Angle of loll is a condition which occurs to a ship in its unstable equilibrium. In an unstable equilibrium, the metacentric height(GM) is negative. That is, the position of metacenter is is below the centre of gravity. In this condition, when a ship heels a capsizing moment is created and it tends the ship to heel further. This is a dangerous situation and may result in the capsizing of the vessel. For reference, States of equilibrium.
For small angles of heel( almost to12 degrees heel) KM (vertical distance from keel to metacenter) is considered as a constant. For small angles the metacenter is considered as a fixed point. For large angles of heel(over 12 degrees) we cannot assume that B (centre of buoyancy) follows an arc of a circle. Here the vertical movement of B is faster than the horizontal movement of B. Hence B does not follow the arc of a circle. So for each angle of heel the metacenter changes and is not a fixed point. The moving metacenter at large angles of heel is called the Pro-metacenter.
Unlike for the small angles of heel, the metacenter is not a fixed point for large angles of heel. Hence KM is not considered as constant for large angles of heel. In unstable equilibrium, when the ship continuously heel to large angles the KM increases, that is Metacenter moves upwards. At a certain point the Metacenter will reach the exact same point as the centre of gravity (GM=0). Here the ship attains a neutral equilibrium at a particular angle of heel. The angle at which this occurs is called Angle of loll.
The formula for angle of loll can be derived from the wall-sided formula( formula for righting lever in large angles of heel)
GZ= sin x [GM + 0.5 BM tan2 x]
At the angle of loll, there is no righting moment. so GZ=0
Hence,
sin x [GM + 0.5 BM tan2 x]=0
That is either sin x = 0 or [GM + 0.5 BM tan2 x]=0
But, sin x cannot be zero since the angle of loll is not zero.
Therefore, [GM + 0.5 BM tan2 x]=0
So, tan2 x = -2GM/BM
Where, x= angle of loll
GM= Initial GM
BM= Metacentric height when upright.
This formula is used to find the angle of loll .
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