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A rectangular floating body is 20 m long and 5 m wide. The water line is 1.5 m above the bottom. If the centre of gravity is 1.8 m from the bottom, then its metacentric height will be approximately
- a)3.3 m
- b)1.65 m
- c)0.34 m
- d)0.30 m
Correct answer is option 'C'. Can you explain this answer?
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A rectangular floating body is 20 m long and 5 m wide. The water line ...
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A rectangular floating body is 20 m long and 5 m wide. The water line ...
Given information:
- Length of the floating body = 20 m
- Width of the floating body = 5 m
- Water line = 1.5 m above the bottom
- Centre of gravity = 1.8 m from the bottom
To find:
- Metacentric height of the floating body
Approach:
- Calculate the displacement of the floating body
- Calculate the moment of inertia of the water plane
- Calculate the moment of inertia of the floating body about an axis passing through the centre of gravity
- Use the formula for metacentric height to calculate the value
Calculation:
1. Displacement of the floating body:
- Volume of the floating body = Length x Width x Depth
- Depth = Water line - Bottom = 1.5 m - 0 m = 1.5 m
- Volume of the floating body = 20 m x 5 m x 1.5 m = 150 m^3
- Density of water = 1000 kg/m^3
- Mass of the floating body = Density x Volume = 1000 kg/m^3 x 150 m^3 = 150000 kg
- Weight of the floating body = Mass x Gravity = 150000 kg x 9.81 m/s^2 = 1471500 N
2. Moment of inertia of the water plane:
- Width of the water plane = 5 m
- Depth of the water plane = Water line - Keel = 1.5 m - 1.5 m = 0 m
- Moment of inertia of the water plane about a horizontal axis passing through the centroid = (1/12) x Width x Depth^3
- Moment of inertia of the water plane = (1/12) x 5 m x 0^3 = 0
3. Moment of inertia of the floating body about an axis passing through the centre of gravity:
- Moment of inertia of the floating body about an axis passing through the centroid = (1/12) x Length x Width^3
- Moment of inertia of the floating body about an axis passing through the centroid = (1/12) x 20 m x 5^3 m = 2083.33 m^4
- Parallel axis theorem: Moment of inertia about an axis passing through the centre of gravity = Moment of inertia about an axis passing through the centroid + Mass x Distance^2
- Distance from the centroid to the centre of gravity = (1/2) x Depth = (1/2) x 1.5 m = 0.75 m
- Moment of inertia of the floating body about an axis passing through the centre of gravity = 2083.33 m^4 + 150000 kg x (0.75 m)^2 = 217083.33 kg.m^2
4. Metacentric height:
- Metacentric height = (Moment of inertia of the water plane) / (Displacement x Distance between the centres of gravity and buoyancy)
- Distance between the centres of gravity and buoyancy = Distance between the centre of gravity and the water line
- Distance between the centre of gravity and the water line = Depth of the water plane + (1/2) x Depth of the floating body = 0 m + (1/2) x 1.5 m = 0.75 m
-
- Length of the floating body = 20 m
- Width of the floating body = 5 m
- Water line = 1.5 m above the bottom
- Centre of gravity = 1.8 m from the bottom
To find:
- Metacentric height of the floating body
Approach:
- Calculate the displacement of the floating body
- Calculate the moment of inertia of the water plane
- Calculate the moment of inertia of the floating body about an axis passing through the centre of gravity
- Use the formula for metacentric height to calculate the value
Calculation:
1. Displacement of the floating body:
- Volume of the floating body = Length x Width x Depth
- Depth = Water line - Bottom = 1.5 m - 0 m = 1.5 m
- Volume of the floating body = 20 m x 5 m x 1.5 m = 150 m^3
- Density of water = 1000 kg/m^3
- Mass of the floating body = Density x Volume = 1000 kg/m^3 x 150 m^3 = 150000 kg
- Weight of the floating body = Mass x Gravity = 150000 kg x 9.81 m/s^2 = 1471500 N
2. Moment of inertia of the water plane:
- Width of the water plane = 5 m
- Depth of the water plane = Water line - Keel = 1.5 m - 1.5 m = 0 m
- Moment of inertia of the water plane about a horizontal axis passing through the centroid = (1/12) x Width x Depth^3
- Moment of inertia of the water plane = (1/12) x 5 m x 0^3 = 0
3. Moment of inertia of the floating body about an axis passing through the centre of gravity:
- Moment of inertia of the floating body about an axis passing through the centroid = (1/12) x Length x Width^3
- Moment of inertia of the floating body about an axis passing through the centroid = (1/12) x 20 m x 5^3 m = 2083.33 m^4
- Parallel axis theorem: Moment of inertia about an axis passing through the centre of gravity = Moment of inertia about an axis passing through the centroid + Mass x Distance^2
- Distance from the centroid to the centre of gravity = (1/2) x Depth = (1/2) x 1.5 m = 0.75 m
- Moment of inertia of the floating body about an axis passing through the centre of gravity = 2083.33 m^4 + 150000 kg x (0.75 m)^2 = 217083.33 kg.m^2
4. Metacentric height:
- Metacentric height = (Moment of inertia of the water plane) / (Displacement x Distance between the centres of gravity and buoyancy)
- Distance between the centres of gravity and buoyancy = Distance between the centre of gravity and the water line
- Distance between the centre of gravity and the water line = Depth of the water plane + (1/2) x Depth of the floating body = 0 m + (1/2) x 1.5 m = 0.75 m
-
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A rectangular floating body is 20 m long and 5 m wide. The water line is 1.5 m above the bottom. If the centre of gravity is 1.8 m from the bottom, then its metacentric height will be approximatelya)3.3 mb)1.65 mc)0.34 md)0.30 mCorrect answer is option 'C'. Can you explain this answer?
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A rectangular floating body is 20 m long and 5 m wide. The water line is 1.5 m above the bottom. If the centre of gravity is 1.8 m from the bottom, then its metacentric height will be approximatelya)3.3 mb)1.65 mc)0.34 md)0.30 mCorrect answer is option 'C'. Can you explain this answer? for Civil Engineering (CE) 2024 is part of Civil Engineering (CE) preparation. The Question and answers have been prepared according to the Civil Engineering (CE) exam syllabus. Information about A rectangular floating body is 20 m long and 5 m wide. The water line is 1.5 m above the bottom. If the centre of gravity is 1.8 m from the bottom, then its metacentric height will be approximatelya)3.3 mb)1.65 mc)0.34 md)0.30 mCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Civil Engineering (CE) 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A rectangular floating body is 20 m long and 5 m wide. The water line is 1.5 m above the bottom. If the centre of gravity is 1.8 m from the bottom, then its metacentric height will be approximatelya)3.3 mb)1.65 mc)0.34 md)0.30 mCorrect answer is option 'C'. Can you explain this answer?.
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