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Metacenter and Stability of Floating Bodies | Buoyancy Principle

Metacenter is a point at which an imaginary vertical line passing through the centre of buoyancy and an imaginary vertical line passing through the centre of gravity of a vessel intersect each other. Before getting into the concept of metacentre it is important to know the concepts of buoyancy and centre of buoyancy all of which are discussed further in this blog.


Buoyancy Principle


Buoyancy is an upward force exerted by a fluid that opposes the weight of a partially or fully immersed object. It is an upward thrust which enables bodies to float over a fluid without sinking.


In the case of fluids, pressure increases with depth as a result of the weight of the overlying fluid. Thus the pressure at the bottom of a fluid column is greater than the pressure at the top of the column. Similarly, pressure at the bottom of a submerged body is greater than the pressure at the top. This pressure difference is what results in a net upward force called the buoyant force on the object.


The magnitude of the force is proportional to the pressure difference and is equivalent to the weight of the fluid that would otherwise occupy the volume of the submerged body. This is also called the displaced fluid. The buoyant force is numerically expressed as shown below.


Fb = - ρ*g*V,


where,

Fb - Buoyant force

ρ - density of the fluid

g - acceleration due to gravity

V - Volume of the displaced fluid


This is the reason why objects whose density is greater than that of the fluid in which it is submerged tend to sink. Know more about Archimedes Principle here.


Center of Buoyancy


The centre of buoyancy is a point at which the centre of mass i.e., the centre of gravity, of the displaced fluid acts. It depends on the shape of the displaced fluid.


Centre of Buoyancy
Centre of Buoyancy

Metacenter and Stability


As said earlier, a metacentre is a point at which an imaginary vertical line passing through the centre of buoyancy and an imaginary vertical line passing through the centre of gravity of a vessel intersect each other. The same is illustrated below.


Metacenter Concept
Metacenter Concept

In the above case, vertical lines of both the centre of gravity and the centre of buoyancy lie on the same line. Therefore, there is no single metacentric point and the ship remains stable. But if the points do not lie in the same line then a metacentric point is formed. The position of this point bears greater importance with regards to the stability of the ship as discussed further.


Case 1: Stable Condition


For the vessel to be stable, the centre of gravity of the vessel must be below the metacentre. This can be understood easily from the picture below.



Metacenter below Center of Gravity
Metacenter below Center of Gravity


The mass of the vessel acts downwards at the point of the centre of gravity as shown by the red line. The buoyancy force acts upwards at the centre of the buoyancy point as shown by the blue line. As both of them act at different points it gives rise to a couple. This couple acts in the anticlockwise direction and tries to bring back the ship to a stable position. Hence, this couple is called the righting moment. Therefore, as long as the metacentre is above the centre of gravity of the vessel, the vessel remains stable.


Case 2: Unstable Condition


The vessel becomes unstable if the centre of gravity of the vessel is above the metacentre. The same is illustrated in the picture below.


Metacenter above Center of Gravity
Metacenter above Center of Gravity

The centre of gravity has moved up because of the huge mass that is now placed over the vessel. The mass of the vessel acts downwards at the point of the centre of gravity as shown by the red line. The buoyancy force acts upwards at the centre of the buoyancy point as shown by the blue line. As both of them act at different points it gives rise to a couple. This couple acts in the clockwise direction and tries to overturn the ship i.e., the ship cannot come back to its stable position. Therefore, if the metacentre is below the centre of gravity of the vessel, the vessel cannot come back to its original stable position making it unstable.



With these, all the important aspects of the metacentre and its relation to the stability of floating bodies have been discussed. If you want to contribute your knowledge to the reading community join the APSEd by filling out the form below.