: If the body does not return to its original position but moves further from it
• Neutral Equilibrium: If the body neither returns to its original position nor increases its displacement further, it will simply adopt its new position

The relative position of the centre of gravity (G) and centre of buoyancy (B) of a body determines the stability of a submerged body.

Stable Equilibrium: B is above G

Unstable Equilibrium: B is below G

Neutral Equilibrium: B coincides with G

AG = (weight of base*distance of C.G from base point A) + (weight of upper part*distance of C.G from point A)/ )weight of base + weight of upper part)
= 14.52
By principle of buoyancy,
Weight of cylinder = Weight of water displaced
h = 38.625
AB = 19.31
BG = 14.25 - 19.31 = -4.79
GM = Metacentric height = I/∀ -BG
= 6.16
As metacentric height is positive, it will float.

When the weight distribution is around the lower part, the centre of gravity is at lower portion and hence below the centre of buoyancy which is condition for stable equilibrium.