Page 97 - statbility for masters and mates

Simpson's Rules for areas and centroids 85
Fig. 10.14
and g by the ®rst rule and the area between the half-ordinates g and h by the `®ve-eight' rule.
In practice, the naval architect divides the ship's length into 10 stations and then subdivides the forward and aft ends in order to obtain extra accuracy with the calculations.
In doing so, the calculations can be made using Simpson's First and Second Rules, perhaps as part of a computer package.
Centroids and centres of gravity
To ®nd the centre of ¯otation
The centre of ¯otation is the centre of gravity or centroid of the water-plane area, and is the point about which a ship heels and trims. It must lie on the longitudinal centre line but may be slightly forward or aft of amidships (from say 3 per cent L forward of amidships for oil tankers to say 3 per cent L aft of amidships for container ships).
To ®nd the area of a water-plane by Simpson's Rules, the half-breadths are used as ordinates. If the moments of the half-ordinates about any point are used as ordinates, then the total moment of the area about that point will be found. If the total moment is now divided by the total area, the quotient will give the distance of the centroid of the area from the point about which the moments were taken. This may be shown as follows:
In Figure 10.15:
Fig. 10.15
Area of strip yEdx L
Area of the 12 WP
yEdx
O