Question

Consider a two-dimensional ship section as shown in the figure. About the point \( O \), let the sway added mass components be \( a_{22} \) and \( a_{24} \) and roll added moment of inertia be \( a_{44} \). The clockwise roll angle is considered positive. The roll added mass due to roll, about \( P \), which is at a distance \( z_p \) above \( O \), is given by: 

• A. \( a_{44} - a_{24}z_P \)
• B. \( a_{44} - a_{22}z_P - a_{24}z_P^2 \)
• C. \( a_{44} - a_{22}z_P^2 + a_{24}z_P \)
• D. \( a_{22} + a_{24} + a_{44} \)

Hint:

For added mass problems, always consider the physical contributions from roll moment and coupling terms. Ensure the distance \( z_P \) is properly incorporated based on the system's geometry.

Answer 1

The Correct Option is A

Step 1: Understand the roll added mass about a point \( P \).
The roll added mass about a point \( P \), located at a distance \( z_P \) from the reference point \( O \), depends on the roll added moment of inertia (\( a_{44} \)) and the coupling term (\( a_{24} \)) related to the distance \( z_P \). Step 2: Derive the expression for roll added mass about \( P \).
The general formula for the roll added mass about \( P \) is given as: \[ \text{Roll added mass} = a_{44} - a_{24}z_P, \] where: - \( a_{44} \): Roll added moment of inertia about \( O \), - \( a_{24} \): Coupling term between sway and roll, scaled by the distance \( z_P \). Step 3: Analyze the options.
Option (A): Correct, as it matches the derived expression for roll added mass about \( P \), \( a_{44} - a_{24}z_P \).
Option (B): Incorrect, as it includes an unnecessary sway term (\( a_{22} \)) and a quadratic term (\( z_P^2 \)) which do not belong to the roll added mass formulation.
Option (C): Incorrect, as it incorrectly modifies the sign of the terms and introduces \( z_P^2 \).
Option (D): Incorrect, as it sums all components without considering their specific relationship to \( P \).
Conclusion: The roll added mass due to roll about \( P \) is \( a_{44} - a_{24}z_P \).