A uniform body of size 4mx2mx1m floats in water. What is the weight of the body if the depth of immersion is 0.6m? Also determine the meta-centric height.

Weight and Metacentric Height of a Floating Body

Problem Statement

A uniform body with dimensions:

Length: 4m
Width: 2m
Height: 1m

The body floats in water with an immersion depth of 0.6m. Determine:

The weight of the body.
The metacentric height (\(GM\)).

Solution

1. Calculate the Weight of the Body

Using the buoyancy principle: \[ \text{Weight of body} = \text{Weight of water displaced} \] \[ W = \gamma_{\text{water}} V_{\text{displaced water}} \] \[ W = 9810 \times (4 \times 2 \times 0.6) \] \[ W = 47088 \text{ N} \]

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

The center of buoyancy is at the centroid of the submerged portion: \[ OB = \frac{\text{Immersion Depth}}{2} = \frac{0.6}{2} \] \[ OB = 0.3 \text{ m} \]

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

Since the body is uniform, its center of gravity is at the midpoint of its height: \[ OG = \frac{\text{Total Height}}{2} = \frac{1}{2} \] \[ OG = 0.5 \text{ m} \]

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

The metacentric height is given by: \[ GM = MB – BG \] First, find the metacentric radius (\(MB\)): \[ MB = \frac{I}{V} \] The moment of inertia about the longitudinal axis: \[ I = \frac{1}{12} \times \text{width} \times (\text{depth})^3 \] \[ I = \frac{1}{12} \times 4 \times (2)^3 \] \[ I = \frac{1}{12} \times 4 \times 8 = \frac{32}{12} = 2.667 \text{ m}^4 \] The displaced volume: \[ V = 4 \times 2 \times 0.6 = 4.8 \text{ m}^3 \] \[ MB = \frac{2.667}{4.8} = 0.556 \text{ m} \] Calculate \( BG \): \[ BG = OG – OB = 0.5 – 0.3 = 0.2 \text{ m} \] Finally, calculate \( GM \): \[ GM = 0.556 – 0.2 \] \[ GM = 0.356 \text{ m} \]

Final Results:

Weight of the body: 47088 N
Metacentric height (\(GM\)): 0.356 m

Explanation

1. Understanding Buoyancy:
The body floats because the buoyant force equals its weight. The volume of displaced water determines the buoyant force.

2. Calculation of Center of Buoyancy:
The center of buoyancy is at the centroid of the displaced volume. Since the submerged depth is 0.6m, its centroid is at 0.3m from the base.

3. Calculation of Metacentric Height:
The metacentric height is a measure of stability. It is determined using the moment of inertia of the waterplane area and the volume of displaced water. A positive \( GM \) means the body is stable.

4. Importance of Metacentric Height:
– If \( GM \) is large, the body is highly stable.
– If \( GM \) is negative, the body is unstable and will capsize.
– A moderate \( GM \) ensures gentle rolling motion, which is preferable in ship design.

Physical Meaning

1. Ship Stability:
The metacentric height is crucial in ship design to ensure vessels remain upright and stable even in rough waters.

2. Industrial Applications:
This principle is used in floating platforms, submarines, and offshore structures to prevent tilting and instability.

3. Floating Object Behavior:
Objects with higher \( GM \) values resist tilting, whereas those with lower \( GM \) values tend to oscillate more.