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Chapter 6: Problem 8

The vessels \(A\) and \(B\) of equal volume and weight are immersed in water to a depth \(h\). The vessel \(A\) has an opening at the bottom through which water can enter. If the work done in immersing \(A\) and \(B\) are \(W_{A}\) and \(W_{B}\) respectively, then (a) \(W_{A}=W_{B}\) (b) \(W_{A}W_{B}\) (d) \(W_{A}>=\left\langle W_{B}\right.\)

Short Answer

Expert verified
The correct answer is (b) \(W_{A}<W_{B}\). Because vessel A allows water to enter through an opening, it displaces less water when fully immersed. This means less work is performed to immerse vessel A compared to vessel B.

Step by step solution

01

Identify the work done formula

Work done (W) is calculated by the formula \(W = F.d \), where F is the force and d is the distance or depth in this case.
02

Find the force exerted by the vessels

The force exerted by the vessels due to their weight is equal to the weight of the water displaced by them.
03

Calculate the work for each vessel

Use the formula from step 1 to calculate the work done on each vessel using the force and depth. Since the vessels have the same volume and weight, they displace the same volume of water, hence the weight of the water displaced is the same. But when immersing vessel A, water enters it through the opening at the bottom, hence it displaces less water compared to vessel B when fully immersed.
04

Compare the work done on each vessel

Because vessel A displaces less water when fully immersed, the work done (force times distance) is less than that for vessel B. Hence, \(W_{A}<W_{B}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Archimedes' Principle
Archimedes' Principle is a fundamental law of fluid mechanics that explains why objects float or sink. The principle states that an object submerged in a fluid experiences an upward force, called the buoyant force, which is equal to the weight of the fluid it displaces. This concept helps in understanding why vessels or any submerged bodies experience a buoyant effect.
For the vessel with an opening, such as vessel A in the exercise, water can enter, thereby reducing the volume of water displaced. Consequently, vessel A experiences a smaller buoyant force compared to vessel B, which does not allow water to enter and thus displaces more water.
  • Buoyant Force: Equal to the weight of displaced fluid.
  • Condition: A vessel loses buoyancy if water enters it.
Understanding this principle allows us to predict that the work done in immersing vessel A will be less than that for vessel B as vessel A experiences a smaller buoyant force due to the lesser volume of displaced water.
Fluid Mechanics
Fluid mechanics is the study of fluids (liquids and gases) and the forces involved with them. It covers a broad range of principles, including viscosity, pressure, and buoyancy. In this exercise, the role of fluid mechanics helps us understand how different configurations of objects immersed in fluids behave.
For vessel A and B, both immersed in water, fluid mechanics allows us to consider how openings affect buoyancy. Vessel A, with its opening, allows water to fill in, reducing the displacement of water. Understanding fluid dynamics in this context helps us see why vessel A does less work upon full immersion than vessel B.
  • Fluid Behavior: Understands both static and dynamic conditions.
  • Effects of Openings: An opening reduces buoyancy due to less displaced fluid.
The understanding of fluid behavior provides insights into the calculations of work done by vividly showing the impact of opening-induced water displacement on buoyancy.
Work Done Calculation
The concept of work done is essential in physics, determining how much energy is transferred when a force is applied over a distance. Mathematically, work done is calculated by the formula, \( W = F \cdot d \), where \( F \) is the force exerted, and \( d \) is the distance moved in the direction of the force.
In the context of the vessels in the exercise, both vessels are submerged to depth \( h \). However, the force exerted in each case differs due to buoyancy differences. Vessel A, having allowed water inside, displaces less water, leading to a smaller force against the immersion compared to Vessel B.
Therefore, we find:
  • Vessel B performs more work than Vessel A because it displaces more water.
  • The work done is a function of distance and the effective force against displacement, modified by buoyancy.
This calculation emphasizes the importance of fluid dynamics in work, impacting energy transfer and force measurements in submerged conditions.

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Most popular questions from this chapter

A force applied by an engine of a train of mass \(2.05 \times 10^{6} \mathrm{~kg}\) changes its velocity from \(5 \mathrm{~m} / \mathrm{s}\) to \(25 \mathrm{~m} / \mathrm{s}\) in 5 minutes. The power of the engine is \(\quad\) [EAMCET 2001] (a) \(1.025 M W\) (b) \(2.05 M W\) (c) \(5 M W\) (d) \(5 M W\)

A particle of mass ' \(m\) ' and charge ' \(q\) ' is accelerated through a potential difference of ' \(V\) 'volt. Its energy is [UPSEAT 2001] (a) \(q V\) (b) \(m q V\) (c) \(\left(\frac{q}{m}\right) V\) (d) \(\frac{q}{m V}\)

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