Metacentric Height

 

Anil Chejara Roll No. 9003022 Group No. 4  

METACENTRIC HEIGHT BACKGROUND: Metacentre: It is defined as the point about which a body starts oscillating

when it is tilted by a small angle. It is defined as the point of intersection of symmetrical axis of floating body and the new line of action of force of Buoyancy.  METACENTRIC HEIGHT: It is the distance between centre of gravity (c) and Metacentre. It is

the measure if static stability of floating body. Large the metacentric height more is the stability. THEORY BEHIND EXPERIMENT:

For a body to be equilibrium W= Fb and both weight of the body (W) and the buoyant force (Fb) are acting along the same vertical v ertical line For a body in equilibrium in liquid surface the two forces gravity force (w) and buoyant force (Fb) must lie in the same vertical line and for stable equilibrium point M must be above G.

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The position of the Metacentre relative to the position of the centre of gravity of a floating body determines the stability of the floating body. 1.  Stable equilibrium: If the point M is above G, the floating body will be in stable equilibrium. 2.  Unstable equilibrium: If the point M is below G, the floating body will be in unstable equilibrium. 3.  Neutral equilibrium: If the point M is at the centre of the gravity of the body,

the floating body will be in neutral equilibrium.  Under equilibrium the moment moment caused by the movement movement of the unbalanced mass mass ‘w’ must be equal to moment caused by the shift of the centre of gravity from G to G1.

AIM: To determine the meta-centric height and position of the meta-centric height using angle of heel and ship model. APPRATUS USED:

  Water Tank   Weights of 1kg and half kg   floating body

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PROCEDURE:

  Make sure that the tank free from dust.

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Fill clean water and ensure that no foreign particles are there. Float the ship model in water check stable equilibrium. Apply the known weight (w) at the centre of model. Place the weight at different places and take readings. Measure the distance moved by the weight applied with the help of scale. Repeat the experiment for different weights.  

CALCULATIONS AND DERIVATION:

Let W be the weight of the Boat plus it's Load. A small load w is moved a distance x and causes a tilt of angle θ. The Boat is now in a new position of equilibrium with B1 and G1

lying along the Vertical through.

The Moment due to the movement of the load is given by

∗ ∗        =

 

=

 

 

 

 

 

 

Hence,



 =

   

 

Where, W = weight of body including w G = centre of gravity of body B = centre of buoyancy b uoyancy of the body M = meta-centre of the body w = applied weight x = distance moved by weight w θ= angle of tilt  DATA GIVEN: 

  Distance of grooves nos. 1, 2, 3, 4, from Centre = 2.5 cm, 5 cm, 7.5 cm,

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10 cm on either sides.

  Weight of ship model = 5.600 kg

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Weight of big strip = 2.640 kg

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Weight of small strips = 1.110 kg (2 No ’s),

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Weight of hanger = 0.144 Kg

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Applied weights = 500 grams and 1000 grams

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w = Weight of hanger + applied weight

W = Weight of ship model + weight of big strip + weight of small strip + w For external weight = 500gm S. No.

w (kg)

W (kg)

Distance, x (cm)

φ 

Meta-centric Height (cm)

1 2

0.644 0.644

11.104 11.104

2.500 5.000

24

0.326

29.5

0.513

3 4

0.644 0.644

11.104 11.104

7.500 10.000

33

0.670

34

0.860

5 6

0.644 0.644

11.104 11.104

-2.500 -5.000

-21

0.378

-29

0.523

7 8

0.644 0.644

11.104 11.104

-7.500 -10.000

-33.5 -34

0.658 0.860

For external weight = 1 kg S. No.

w (kg)

W (kg)

Distance, x (cm)

φ 

Meta-centric Height (cm)

1

1.144

11.604

2.500

8.5 

1.650

2

1.144

11.604

5.000

14 

1.978

3

1.144

11.604

7.500

18 

2.277

4

1.144

11.604

-2.500

20.5 

2.638

5

1.144

11.604

-5.000

-6 

2.346

6

1.144

11.604

-7.500

-12.5 

2.225

CONCLUSION

  Meta-centric height is observed by keeping the weights at different distance from

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the centre of the body which create different torque value in the body.

ERROR: 

  Experiment has slight oscillations while taking the readings.   Manual checking of protractor

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