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A cylindrical vessel having its height equal to its diameter is filled with liquid and moved horizontally at acceleration equal to acceleration due to gravity. The ratio of the liquid left in the vessel to the liquid at static equilibrium condition is:
- a)0.2
- b)0.4
- c)0.5
- d)0.75
Correct answer is option 'C'. Can you explain this answer?
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A cylindrical vessel having its height equal to its diameter is filled...
Ans. (c)
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A cylindrical vessel having its height equal to its diameter is filled...
Given data:
Height (h) = Diameter (d)
Acceleration (a) = Acceleration due to gravity (g)
To find: Ratio of liquid left in the vessel to the liquid at static equilibrium condition
Solution:
Let the height and radius of the cylindrical vessel be h and r respectively.
Volume of the cylindrical vessel = πr²h
At static equilibrium condition:
The weight of the liquid inside the vessel is balanced by the buoyant force and hence, the liquid does not spill out of the vessel.
Weight of the liquid = Volume x Density x g
= πr²hρg
Buoyant force = Weight of the liquid displaced by the vessel = Volume of the vessel x Density of the liquid x g
= πr²hρg
As the weight of the liquid and the buoyant force are equal, the liquid in the vessel does not spill out.
When the vessel is moved horizontally with an acceleration equal to acceleration due to gravity:
The liquid experiences an apparent weight due to the acceleration.
Apparent weight of the liquid = Weight of the liquid + ma
where m = Mass of the liquid
= Volume of the liquid x Density of the liquid
= πr²hρ
Apparent weight of the liquid = πr²hρg + πr²hρa
= πr²hρ(g+a)
As the vessel is moved with an acceleration equal to acceleration due to gravity, a = g.
Apparent weight of the liquid = πr²hρ(g+g)
= 2πr²hρg
The vessel will spill out the liquid if the apparent weight of the liquid is greater than the buoyant force acting on it. Therefore, the equilibrium condition is given by:
2πr²hρg > πr²hρg
2 > 1
Hence, the liquid spills out of the vessel when it is moved horizontally with an acceleration equal to acceleration due to gravity.
Let x be the ratio of liquid left in the vessel to the liquid at static equilibrium condition.
Therefore, the mass of the liquid left in the vessel is:
m' = xπr²hρ
Apparent weight of the liquid left in the vessel = m'g + ma
= xπr²hρg + xπr²hρg
= 2xπr²hρg
The vessel will not spill out the liquid if the apparent weight of the liquid left in the vessel is less than or equal to the buoyant force acting on it. Therefore, the equilibrium condition is given by:
2xπr²hρg ≤ πr²hρg
2x ≤ 1
x ≤ 0.5
Hence, the ratio of the liquid left in the vessel to the liquid at static equilibrium condition is 0.5 or 1/2 or option C.
Height (h) = Diameter (d)
Acceleration (a) = Acceleration due to gravity (g)
To find: Ratio of liquid left in the vessel to the liquid at static equilibrium condition
Solution:
Let the height and radius of the cylindrical vessel be h and r respectively.
Volume of the cylindrical vessel = πr²h
At static equilibrium condition:
The weight of the liquid inside the vessel is balanced by the buoyant force and hence, the liquid does not spill out of the vessel.
Weight of the liquid = Volume x Density x g
= πr²hρg
Buoyant force = Weight of the liquid displaced by the vessel = Volume of the vessel x Density of the liquid x g
= πr²hρg
As the weight of the liquid and the buoyant force are equal, the liquid in the vessel does not spill out.
When the vessel is moved horizontally with an acceleration equal to acceleration due to gravity:
The liquid experiences an apparent weight due to the acceleration.
Apparent weight of the liquid = Weight of the liquid + ma
where m = Mass of the liquid
= Volume of the liquid x Density of the liquid
= πr²hρ
Apparent weight of the liquid = πr²hρg + πr²hρa
= πr²hρ(g+a)
As the vessel is moved with an acceleration equal to acceleration due to gravity, a = g.
Apparent weight of the liquid = πr²hρ(g+g)
= 2πr²hρg
The vessel will spill out the liquid if the apparent weight of the liquid is greater than the buoyant force acting on it. Therefore, the equilibrium condition is given by:
2πr²hρg > πr²hρg
2 > 1
Hence, the liquid spills out of the vessel when it is moved horizontally with an acceleration equal to acceleration due to gravity.
Let x be the ratio of liquid left in the vessel to the liquid at static equilibrium condition.
Therefore, the mass of the liquid left in the vessel is:
m' = xπr²hρ
Apparent weight of the liquid left in the vessel = m'g + ma
= xπr²hρg + xπr²hρg
= 2xπr²hρg
The vessel will not spill out the liquid if the apparent weight of the liquid left in the vessel is less than or equal to the buoyant force acting on it. Therefore, the equilibrium condition is given by:
2xπr²hρg ≤ πr²hρg
2x ≤ 1
x ≤ 0.5
Hence, the ratio of the liquid left in the vessel to the liquid at static equilibrium condition is 0.5 or 1/2 or option C.
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A cylindrical vessel having its height equal to its diameter is filled with liquid and moved horizontally at acceleration equal to acceleration due to gravity. The ratio of the liquid left in the vessel to the liquid at static equilibrium condition is:a)0.2b)0.4c)0.5d)0.75Correct answer is option 'C'. Can you explain this answer?
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A cylindrical vessel having its height equal to its diameter is filled with liquid and moved horizontally at acceleration equal to acceleration due to gravity. The ratio of the liquid left in the vessel to the liquid at static equilibrium condition is:a)0.2b)0.4c)0.5d)0.75Correct answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A cylindrical vessel having its height equal to its diameter is filled with liquid and moved horizontally at acceleration equal to acceleration due to gravity. The ratio of the liquid left in the vessel to the liquid at static equilibrium condition is:a)0.2b)0.4c)0.5d)0.75Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A cylindrical vessel having its height equal to its diameter is filled with liquid and moved horizontally at acceleration equal to acceleration due to gravity. The ratio of the liquid left in the vessel to the liquid at static equilibrium condition is:a)0.2b)0.4c)0.5d)0.75Correct answer is option 'C'. Can you explain this answer?.
A cylindrical vessel having its height equal to its diameter is filled with liquid and moved horizontally at acceleration equal to acceleration due to gravity. The ratio of the liquid left in the vessel to the liquid at static equilibrium condition is:a)0.2b)0.4c)0.5d)0.75Correct answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about A cylindrical vessel having its height equal to its diameter is filled with liquid and moved horizontally at acceleration equal to acceleration due to gravity. The ratio of the liquid left in the vessel to the liquid at static equilibrium condition is:a)0.2b)0.4c)0.5d)0.75Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A cylindrical vessel having its height equal to its diameter is filled with liquid and moved horizontally at acceleration equal to acceleration due to gravity. The ratio of the liquid left in the vessel to the liquid at static equilibrium condition is:a)0.2b)0.4c)0.5d)0.75Correct answer is option 'C'. Can you explain this answer?.
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