A body of size 3mx2mx2m floats in water. Find the limit of weight of the body for stable equilibrium.

Stable Equilibrium Limit of a Floating Body

Problem Statement

A body with the following dimensions floats in water:

Length: 3m
Width: 2m
Height: 2m

Determine the limits of weight for stable equilibrium.

Solution

1. Define the Equilibrium Condition

The weight of the body must be equal to the weight of the displaced water: \[ \gamma_{\text{body}} V_{\text{body}} = \gamma_{\text{water}} V_{\text{displaced water}} \] \[ S \times 9810 \times 3 \times 2 \times 2 = 9810 \times 3 \times 2 \times h \] \[ h = 2S \]

2. Calculate the Center of Buoyancy (\(OB\))

The center of buoyancy is at the centroid of the submerged volume: \[ OB = \frac{h}{2} = \frac{2S}{2} \] \[ OB = S \]

3. Calculate the Center of Gravity (\(OG\))

The center of gravity for a uniform body is at the midpoint of its height: \[ OG = \frac{2}{2} = 1 \text{ m} \] \[ BG = OG – OB = 1 – S \]

4. Calculate the Metacentric Height (\(GM\))

The metacentric radius (\(MB\)) is given by: \[ MB = \frac{I}{V} \] The moment of inertia about the longitudinal axis: \[ I = \frac{1}{12} \times 3 \times 2^3 \] \[ = \frac{1}{12} \times 3 \times 8 = 2 \text{ m}^4 \] The displaced volume: \[ V = 3 \times 2 \times 2S \] \[ V = 12S \text{ m}^3 \] \[ MB = \frac{2}{12S} \] \[ MB = \frac{0.166}{S} \] \[ GM = MB – BG \] \[ GM = \frac{0.166}{S} – (1 – S) \] For stable equilibrium, \( GM > 0 \): \[ \frac{0.166}{S} – 1 + S > 0 \] \[ S^2 – S + 0.166 > 0 \] Solving for \(S\), the values are \(0.21\) and \(0.79\).

5. Calculate the Limits of Weight

Lower limit: \[ W_{\text{lower}} = \rho_{\text{body}} g V_{\text{body}} \] \[ = 0.21 \times 1000 \times 9.81 \times 3 \times 2 \times 2 \] \[ = 24721 \text{ N} = 24.72 \text{ kN} \] Upper limit: \[ W_{\text{upper}} = 0.79 \times 1000 \times 9.81 \times 3 \times 2 \times 2 \] \[ = 92999 \text{ N} = 92.99 \text{ kN} \]

Final Results:

Lower limit of weight: 24.72 kN
Upper limit of weight: 92.99 kN

Explanation

1. Stability Condition:
A floating body is stable if its metacentric height (\(GM\)) is positive. The condition for stability provides two critical values of \(S\) that determine the range of possible stable weights.

2. Why are there two limits?
– If the weight is too low, the center of gravity is too high, leading to instability.
– If the weight is too high, the center of buoyancy shifts too much, again leading to instability.

3. Importance of Stability Analysis:
– Ensuring a floating structure remains upright under various loading conditions.
– Used in shipbuilding, offshore platforms, and industrial flotation devices.

Physical Meaning

1. Engineering Applications:
The concept is crucial in designing ships, submarines, and floating structures to prevent capsizing.

2. Industrial and Real-World Uses:
– Used in the design of floating bridges.
– Helps in constructing stable offshore drilling rigs.