# Stability and Metacentric Height

**Contents**

### Key Facts

**Gyroscopic Couple**: The rate of change of angular momentum () = (In the limit).

- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.

## Introduction

In 1628 the Swedish warship Vasa was launched in Stockholm harbour.##### MISSING IMAGE!

**13108/img_0001_14.jpg** cannot be found in /users/13108/img_0001_14.jpg. Please contact the submission author.

## Centre Of Buoyancy And Stability

**Center of gravity**refers to the mean location of the gravitational force acting on a body.

**The Centre of Buoyancy**There are three Types of Equilibrium:

**Stable**. The body returns to it's original position if given a small angular displacement.**Neutral**. The body remains in a new position if given a small angular displacement.**Unstable**. The body heals further over if given a small angular displacement.

## The Stability Of Fully Submerged Bodies

##### MISSING IMAGE!

**23287/stability-and-metacentric-s1.png** cannot be found in /users/23287/stability-and-metacentric-s1.png. Please contact the submission author.

**Let:**

- = Volume of Body.
- = Specific weight of the fluid.
- = Mass of the Body.
- is the Centre of Gravity.
- is the Centre of Buoyancy and is the centre of gravity of the displaced liquid.

**If:**

- and are coincident then the Body will be in
**Neutral**equilibrium. - is below then the Body is in
**Unstable**equilibrium. - is above then the body is in
**Stable**equilibrium.

## The Stability Of Partially Submerged Bodies

##### MISSING IMAGE!

**23287/stability-and-metacentric-s2.png** cannot be found in /users/23287/stability-and-metacentric-s2.png. Please contact the submission author.

**"META CENTRE"**and is defined as the point where the vertical through the new Centre of Buoyancy meets the original vertical through the Centre of Gravity after a very small angle of rotation.

**METACENTRIC HEIGHT**.

##### MISSING IMAGE!

**23287/stability-and-metacentric-s3.png** cannot be found in /users/23287/stability-and-metacentric-s3.png. Please contact the submission author.

**stable**equilibrium for a

**floating Partially Submerged Body**the Meta centre

**must**be above the Centre of Gravity . If the Metacentric height is zero the Body will be in Neutral equilibrium. In ship design the choice of the Metacentric height is a compromise between stability and the amount that the ship rolls. In British Dreadnaught Battle ships, for instanace, the metacentric height was so great that they had a tendency to roll badly, even with large bilge keels. The Righting couple

### Experimentally

- Let be the weight of the Boat plus it's Load. A small load is moved a distance and causes a tilt of angle . The Boat is now in a new position of equilibrium with and lying along the Vertical through . The Moment due to the movement of the load is given by:
##### MISSING IMAGE!

**23287/stability-and-metacentric-s4.png**cannot be found in /users/23287/stability-and-metacentric-s4.png. Please contact the submission author.

Moment due to movement of of

### Theory

- The Ship tilts from it's old waterline to a new waterline as it moves through an angle . Due to the movement of the wedge of water from to , the Centre of Buoyancy moves from to . The Change in the moment of the buoyancy Force = where is small The Volume of the Wedge Therefore the Moment of the Couple due to the movement of the wedge Where is the Second Moment of Area of the Water Plane Section and is the volume of water Displaced. Thus if the positions of and are known or can be calculated , then the distance can be determined since:
##### MISSING IMAGE!

**23287/stability-and-metacentric-s5.png**cannot be found in /users/23287/stability-and-metacentric-s5.png. Please contact the submission author.There are in fact two Metacentric heights of a ship. One for Rolling and the other for Pitching. The former will always be less than the latter and unless otherwise stated, the Metacentric given will be for Rolling.##### MISSING IMAGE!

**23287/stability-and-metacentric-s6.png**cannot be found in /users/23287/stability-and-metacentric-s6.png. Please contact the submission author.

##### Example - Example 1

##### MISSING IMAGE!

**23287/Stability-and-Metacentric-s7.png** cannot be found in /users/23287/Stability-and-Metacentric-s7.png. Please contact the submission author.

Find it's

**metacentric height**and establish the

**angular tilt**which will result if the load is moved by one ft. sideways.

The height, And: But the Metacentric height The Moment due to the Movement of the Load = 8 ft. tons The Moment due to the movement of the of = = 20 X 2.40

- The
**metacentric heigh**is - The
**angle**is