Metacentric Height

IIT GANDHINAGAR

METACENTRIC HEIGHT FLUID MECHANICS LAB REPORT Ravi Agarwal (09003017)

OBJECTIVE: To study force balances in a static system. AIM: To determine the meta-centric height and position of the meta-centric height with angle of heel of ship model. APPARATUS REQUIRED:  Water tank  Ship model (floating body)  Weights ( 491 gm and 995 kg) INTRODUCTION AND THEORY: The metacentric height is a measurement of the static stability of a floating body. It is calculated as the distance between the centre of gravity of a ship and its metacentre (GM). A larger metacentric height implies greater 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. Metacentre is defined as the point about which a body starts oscillating when the body is tilted by a small angle. The meta- centre may also be defined as the point at which the line of action of the force of buoyancy will meet the normal axis of the body when the body is given a small angular displacement .It is denoted by M. The distance between the meta-centre (M) of a floating body and the centre of gravity (G) of the body is called meta-centric height (GM). For a body to be in equilibrium on the liquid surface the two forces gravity force (w) and buoyant force (Fb) must lie in the same vertical line. If the point M is above G, the floating body will be in stable equilibrium. If slight angular displacement is given to the floating body in clockwise direction, the centre of buoyancy shifts from B to B1 such that the line of action of Fb through B1 cuts the axis at M, which is called the meta – centre and the distance GM is called the meta-centric height. The buoyant force Fb through B1 and weight W through G constitute a couple acting in anti- clockwise direction and thus bringing the floating body in the original position.

DETERMINING META-CENTRIC HEIGHT EXPERIMENTALLY: Let W be the weight of the Boat plus its Load. A small load w is moved a distance ‘x’ 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 wx. And moment due to the movement of G= W x GM tan EQUATING BOTH, we get

PROCEDURE:  Make the tank free from dust.  Fill tank ¾ with clean water and ensure that no foreign particles are there.  Weigh the ship model to find W.  Float the ship model in water and ensure that it is in stable equilibrium.  Apply the known weight (w) at the center of model.  Give the model a small angular displacement in clockwise direction.  Measure the distance moved by the weight applied with the help of scale.  Measure the angle of tilt on the graduated arc.  Repeat the experiment for different weights. OBSERVATIONS:        

Distance of grooves nos. 1, 2, 3, 4, from centre = 2.5 cm, 5 cm, 7.5 cm, 10 cm. Weight of ship model = 5.600 kg Weight of big strip = 2.640 kg Weight of small strips = 1.110 kg (2 No’s), Weight of hanger = 0.144 Kg Applied weights = 491 grams and 995 grams w = Weight of hanger + applied weight W = Weight of ship model + weight of big strip + weight of small strip + w

FOR 491 grams Sr No.

w (Kg)

W (Kg)

Distance x (cm)

Tilt (θ) (°C)

Metacentric Height (cm)

1

0.635

11.095

2.5

10

0.81

2

0.635

11.095

5

14

1.15

3

0.635

11.095

7.5

19

1.25

4

0.635

11.095

10

21

1.49

5

0.635

11.095

-10

20

1.57

6

0.635

11.095

-7.5

17

1.40

7

0.635

11.095

-5

14

1.15

8

0.635

11.095

-2.5

10

0.81

FOR 995 grams Sr No.

w (Kg)

W (Kg)

Distance x (cm)

Tilt (θ) (°C)

Metacentric Height (cm)

1

1.139

11.599

2.5

25

0.53

2

1.139

11.599

5

30

0.85

3

1.139

11.599

7.5

33

1.13

4

1.139

11.599

10

35

1.40

5

1.139

11.599

-10

37

1.30

6

1.139

11.599

-7.5

33

1.13

7

1.139

11.599

-5

29

0.89

8

1.139

11.599

-2.5

24

0.55

SOURCES OF ERROR:  The error in the measurement of angle because of the setup could not be brought to rest completely.  Error in noting down the exact angle. CONCLUSIONS: Force balancing of a system floating in water was studied and the position of the meta-centric height was determined for different values of load.