https://file.aiquickdraw.com/mathbot/file/1726216639125au9y9w9f.jpg

Solved this problem using ship stabilty formula

To solve this problem using ship stability principles, we will follow these steps:

### Given Data:
- **Vessel Dimensions**:  
  Length \( L = 78 \, \text{m} \)  
  Breadth \( B = 12 \, \text{m} \)  
  Initial draft \( T_i = 4.0 \, \text{m} \)
  
- **Deep Tank amidships dimensions**:  
  Length \( L_{tank} = 14 \, \text{m} \)  
  Breadth \( B_{tank} = 12 \, \text{m} \)  
  Depth \( D_{tank} = 4.2 \, \text{m} \)
  
- **Cargo Density**: \( \rho = 1.050 \, \text{T/m}^3 \)

- **Stowage Factor (SF)**: \( 1.5 \, \text{m}^3/\text{T} \)

- **Water Density (Seawater)**: \( 1.025 \, \text{T/m}^3 \) (standard for seawater)

### Step 1: Calculate the Displacement of the Ship
The displacement ( \( \Delta \) ) of the vessel in seawater is given by:
\[
\Delta = L \times B \times T_i \times \text{Water density}
\]
Substitute the values:
\[
\Delta = 78 \times 12 \times 4.0 \times 1.025 \, \text{T}
\]
This will give the initial displacement of the vessel.

### Step 2: Volume and Weight of the Cargo in the Tank
The volume of the tank is:
\[
V_{\text{tank}} = L_{\text{tank}} \times B_{\text{tank}} \times D_{\text{tank}}
\]
Substitute the values:
\[
V_{\text{tank}} = 14 \times 12 \times 4.2 \, \text{m}^3
\]

Now, to find the weight of the cargo in the tank, we use the cargo density:
\[
W_{\text{cargo}} = \frac{V_{\text{tank}}}{\text{SF}} \times \rho
\]

### Step 3: Total Displacement after Cargo Loading
The final displacement \( \Delta_{\text{final}} \) will be:
\[
\Delta_{\text{final}} = \Delta + W_{\text{cargo}}
\]

### Step 4: Final Draft Calculation
To calculate the final draft \( T_f \), use the displacement formula again with the final displacement:
\[
T_f = \frac{\Delta_{\text{final}}}{L \times B \times \text{Water density}}
\]

Now let's calculate everything step by step to find the final draft. I'll proceed with the calculations.### Results:
1. **Initial Displacement of the Vessel**:  
   \( \Delta = 3837.60 \, \text{T} \)
   
2. **Weight of the Cargo in the Tank**:  
   \( W_{\text{cargo}} = 493.92 \, \text{T} \)

3. **Final Displacement of the Vessel**:  
   \( \Delta_{\text{final}} = 4331.52 \, \text{T} \)

4. **Final Draft of the Vessel**:  
   \( T_f = 4.51 \, \text{m} \)

So, the final draft of the vessel after the cargo is loaded will be approximately **4.51 meters**.

Would you like more details or have any questions?

Here are some related questions for further exploration:
1. How does the stowage factor affect the total displacement of the vessel?
2. What would the final draft be if the cargo density were different, say \( 1.2 \, \text{T/m}^3 \)?
3. How would the draft change if the length or breadth of the vessel increased?
4. What are the key factors influencing the stability of a vessel when loading cargo?
5. How does the water density (seawater vs. freshwater) affect the draft?

**Tip**: Always ensure the cargo's weight is well distributed to maintain the vessel's stability and prevent excessive heel or trim.

A box-shaped vessel 78 m length, 12 m breadth, floats at 4.0 m in seawater. A deep-tank amidships (14 m x 12 m x 4.2 m) cargo density = 1.050 T/m³, SF 1.5 m³/T on the centerline is bilged. Find the final draft.

Learn how to calculate the final draft of a vessel after cargo is loaded in a deep tank amidships. This problem covers hydrostatic principles and displacement calculations, using formulas for weight and buoyancy.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation