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# METACENTRIC HEIGHT

## Ex. No: DATE:

AIM:

To determine the Metacentric height of a floating body (i.e. a model of ship).

APPARATUS REQUIRED:

1. Metacentric height apparatus
2. Water tank

BASIC THEORY

A floating body is generally called pontoon. The body is stable if, when displaced, it
returns to its equilibrium position. A small rotation can result in either a restoring or
overturning couple. For a completely immersed body, if the center of gravity (G) is below
the center of buoyancy (B), a rotation from its equilibrium position will create a restoring
couple and the body rotates back to its original position. Thus, it is stable. If the center of
gravity is above the center of buoyancy, the resulting couple of weight and buoyancy
force will overturn the body.
For floating bodies, if the center of gravity lies above center of buoyancy, it may be either
stable or unstable depending on whether the couple is restoring or overturning.

Buoyancy
It is the tendency of fluid to lift a submerged body.

Buoyancy force (FB)
It is the resultant upward force exerted by a fluid on a submerged body. According to
Archimedes principle,
FB = weight of the fluid displaced by the body and it acts through the centroid of the
displaced volume.

Center of gravity (G)
It is the point where the weight of the body is acting.

Center of buoyancy (B)
It is the point at which force of Buoyancy is acting. Center of Buoyancy is the centroid of
liquid displace by the body. In 2-D case, we can say it is the center of area of immersed
section.

Metacenter (M)
It is the point of intersection of symmetrical axis of floating body and the new line of
action of force of Buoyancy.

Metacentric height (GM)

It is the distance between center of gravity (G) and Metacenter (M). It is the measure of
static stability of floating body. Larger the metacentric height more is the stability. Very
large Metacentric height means shorter period of rolling that would be uncomfortable for
the passengers in the ship. That is why ship meant for passengers are designed for
relatively small metacentric height, (but sufficient enough to give comfort), so that both
stability and comfort can be provided.

SPECIFICATIONS:

Internal plan area of tank, A = 36 cm 36 cm
1
= 0.20 kg
Hanging load, m = 0.10 kg (each)

FORMULA USED:
mX
Metacentric height = ---------
W tan

where, m = hanging load (kg)
X = distance through which the hanging is moved (m)
= angle through which the pontoon is tilted ()
W = w
1
+ w
0
+ m (kg)
w
1
w
o
= weight of the floating body (area h density of water)
Area = internal plan area of the collecting tank (m
2
)
h = rise in water level (m)

PROCEDURE:

1. Note the initial height of the water in the vessel (h
1
) before placing the ship.
2. Place the model ship in the water.
3. Adjust the counter weight to keep the ship in the equilibrium.
4. Note the height of the water level (h
2
).
5. Now hang the hook gauge at any one side.
6. Place the weight on the hook.
7. Note down the reading shown by the needle.
8. By moving the hook gage to X away from the center of the ship by horizontally,
observe the needle position.
9. Repeat the above procedure for the different distances and hanging loads, note down
10. The metacentric height of the ship is calculated.

OBSERVATIONS:

Initial height, h
1
=
Final height, h
2
=
Difference in height, h =
TABULATIONS:
1
= 0.2 kg
Sl.
No.
m (kg)
X
(m)

(degrees)
Metacentric height
H (m)
1.
2.
3.
4.
5.

Mean =

MODEL CALCULATIONS:

RESULT:

The Metacentric height of a floating body is ___________.
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