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Moments of statical stability 125
The perpendicular distance between the lines of action of the forces (GZ) is called the righting lever. Taking moments about the centre of gravity, the moment of statical stability is equal to the product of the righting lever and the displacement, or:
Moment of statical stability W GZ
The moment of statical stability at a small angle
of heel
At small angles of heel the force of buoyancy may be considered to act vertically upwards through a ®xed point called the initial metacentre (M). This is shown in Figure 14.2, in which the ship is inclined to a small angle (y degrees).
Fig. 14.2
Moment of statical stability W GZ But in triangle GZM: GZ GM sin y
; Moment of statical stability W GM sin y
From this formula it can be seen that for any particular displacement at small angles of heel, the righting moments will vary directly as the initial metacentric height (GM). Hence, if the ship has a comparatively large GM she will tend to be `stiff', whilst a small GM will tend to make her `tender'. It should also be noticed, however, that the stability of a ship depends not only upon the size of the GM or GZ but also upon the displacement. Thus two similar ships may have identical GM's, but if one is at the light displacement and the other at the load displacement, their respective states of stability will be vastly different. The ship which is at the load displacement will be much more `stiff' than the other.
Example 1
A ship of 4000 tonnes displacement has KG 5.5 m and KM 6.0 m. Calculate the moment of statical stability when heeled 5 degrees.