Question

In the following "GZ (righting lever arm)" versus "angle of heel" curve, the point 'X' indicates 

 

• A. angle of loll
• B. angle of vanishing stability
• C. deck edge immersion angle
• D. trim angle

Hint:

Remember the key features of a GZ curve:
- {Initial slope:} Proportional to GM. If the slope is negative, GM is negative.
- {Negative GZ near origin:} Indicates an angle of loll.
- {Maximum GZ:} Angle of maximum righting moment.
- {GZ becomes zero again:} Angle of vanishing stability (capsize point).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The curve shown is a stability curve (or GZ curve) for a ship. It plots the righting lever arm (GZ) against the angle of heel (\(\theta\)). The righting lever is the horizontal distance between the forces of gravity (acting downwards through the center of gravity, G) and buoyancy (acting upwards through the center of buoyancy, B). The righting moment, which acts to restore the ship to an upright position, is \(M = \Delta \times GZ\), where \(\Delta\) is the ship's displacement. - Positive GZ: Indicates a positive righting moment that will return the ship to upright. - Negative GZ: Indicates a capsizing moment that will cause the ship to heel further.
Step 2: Detailed Analysis of the Curve and Point X:
- The curve starts at GZ=0 for \(\theta=0\) (upright).
- For small angles of heel, the GZ becomes negative. This means the ship has an initial negative metacentric height (GM). A ship with a negative GM is unstable when upright.
- If disturbed, the ship will heel over to one side. As it heels, the GZ value increases from its negative value, eventually reaching GZ=0 at a certain angle of heel, marked by point 'X'.
- At this angle 'X', the righting moment is zero, and the ship will come to rest in a stable, heeled position. This angle of static heel, caused by an initial negative GM, is called the angle of loll.
- If the ship is heeled further, a positive GZ is generated, creating a righting moment that will try to return the ship to the angle of loll.
Now let's evaluate the options:
- (A) angle of loll: This is the angle of heel at which a ship with an initial negative GM finds a stable equilibrium. Point X, where the GZ curve crosses from negative to positive, perfectly represents this.
- (B) angle of vanishing stability: This is the angle where the GZ curve crosses back to zero after reaching its maximum positive value. At this angle, the righting moment vanishes, and any further heel will lead to capsizing. This point is much further along the curve.
- (C) deck edge immersion angle: This is the angle of heel at which the edge of the main deck first touches the water. It is typically marked on the GZ curve, often corresponding to a point of inflection where the rate of increase of GZ changes, but it is not point X.
- (D) trim angle: Trim refers to the difference in draft between the bow and stern (a longitudinal inclination), whereas heel is a transverse (side-to-side) inclination. The GZ curve is for transverse stability.
Step 3: Why This is Correct:
The negative GZ at small angles indicates initial instability. Point X is the first non-zero angle where GZ becomes zero again, defining a new stable equilibrium position. This is the definition of the angle of loll.