Condition of Equilibrium and Stability of Floating Bodies- Stable, Neutral, Unstable Equilibrium [PDF]

Introduction

In this article, we will delve into the fascinating world of floating bodies and explore the concepts of equilibrium and stability. When a body is placed on a fluid surface, it experiences various forces that determine whether it will float, sink or stay in equilibrium. The stability of a floating body depends on its center of gravity and the position of the center of buoyancy. Understanding the conditions of stable, neutral, and unstable equilibrium is critical for engineers and designers who work with floating structures, such as ships, boats, and oil rigs. By the end of this article, you will have a deeper understanding of the principles that govern the stability of floating bodies and the factors that affect them.

Archimedes principle

The Archimedes principle explains about the force acting on a floating body in a liquid. It states that “The body immersed or floating in a liquid are acted upon by a vertical upward liquid force equal to the weight of the liquid displaced”. This vertical upward force is called buoyancy or buoyant force. The point, through which this force acts, is known as a center of buoyancy.

A body floating or immersed in a liquid will lose its weight equal to the buoyant force of the liquid. The body has less weight in a liquid than outside. Archimedes principle is applicable to bodies floating or immersed in a fluid. This has been used by man for about 2200 years, for the problem of general floatation and naval architectural design.

Stability of Floating Bodies

When a body floats, it is subjected to two parallel forces which are as follows:

  1. The downward force of gravity acting on each of the particles.
  2. The upward buoyant force of the liquid acting on various elements of the submerged surface.

If the body is to float in equilibrium in an upright position the resultant of these two forces must be collinear, equal and opposite.

Stable , neutral and unstable equilibrium

Hence center of gravity of the floating body and center of buoyancy must lie in the same vertical line. B is the center of buoyancy which is the center of gravity of the area ACO, and G is the center of gravity of the body. If the ship in the (figure above) heels through an angle θ (fig b), due to tilting moments caused by wind or wave action or due to movement of loads across the deck, portion A’C’O’ will now stand immersed in water.

The center of buoyancy will shift from B to B’. The buoyant force will act through B’. The center of gravity G will of course not change and W will continue to act through it. 

What is Metacenter?

A vertical line through the new center of buoyancy intersects the inclined axis of the ship at M which is known as Metacenter. The term metacenter is defined as the point at which a vertical line through the center of buoyancy intersects the vertical center line of the ship section, after a small angle of heal.

What is Metacentric Height?

The distance between the center of gravity and metacenter is called metacentric height. The metacentric height is a measure of the static stability of the ship. For the small angle of inclination, the position of M does not change materially and the metacentric height is approximately constant. Hence the ship may be considered as rotating about M. In other words, the ship may be considered as behaving like a pendulum suspended at M, the point G, corresponding to bob.

What do you mean by equilibrium?

Equilibrium of floating bodies refers to the state of balance or stability that is achieved when a body is submerged in a fluid and is neither sinking nor rising. In other words, it is the state in which the weight of the object is equal to the buoyant force acting on it. The buoyant force is the force exerted by the fluid on the object, which is equal to the weight of the fluid displaced by the object. This phenomenon is known as Archimedes’ principle.

When a body is placed in a fluid, it experiences an upward force, which is known as the buoyant force. This force is equal to the weight of the fluid displaced by the object. If the weight of the object is less than the buoyant force, it will float on the surface of the fluid. However, if the weight of the object is greater than the buoyant force, it will sink.

what is equilibrium of floating bodies?

In the case of floating bodies, the weight of the object is equal to the buoyant force. This means that the object is in a state of balance or equilibrium. If the object is pushed down, the buoyant force will increase, and if the object is pushed up, the buoyant force will decrease. As a result, the object will return to its original position of equilibrium.

The equilibrium of floating bodies has many practical applications, such as in the design of boats and ships. In order to ensure that a boat or ship is stable and does not tip over, the weight of the vessel must be distributed evenly and the center of gravity must be located below the center of buoyancy. This ensures that the buoyant force acting on the vessel is greater than its weight, and it remains in a state of equilibrium.

The condition of Equilibrium of Floating Bodies:

By the condition of equilibrium of floating bodies, we mean the possible state of stability or instability of floating bodies under all odds. Following are the three conditions of equilibrium of floating bodies:

  • Stable Equilibrium
  • Neutral Equilibrium
  • Unstable Equilibrium

Let us discuss each condition in detail.

Stable Equilibrium:

In the above figure (b) we found that when the ship was subjected to turning moments, the center of buoyancy changed from B to B’. Further, From the same diagram, we also notice that W and F are two equal and opposite parallel forces acting at a distance X apart. Naturally, this causes anticlockwise couples WX, tending to restore the ship to its original position.

In this case, the ship is said to be in stable equilibrium. Hence it may be stated that a floating body is said to be in a state of stable equilibrium which, when subjected to turning moments, leads to regaining its original position and that M lies above G or BM>BG.

Neutral Equilibrium:

If the ship in figure (b) is tilted or rolled over such that the new center of buoyancy B’ lies on the line of action of W, the buoyant force F and weight W would be collinear, equal and opposite and would not exert any restoring moment. In that case, the ship would neither tend to regain its original position nor would tend to heel over further.

Hence it may be stated that “a floating body is said to be in a state of neutral equilibrium which, when subjected to turning moment, neither tends to regain its original position, nor tend to heel over further but instead keeps on in the tilted position, and that M coincides with G or BM=BG”.

Unstable Equilibrium:

In this case, the new center of buoyancy B’ lies in between B and the line of action of W. The vertical upward buoyant force F passing through B’ will intersect the inclines central line at the point M below G. The couple thus formed by W and F will be clockwise; further helping the turning moment of tilt over the ship further.

Video on Ship Stability to understand the above Conditions:

Source : MARINE ENGINEERING WORLD

FAQ’s

What is the condition of equilibrium for a floating body?

It is the condition for a floating body when the body is neither sinking nor rising and is stable in its position on the fluid surface. This state occurs when the weight of the floating body is equal to the buoyant force acting on it.

What are the factors that affect the stability of a floating body?

The stability of a floating body depends on several factors, including the position of the center of gravity, the position of the center of buoyancy, the shape and size of the body, and the density of the fluid.

What is metacentric height, and why is it important for determining stability?

Metacentric height is the distance between the metacenter (the point at which a floating body’s center of gravity will be when it tilts) and the center of buoyancy (the center of mass of the displaced fluid). It is an important parameter for determining the stability of a floating body, as a higher metacentric height indicates greater stability.

What is the difference between stable, neutral, and unstable for a floating body?

In stable equilibrium, the body will return to its original position when it is slightly tilted, whereas in neutral equilibrium, the body will remain in the new position after being tilted. Unstable equilibrium occurs when the body will continue to tilt until it capsizes, and it cannot return to its original position.

Conclusion

In conclusion, the condition of equilibrium and stability of floating bodies is a crucial concept in physics. It is essential to understand the forces acting on a body and their effects on its stability to prevent accidents and ensure safety in various activities involving floating bodies. By understanding the frequently asked questions about this topic, we can develop a better understanding of the principles involved and apply them to practical situations.

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