FMC610S
BOUYANCY AND FLOATATION
Contents/Objectives
• Buoyant force-description
• Centre buoyancy
• Metacentre and metacentric height
• Analytical method and experimental method of determining the metacentric height
• Condition of equilibrium of submerged bodies
Target/Outcomes
• Understand and describe buoyancy and center of buoyancy
• Understand metacentre, metacentric height and how they are determined.
• Understand conditions of equilibrium of submerged bodies
BUOYANT FORCE
• When a body is immersed in fluid, an upward force is exerted by the fluid on the body.
• This upward force is equal to the weight of the fluid displaced by the body and is called buoyant force
• The force exerted by the surrounding fluid on the object is immersed causes buoyant force.
FORCES ACTING ON BUOYANCY
• Gravitational
• Buoyant force
NOTE
• Since the pressure at the bottom is always greater than at
the top, every object submerged in a fluid feels an upward
buoyant force.
• Buoyant force is measured in Newtons (N)
HOW DO WE CALCULATE BUOYANT FORCE
• Buoyant Force = Weight of displaced fluid
• Buoyant = Weight of object in air - Weight of object in water
𝐅𝐁 = 𝛒𝐕𝐠
FB = buoyant force
𝜌 = density of the fluid
V= volume of fluid displaced
g = acceleration due to gravity
CENTRE OF BUOYANCY
• The point through which the force of buoyancy is
supposed to act is known as Centre of Buoyancy.
NOTE:
• The force of buoyancy is a vertical force and it
is equal to the weight of the fluid displaced.
• The centre of buoyancy is therefore the
centre of gravity of the fluid displaced.
Example:
Find the volume of water displaced and the
position of centre of buoyancy for a wooden
block of width 2.5 m and of depth 1.5 m, when
it floats horizontally in water. the density of the
block is 650 𝑘𝑔 𝑚3 and its length 6.0 m.
META-CENTRE
• It is defined as the point about which a body starts oscillating when the body is tilted by a small angle.
• It is the point at which the line of action of the force of buoyancy will meet the normal axis of the
body when the body is given small angular displacement.
• The line of action of the buoyant force FB that passes
through B1 and intersects the normal axis of the body at
some point say M.
• Position of M on the normal axis is called the metacentre.
METACENTRIC HEIGHT
NOTE:
• At equilibrium G and B lie on normal axis which is
vertical
• Upon small angular displacement in the clockwise
direction,
• B which is the centre of gravity of the displaced liquid
or centre of gravity of the portion of the body submerged in the liquid shifts to B1 towards the right.
• The distance between the meta-centre of a
floating body and the centre of gravity of the
body is called the metacentric height (MG)
• The metacentric height of a floating body is a
direct measure of its stability.
• More the metacentric height of a floating
body, more it will be stable
METACENTRIC HEIGHT
Transverse stability- when a vessel is floating
upright (at equilibrium) in still water
Conditions of Equilibrium of a Floating Body based on the metacentric position
A body is said to be in equilibrium, when it remains in steady state while floating in a liquid
• Stable Equilibrium
• Unstable Equilibrium
• Neutral Equilibrium
Stable Equilibrium
• A body is said to be in stable equilibrium, if it returns
to its original position, when given a small angular
displacement.
• This happens when the metacentre (M) is higher than
the centre of gravity (G) of the floating body.
Un-Stable Equilibrium
• A body is said to be in stable equilibrium, if it does not
return back to its original position and heels further
away, when given a small angular displacement.
• This happens when the metacentre (M) is lower than
the centre of gravity (G) of the floating body.
Neutral Equilibrium
• A body is said to be in stable equilibrium, if it occupies a
new position and remains at rest in this new position,
when given a small angular displacement.
• This happens when the metacentre (M) coincides with
the centre of gravity (G) of the floating body.
Determination of metacentric height
• Analytical method &
• Experimental Method
Analytical method
Based on the analytical method, the metacentric
height (GM) is computed thus:
𝑰
𝑮𝑴 = − 𝑩𝑮
𝑽
𝐼 = moment of inertia of plane
V = volume of displaced liquid
BG = distance between centre of buoyancy and
centre of gravity
EXAMPLE: A rectangular pontoon is 5 m long , 3 m wide and
1.2 m heigh . The depth of the immersion of the ponton is
0.8 m in sea water. If the centre of gravity is 0.6 m above the
bottom of the pontoon , determine the metacentric height.
The density for sea water = 1025 𝑘𝑔 𝑚3