To determine experimentally the metacentric height of a ship model

Objective:

To determine metacentric height of floating body

Equipement

Introduction and Theory:

Whenever a body, floating in a liquid is given a small angular displacement, it starts oscillating about the same point, this point about which the body starts oscillating is called metacentre. The distance between the center of gravity of a floating body and the metacentre is called metacentre height. As a matter of fact the metacentre height of a floating body is a direct measure of its stability or in otherwards more the metacentric height of a floating body more it will be stable. A body is said to be in equilibrium when it remains in a steady state, while floating in a liquid

Figure: Metacenter and Metacentric Height. FA=Buoyancy Force, FG=Dead weight. S = Center of gravity, A = center of buoyancy, M = metacenter location, zm = distance between center of gravity and metacenter

Following are the three conditions of equilibrium of a

floating body,

1. Stable equilibrium
2. Unstable equilibrium and
3. Neutral equilibrium

Stable Equilibrium:A body is said to be in a stable equilibrium if it returns back to its original position when given a small angular displacement. This happens when the metacentre is higher than the center of gravity of the floating body.

Unstable Equilibrium:A body is said to be in an unstable equilibrium if it does not return back to its original position and heels further away when given a small angular displacement. This happens when the metacentre is lower than the center of gravity of the floating body.

Neutral Equilibrium:A body is said to be in a Neutral Equilibrium if it occupies a new position and remains rest in this new position when given a small angular displacement. This happens when the metacentre coincides with the centre of gravity of the floating body. In the experimental set up the variation of metacentric height for different types of loading of a floating vessels can be determined.8 When a floating body is tilted through a small angle its centre of buoyancy will be shifted to a new position the point of intersection of the vertical line drawn through the new centre of buoyancy and centre of buoyancy is called metacentric height.

Experimental Setup:The experimental setup essentially consists of a pontoon and a water tank as a floating vessel. The rectangular pontoon is tilted with a vertical sliding wight to permit adjustment of the height of the center of gravity and a horizontal sliding weight to generate a defined as heeling moment. The sliding weights can be fixed in position using knurled screws. The positions of the sliding weights and draught of the pontoon can be read off scales. A heel indicator with scale in degrees is also provided.

Weight of the model (without horizontal and vertical weights) = 1325g

Horizontal weight = 200g

Vertical weight = 500gm

Vertical height = 63mm

Experimental Procedure:

Calculation of center of gravity

The first step is to determine the position of the overall center of gravity xs, zs from the set position of the sliding weights

The horizontal position is referenced to the center line:

xs = mh.x/(m+mv+mh)

where,

mh =. Vertical sliding mass

m = total mass (not including sliding mass)

x = distance of sliding weight from the center

The vertical position is referenced to the underside of the floating body.

Zs = (mvz+m+mhzg)/(m+mv+mh)

Zg = vertical center of gravity without the sliding weight

Zs = vertical center of gravity

Stability gradient

Dxs/dalpha = xs/alpha

Performance of the experiment:

Set horizontal sliding weight to position x = 8cm

Move vertical sliding weight to the bottom position

Fill the tank provided with water and insert the floating body

Gradually raise the vertical sliding weight and read off angle on heel indicator. Read off height of the sliding weight at the top edge of the weight and enter in table together with angle