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OBJICTIVE :To determine the metacentric height. The position of the so -called metacentric, the metacentric height is of crucial significance to the stability of a floating body. The metacentric height is an essential factor when assessing the of a ship in waves stability.

APPARATUS :

i. Water tankii. Metacentric Height Apparatus

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The unit shown in Fig. 1 consists of a pontoon (1) and a water tank as float vessel. The rectangular pontoon is fitted with a vertical sliding weight (2) which permits adjustment of the height of the centre of gravity and a horizontal sliding weight (3) that generates a defined tilting moment. The sliding weights can be fixed in any positions using knurled

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screws. The positions (4, 5) of the sliding weights and the draught (6) of the pontoon can be measured using the scales. A heel indicator (7) with scale in degrees is also provided.

THEORY :

3.1 Buoyancy

Fig 3.1 Buoyancy

A body floats in liquid if the buoyancy of the fully immersed body is greater than its weight. It will only sink into the liquid until the buoyancy FA correspond exactly to its dead weight FG. The buoyancy is the weight of the water displaced by the body. The centre of gravity of the displaced water mass is referred to as the centre of buoyancy A. The centre of gravity of the body is known as the centre of mass S.

3.2 Stability of Floating Body

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Figure 3.2- Metacentre and metacentric height

For a floating body to be stable buoyancy FA and dead weight FG must have the same line of action and be equal and opposite (Fig;3.1). Stability does not necessarily demand that the centre of mass S be below the centre of buoyancy A. Of far greater importance is the existence of a stabilising, resetting moment in the event of deflection or heel α out of the equilibrium position. Dead weight FG and buoyancy FA then form a force couple with distance b which provides a righting moment. This distance or the distance between the centre gravity and the point of intersection of line of action buoyancy and gravity axis is a measure of stability. This point of intersection is referred to as the metacentre M and the distance between the centre of gravity and the metacentre is called metacentric height Zm.

The following conditions the apply to stable floating :-

Stable floating of a body occurs when the metacentric height Zm is positive ,i.e the metacentre M is above the centre of gravity S (fig3.3.)

Zm > 0

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Unstable floating of a body occurs when the metacentric height Zm is negative , ie. The metacentre M is below the centre of gravity S (fig3.3.)

Zm < 0

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3.3 Determine of Metacentre Position

The position of the metacentre is not gorverned by the position of the centre of gravity. It merely on the shape of the portion of the body under water and the displacement. There are two method of determining the position by way of experiment. In the first method , the centre of gravity is laterally shifted by a certain constant distance Xs using an additional weight , thus causing heeling to occur. Further vertical shifting of the centre of gravity alters the heel α . A stability gradient formed the derivation dXs/dα is then defined. The stability gradient decreases as the vertical centre of gravity position approaches the metacentre. If centre of gravity position and metacentre coincide the stability gradient is equal to zero and the system is metastable. This problem is most easiy solved using graph (fig3.4). The vertical centre of gravity position is plotted versus the stability gradient. A curve is drawn through the measurement points and extended as far as the vertical axis then gives the position wuth the vertical axis the gives the position of the metacentre. With the second method of determining the metacentre it is assumed that given a stable intersection of this line of action with the central axis gives he metacentre M(fig3.5). The heel angle α and the lateral displacement of the centre of gravity Xs yield the following for the metacentric height Zm.

Zm = Xs cot α

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Figure 3.4- Graphical determination of metacentre

The first step is to determine the position of the overall centre of gravity Xs , Zs from the setting positions of the sliding weights. The horizontal position is referenced to the centre line:

The vertical position is refenced to the underside of the floating body :-

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Fig.4.1- Position and size of sliding weights

And the stability gradient is:

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PROCEDURES

1. The horizontal sliding weight was setting to position x=8cm.

2. The vertical sliding weight moved to bottom position.

3. Fill tank provided with water and the floating body inserted.

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4. The vertical sliding weight rose gradually and took the reading of angle on heel

indicator. Recorded the reading of height of sliding at top edge of weight and an

angle when the floating body stop moving after it inserted into a water tank.

5. The step 1 until 4 is repeated by replaced x=6cm.

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QUESTION

1. The stated formula are used to calculate the centre of gravity position and stability gradient

and plot them on a graph.

horizontal position of centre of gravity Xs = 0.055x= 0.44

i) the centre of gravity position

z = 3 cm

formula:

Zs = mvz + (m+mn)zg/m+mv+mn

= 5.364 + 0.156 z

= 5.364 + 0.156 (3)

= 5.832 cm

stability gradient,

xs = 0.44

α = 12

formula : dxs/dα = xs/α

= 0.44/12

= 0.037 cm

ii) the centre of gravity position

z = 5 cm

formula:

Zs = mvz + (m+mn)zg/m+mv+mn

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= 5.364 + 0.156 z

= 5.364 + 0.156 (5)

= 6.144 cm

stability gradient,

xs = 0.44

α = 15

formula : dxs/dα = xs/α

= 0.44/15

= 0.029

iii) i) the centre of gravity position

z = 7 cm

formula:

Zs = mvz + (m+mn)zg/m+mv+mn

= 5.364 + 0.156 z

= 5.364 + 0.156 (7)

= 6.456 cm

stability gradient,

xs = 0.44

α = 18

formula : dxs/dα = xs/α

= 0.44/14

= 0.024

iv) the centre of gravity position

z = 9 cm

formula:

Zs = mvz + (m+mn)zg/m+mv+mn

= 5.364 + 0.156 z

= 5.364 + 0.156 (9)

= 6.768 cm

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stability gradient,

xs = 0.44

α = 21

formula : dxs/dα = xs/α

= 0.44/21

= 0.021

horizontal position of centre of gravity Xs = 0.055x=0.33

i) the centre of gravity position

z = 3 cm

formula:

Zs = mvz + (m+mn)zg/m+mv+mn

= 5.364 + 0.156 z

= 5.364 + 0.156 (3)

= 5.832 cm

stability gradient,

xs = 0.33

α = 11

formula : dxs/dα = xs/α

= 0.33/11

= 0.03

ii) the centre of gravity position

z = 5 cm

formula:

Zs = mvz + (m+mn)zg/m+mv+mn

= 5.364 + 0.156 z

= 5.364 + 0.156 (5)

= 6.144 cm

stability gradient,

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xs = 0.33

α = 13

formula : dxs/dα = xs/α

= 0.33/13

= 0.025

iii) i) the centre of gravity position

z = 7 cm

formula:

Zs = mvz + (m+mn)zg/m+mv+mn

= 5.364 + 0.156 z

= 5.364 + 0.156 (7)

= 6.456 cm

stability gradient,

xs = 0.33

α = 14

formula : dxs/dα = xs/α

= 0.33/14

= 0.024

iv) the centre of gravity position

z = 9 cm

formula:

Zs = mvz + (m+mn)zg/m+mv+mn

= 5.364 + 0.156 z

= 5.364 + 0.156 (9)

= 6.768 cm

stability gradient,

xs = 0.33

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α = 17

formula : dxs/dα = xs/α

= 0.33/17

= 0.019

2. Determine and show the buoyant force of body (weight of metacentric height apparatus) is

equal the weight of water displaced.

The metacentric height (GM) is a measurement of the initial static stability of a floating

body. It is calculated as the distance between the centre of gravity of a ship and its

metacentre. A larger metacentric height implies greater initial stability against overturning.

Metacentric height also has implication on the natural period of rolling of a hull, with very

large metacentric heights being associated with shorter periods of roll which are

uncomfortable for passengers. Hence, a sufficiently high but not excessively high

metacentric height is considered ideal for passenger ships.

When a ship is heeled, the centre of buoyancy of the ship moves laterally. It may also move

up or down with respect to the water line. The point at which a vertical line through the

heeled centre of buoyancy crosses the line through the original, vertical centre of buoyancy

is the metacentre. The metacentre remains directly above the centre of buoyancy by

definition.

GB

Hh

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