Stability of a floating body:

• The question of the stability of a body such as a ship which floats on the surface of a liquid is of importance. Whether the equilibrium is stable, neutral or unstable is determined by the height of its center of gravity, and in this experiment the stability of a pontoon may be determined with its center of gravity at various heights.
  

Apparatus Required:

• Metacentric height apparatus
  

Theory:

The arrangement of the apparatus is shown on Fig. 2.1. A plastic pontoon of a rectangular form floats in water and carries a single mast. From this a plumb-bob is suspended so that the angle of list of the pontoon may be read off a scale marked in degrees. The height of the center of gravity of the floating body may be varied by an adjustable weight which slides up and down the system. A jockey weight is arranged to slide along a bar fixed on the pontoon parallel to its base; as this weight is moved by known intervals, the change in angle of list is noted, and the stability of the pontoon thereby measured.

Fig. 2.1 Arrangement of floating pontoon.

Consider the rectangular pontoon shown floating in equilibrium on even keel as shown in cross- section on Fig. 2.2 (a). the weight of the floating body acts vertically downwards through its center of gravity G and this balanced by an equal and opposite buoyancy force acting upwards through the center of buoyancy B, which lies at the center of gravity of the liquid displaced by the pontoon.

Fig. 2.2. Derivation of stability of floating pontoon.

To investigate the stability of the system, consider a small angular displacement θ from the equilibrium position as shown in Fig. 2.2 (b). The centre of gravity of the liquid displaced by the pontoon shifts from B to B1. The vertical line of action of the buoyancy force is shown on the figure and intersects the extension of line BG in M, the metacentre.

The equal and opposite forces through G and B1 exert a couple on the pontoon, and provided that M lies above G (as shown in Fig. 2.2 (b)) this couple acts in the sense of restoring the pontoon to even keel, i.e. the pontoon is stable. If, however, the metacentre M lies below the centre of gravity G, the sense of the couple is to increase the angular displacement and the pontoon is unstable. The special case of neutral stability occurs when M and G coincide.

Fig. 2.2 (b) shows how the metacentric height GM may be established experimentally using the jockey weight to displace the center of gravity sideways from G. For suppose the jockey weight w is moved a distance x from its central position, and wt. of the whole floating assembly is W, then the corresponding movement of the centre of gravity of the whole, in a direction parallel to the base of the pontoon, is (w/W) x. If this movement produces a new equilibrium position at an angle of list θ, then in Fig. 2.2 (b), G1 is the new position of the centre of gravity of the whole, i.e.

GG1 = (w / W) . x (3.1)

Now, from the geometry of the figure,

GG1 = GM. tan θ

= G M. θ^c as θ (radians) is small (3.2)

From eq. (2.1) & (2.2),

metacentric height GM = (w/W) . x / θ^c (3.3)

Procedure:

Observations:

Total wt. of floating assembly, W= 1.690 kg

Jockey wt., w = 0.304 kg

Results & Comments:

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Introduction:

The Stability of any vessel which is to float on water, such as a pontoon or ship, is of paramount importance. The theory behind the ability of this vessel to remain upright must be clearly understood at the design stage. Archimedes’ principle states that the buoyant force has a magnitude equal to the weight of the fluid displaced by the body and is directed vertically upward. Buoyant force is a force that results from a floating or submerged body in a fluid which results from different pressures on the top and bottom of the object and acts through the centroid of the displaced volume.

Apparatus:

1. Flat bottomed pontoon
   
2. Hydraulic bench.
   
3. Water tank
   
4. Weights ( 491 gm and 995 kg)
   

Equipment set up:

The flat bottomed pontoon is constructed from non-ferrous materials and has a detachable bridge piece and loading system. Provision is made to alter the keel weight and the mast weight so obtaining a variety of loading conditions. For off balance loadings, the degree of list can be directly measured using the plumb-bob line attached to the mast and swinging over a scale mounted on the bridge piece. The floatation experiments can be carried out using the measuring tank of the hydraulics bench.

Floatation characteristics of flat bottomed pontoon. Depth = mm.

Length =mm.

Width =mm.

Distance from pontoon center line to added weight = 90mm.

Center of gravity of vessel with mast = 90 mm approximately from outer surface of vessel base. Weight of vessel with mast= 2.12 Kg.

Height of mast loading position above water surface of vessel base =mm.

Theory:

Consider a ship or pontoon floating as shown in figure 2. The center of gravity of the body is at Gand the center of buoyancy is at B . For equilibrium, the weight of the floating body is equal to the weight of the liquid it displaces and the center of gravity of the body and the centroid of the displaced liquid are in the same vertical line. The centroid of the displaced liquid is called the “center of buoyancy”. Let the body now be heeled through an angle θ, B1 will be the position of the center of buoyancy after heeling. A vertical line through B1 will intersect the center line of the body at M and this point is known as the metacenter of the body when an angle θ is diminishingly small. The distance GM is known as the metacentric height. The force due to buoyancy acts vertically up through B1 and is equal to W . The weight of the body acts downwards through G.

Stability of submerged objects:

Stable equilibrium: if when displaced, it returns to equilibrium position. If the center of gravity is below the center of buoyancy, a righting moment will produced and the body will tend to return to its equilibrium position (Stable). Unstable equilibrium: if when displaced it returns to a new equilibrium position. If the Centre of Gravity is above the centre of buoyancy, an overturning moment is produced and the body is (unstable). Note : As the body is totally submerged, the shape of displaced fluid is not altered when the body is tilted and so the center of buoyancy unchanged relative to the body.

Stability of floating objects:

Metacenter point M: the point about which the body starts oscillating.

Metacentric height GM: is the distance between the center of gravity of floating body and the metacenter.

If M lies above G a righting moment is produced, equilibrium is stable and GM is regarded as positive. If M lies below G an overturning moment is produced, equilibrium is unstable and is GM regarded as negative.

If M coincides with G, the body is in neutral equilibrium.

Determination of Metacentric height

1. Practically
   

Where x= distance from pontoon centerline to added weight.

W= weight of the vessel including .

Theoretically

This immersed volume, V, can be determined by calculation. Since the buoyancy force (upthrust) is equal to the total weight, W, of pontoon and its load then

The depth of immersion (di) can then be found from