A rectangular open box, 7.6m by 3m in plan and 3.7m deep, weighs 350KN and is launched in fresh water. (a) How deep will it sink? (b)If the water is 3.7m deep, what weight of stone placed in the box will cause it to rest on the bottom?

A rectangular open box, 7.6m by 3m in plan and 3.7m deep, weighs 350KN and is launched in fresh water.

(a) How deep will it sink?

(b)If the water is 3.7m deep, what weight of stone placed in the box will cause it to rest on the bottom?

Buoyancy Analysis of Rectangular Box

Problem Statement

A rectangular open box with dimensions:

Length: 7.6m
Width: 3m
Depth: 3.7m
Weight: 350 kN

The box is launched in fresh water. Determine:

How deep it will sink.
The additional weight of stone required to sink it completely when the water depth is 3.7m.

Solution

1. Calculate Depth of Immersion (\( h \))

Using the buoyancy principle: \[ \text{Weight of box} = \text{Weight of displaced water} \] \[ 350000 = 9810 \times 7.6 \times 3 \times h \] Solving for \( h \): \[ h = \frac{350000}{9810 \times 7.6 \times 3} \] \[ h = 1.56 \text{ m} \]

2. Determine Weight of Stone to Sink the Box Completely

\[ \text{Total weight required} = \text{Weight of water displaced at full depth} \] \[ 350000 + W_{\text{stone}} = 9810 \times 7.6 \times 3 \times 3.7 \] Solving for \( W_{\text{stone}} \): \[ W_{\text{stone}} = 9810 \times 7.6 \times 3 \times 3.7 – 350000 \] \[ W_{\text{stone}} = 477572 \text{ N} = 477.57 \text{ kN} \]

Final Results:

Depth of immersion: 1.56 m
Weight of stone required: 477.57 kN

Explanation

1. Floating Equilibrium:
A floating object displaces a volume of liquid equal to its own weight. The box will sink until it displaces enough water to balance its own weight.

2. Depth of Immersion Calculation:
The weight of the box is given, and using the formula for buoyancy, we solve for \( h \) to determine how much of the box is submerged.

3. Weight Required for Complete Submersion:
When the box is fully submerged, it will displace the maximum possible volume of water. We calculate this volume and subtract the initial weight to determine the additional weight needed.

4. Importance of Maximum Submersion Calculation:
Engineers must ensure that floating structures do not exceed their buoyancy limits. If an excessive load is placed inside, the structure may sink completely.

Physical Meaning

1. Application in Marine Engineering:
The principles used here are fundamental in designing boats, ships, and floating platforms. Proper weight distribution and buoyancy calculations ensure stability.

2. Stability and Load Management:
A floating structure must balance its weight with the water it displaces. Additional loads, such as cargo or equipment, must be carefully calculated to avoid sinking.

3. Real-World Relevance:
The same method is applied in ship loading, where engineers determine how much cargo a vessel can carry while maintaining safe buoyancy.