Buoyancy and Floatation Concepts and Formulas
When a body is immersed in a fluid, an upward force is exerted by the fluid on the body which is equal to the weight of the fluid displaced by the body. This upward force is called buoyant force and the phenomenon is called buoyancy.
- When a body is submerged either fully or partially then it is acted upon by a Force of Buoyancy in the vertical direction which is equal to the weight of the liquid displaced by the body.
- This Buoyant Force always acts through the centroid of liquid displaced.
- Center of Buoyancy is the point through which buoyant force acts.
Principle of Flotation
- According to this principle, if the weight body is equal to the buoyant force then, the body will float.
HρBody = hρfluid
H = Height of Body
h = height of body submerged in fluid
Centre of Buoyancy
- The point at which force of buoyancy acts is called the center of buoyancy.
- It lies at the center of gravity (G) of the volume of water displaced.
If a body that is floating in a liquid is given small angular displacement, it starts oscillating about some point, ‘M’. This point is called the Metacenter.
Metacentric Height (GM or MG)
It is the distance between the gravity center and the metacenter.
Condition for Equilibrium for Floating/Submerged Body
For Stable Equilibrium
- In the case of a floating body, the metacenter should be above the center of gravity.
- In the case of a submerged body, the center of buoyancy (B) should be above the center of gravity (G).
For Unstable Equilibrium
- In the case of a floating body, Metacenter (M) lies below the Center of Gravity (G).
- In the case of a submerged body, the Center of Buoyancy (B) lies below the Center of Gravity (G).
For Neutral Equilibrium
- In the case of a floating body, M and G both coincide.
- In the case of a submerged body, ‘B’ and ‘G’ both coincide.
Distance between metacenter and center of buoyancy
BM = Imin/Vimmersed
Where Imin = Moment of Inertia of the top view of the floating body about the longitudinal axis
V = Volume of the body immersed in liquid
The relation between B, G and M is given by
GM = I/V – BG
BG = Distance between CG of the whole body and CG of the submerged part.
- GM > 0 (Stable Equilibrium)
- GM < 0 (Unstable Equilibrium)
- GM = 0 (Neutral Equilibrium)
- Metacenter height for rolling condition will be less than metacentric height for pitching condition.
Time Period of Oscillation
- If a floating body oscillates then it’s time period of transverse oscillation is given by-
GM = Metacentric height= Least Radius of Gyration