# Stability with liquid loads

**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

**Tankers**are designed to transport liquid loads, but most large vessels have fuel tanks needed for their propulsion.

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*Note:*This page should be read in conjunction with Stability and Metacentric Height

## The Metacentric Height For A Vessel With Liquid Ballast.

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

The

**center of buoyancy**is the centre of the volume of water which the hull displaces.

**Where**

- is the original Waterline.
- is the Centre of Gravity of the Ship and Ballast.
- is the Centre of Buoyancy.

The Ship is tilted through a small angle clockwise giving a new waterline of . Due to the movement of the water wedge the Centre of Buoyancy of the Ship ( i.e. The of of the displaced liquid) moves to . By the

**Previous Theory**where is the second Moment of Area of the Water plane area and is the Total displacement by the Ship and it's contents. Due to the movement of the liquid in the Ballast tank, the of of the Ship and Ballast moves to . Now the vertical through cuts the old vertical centre line at . The Stability of the ship depends upon whether is above or below . The Metacentric Height is now Consider the Liquid in the Ballast Tank.

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- = The second moment of area of the liquid surface about it's centre line.
- = the Volume of liquid in the Tank.
- = the specific weight of the liquid.
- = The movement of the of in the tank due to the tilting of the vessel.

## Divided Ballast Tanks

If the**Ballast Tank**is

**divided**into two by a longitudinal Partition, the above equation becomes:

and

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1) Single Tank of width . 2) Same tank divided into two equal compartments.

## The Righting Couple With A Liquid Ballast.

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**Righting couple**is a

**force**which normally restores a ship to

**equilibrium**once a heel has changed the relationship between her center of buoyancy and her center of gravity.

**Righting Couple**due to the Buoyancy Force acting upwards through and and the weight of the ship acting downwards through and . where is small.

This is the same as in the simple case without a liquid balance but it must be remembered that the

**Metacentric Height**is now and

**not**.

##### Example - Exemple 1

The Pontoon and its load weigh 80 tons.

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Find:

- a)The
**metacentric height**. - b)The
**angle**through which the pontoon will heel if 2 tons of deck cargo are moved 10 ft. from the centre to edge.

**Two Tons of Deck cargo are moved ten feet laterally.**Moment due to the movement of the load = The righting Couple. Thus:

- The
**metacentric height**is - The
**angle**