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. It is the distance between center of gravity (c) and Metacenter. It is the measure if static stability of floating body. Large the metacentric height more is the stability. 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.
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Suppose that the body now turns through a small angle about the -axis.
As is easily demonstrated, the body’s new draft becomes. Hence, the new submerged volume is.
Stability and Metacentric Height
It follows, from the previous analysis, that if the floating body under consideration turns through a small angle geight the -axis then its center of buoyancy shifts horizontally a distance in the plane perpendicular to the axis of rotation. Simple trigonometry reveals that assuming that is small.
Now, is the height of the metacenter relative to the center of buoyancy. However, the center of buoyancy lies a depth below the surface of the water which corresponds to the plane.
Hence, the -coordinate of the metacenter is.
Finally, if and are the -coordinates of the unperturbed centers of gravity and buoyancy, respectively, then. A simple extension of the previous argument reveals that if the body turns through a small angle about the -axis then the metacentric height is. As an example, consider a uniform rectangular block of specific gravity floating such that its sides of length, and are parallel to the – – and -axes, respectively.
To Determine The Metacentric Height Of a Ship Model
Such a block can be thought of as a very crude model of a ship. The volume of the block is. Hence, the submerged metadentric is. The cross-sectional area of the block at the waterline is. It is easily demonstrated that.
Thus, the center of buoyancy lies a depth below the surface of the water. Hence, and.
Consider the stability of the block to small amplitude angular displacements about the -axis. The metacentric height of a conventional ship whose length greatly exceeds its width is typically much less for rolling i.
For this reason, maritime regulatory agencies, such as the International Maritime Organization, specify minimum metacentric heights for various different types of sea-going vessel. A relatively large metacentric height, on the other hand, generally renders a ship uncomfortable for passengers and crew, because the ship executes short period rolls, resulting in large g-forces.
Such forces also increase the risk that cargo may break loose or shift. We shall now demonstrate that these positions are either metacrntric stable, or both unstable, provided the body is of uniform density.
Determination of Metacentric Height
Let and be the volumes that are above and below the waterline, respectively, in the first position. Let and be the mean centers of these two volumes, and that of the whole volume.
Jeight follows that is the center of buoyancy in the first position, the center of buoyancy in the second inverted position, and the center of gravity in both positions.