**Before going to study all the three equilibrium let’s have brief knowledge about the Archimedes principle and Stability of floating bodies. So,**

Contents

## Archimedes principle:

The Archimedes principle 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 liquid. 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, they are:

- The downward force of gravity acting on each of the particles that go to makes up the body.
- 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.

Hence center of gravity of the floating body and center of buoyancy must lie in the same vertical line as shown in the above diagram.

**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.

*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 metacenter is the point at which a vertical through the center of buoyancy intersects the vertical center line of the ship section, after a small angle of heal.*

#### 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 regarded 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.

## 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. There is three condition equilibrium of floating bodies:

### Stable, Neutral and, Unstable Equilibrium:

### 1. Stable Equilibrium:

In the above fig 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 may 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.

### 2. Neutral Equilibrium:

If the ship in fig 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”.

### 3. Unstable Equilibrium:

In this case, the new center of buoyancy B’ lies in between B and the line of action od 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.