Question

Comparing a catamaran (with a separation between demi-hulls) and a mono-hull craft of the same displacement and water plane area, the initial metacentric radius of the catamaran will be

• A. same as that of the mono-hull
• B. one-half of the mono-hull
• C. greater than that of the mono-hull
• D. one-third of the mono-hull

Hint:

Think of stability like balancing. A catamaran is like a person standing with their feet wide apart, while a monohull is like a person with their feet together. The wide stance (separated hulls) provides a much more stable base, which in naval architecture translates to a larger \(I_T\) and greater initial stability.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The question asks to compare the initial metacentric radius (BM) of a catamaran and a monohull. The metacentric radius is a measure of the initial stability of a floating body. It represents the vertical distance between the center of buoyancy (B) and the initial metacenter (M).
Step 2: Key Formula or Approach:
The transverse metacentric radius is calculated using the formula: \[ BM_T = \frac{I_T}{\nabla} \] where:
- \(I_T\) is the second moment of area (or moment of inertia) of the ship's waterplane area about its longitudinal centerline.
- \(\nabla\) is the displaced volume of water (related to displacement by \( \Delta = \rho \nabla \)). We are given that the displacement (and thus \(\nabla\)) and the total waterplane area are the same for both the catamaran and the monohull. The key difference lies in how that area is distributed, which affects \(I_T\).
Step 3: Detailed Analysis:
Let's consider the calculation of \(I_T\).
- For a monohull, the waterplane area is a single shape, and \(I_T\) is calculated about its centerline.
- For a catamaran, the total waterplane area is split into two separate demi-hulls. Let the area of each demi-hull be \(A_h\), so the total area is \(A = 2A_h\).
Let the centerline of each demi-hull be separated by a distance 's' from the overall vessel centerline. The total second moment of area for the catamaran is found using the parallel axis theorem: \[ I_{T,cat} = I_{T,demi-hull-1} + I_{T,demi-hull-2} \] Applying the parallel axis theorem to each demi-hull (where \(I_{c}\) is the second moment of the demi-hull's area about its own centroidal axis): \[ I_{T,cat} = (I_{c1} + A_h . s^2) + (I_{c2} + A_h . s^2) \] \[ I_{T,cat} = 2I_c + 2A_h s^2 = 2I_c + A s^2 \] The term \(I_{T,mono} = I_{T,total}\) is the second moment of the total area \(A\) if it were a single shape. The catamaran's total \(I_T\) is \(2I_c\), which is the sum of the individual moments of inertia of the two demi-hulls about their own centerlines, plus a large transfer term \(A s^2\).
Because the two demi-hulls are separated by a significant distance (\(s>0\)), the term \(A s^2\) is large and positive. This makes the total second moment of area for the catamaran, \(I_{T,cat}\), substantially larger than the second moment of area for a monohull, \(I_{T,mono}\), of the same total waterplane area.
Since \(BM = I_T / \nabla\), and \(\nabla\) is the same for both vessels, the vessel with the larger \(I_T\) will have the larger BM. Therefore, \(I_{T,cat} \gg I_{T,mono}\), which implies \(BM_{cat} \gg BM_{mono}\). The metacentric radius of the catamaran will be greater than that of the mono-hull.
Step 4: Why This is Correct:
The large separation between the catamaran's demi-hulls dramatically increases the second moment of area of the waterplane due to the parallel axis theorem. With the same displacement volume, this directly results in a much larger metacentric radius (BM) and consequently a much larger initial metacentric height (GM), which is why catamarans are known for their very high initial stability.