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Chapter 14: Problem 18

A boat floating in fresh water displaces water weighing \(50.1 \mathrm{kN}\). (a) What is the weight of the water this boat displaces when floating in salt water of density \(1.10 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) ? (b) What is the difference between the volume of fresh water displaced and the volume of salt water displaced?

Short Answer

Expert verified
(a) 50.1 kN; (b) 0.47 m鲁 difference in displaced volumes.

Step by step solution

01

Understanding Buoyancy

According to Archimedes' principle, a floating object displaces a volume of fluid equal to the weight of the object. Hence, the weight of the object is equal to the weight of the fluid displaced by it. This means that the weight of the fresh water displaced is equal to the weight of the boat, which is given as 50.1 kN.
02

Calculating Weight in Salt Water

When the boat floats in salt water, it will still displace water weighing exactly equal to its weight due to buoyancy. Hence, the weight of the salt water displaced is also 50.1 kN, as the boat itself hasn't changed in weight.
03

Determining Volume of Fresh Water Displaced

We can determine the volume of fresh water displaced using the equation: weight = volume 脳 density 脳 gravitational acceleration. Given the density of fresh water as 1000 kg/m鲁, and gravitational acceleration as 9.8 m/s虏:\[ \text{Weight in N} = \text{Volume} \times 1000 \times 9.8 \]Rearrange to solve for volume:\[ \text{Volume} = \frac{50,100 \text{ N}}{1000 \times 9.8} \approx 5.11 \text{ m}^3 \].
04

Calculating Volume of Salt Water Displaced

Similarly, the volume of the salt water displaced can be found using the same principle but with the salt water density, which is 1100 kg/m鲁:\[ \text{Volume} = \frac{50,100 \text{ N}}{1100 \times 9.8} \approx 4.64 \text{ m}^3 \].
05

Comparing Displaced Volumes

Now, find the difference in volume between the fresh and salt water displaced:\[ \text{Difference} = 5.11 \text{ m}^3 - 4.64 \text{ m}^3 = 0.47 \text{ m}^3 \].

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Key Concepts

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

Buoyancy
The concept of buoyancy is key in understanding why objects float or sink in a fluid. According to Archimedes' Principle, a body immersed in a fluid experiences a buoyant force equal to the weight of the fluid displaced by it. This principle explains why the boat in our exercise displaces an amount of water that weighs exactly 50.1 kN.
  • Buoyant force acts upwards, counteracting the object's weight.
  • It allows objects to float if their weight is less than or equal to the weight of the fluid displaced.
In static equilibrium, the buoyant force matches the weight of the object, resulting in floating equilibrium. Hence, despite changing from fresh to salt water, the weight of the displaced water remains constant because the boat's weight remains unchanged at 50.1 kN.
Fluid Displacement
Fluid displacement refers to how much fluid a floating object moves out of the way. When an object floats, it sinks until it displaces a volume of fluid whose weight matches its own. The volume displaced directly relates to the fluid's density and the weight of the object.
  • Displacement increases with the object's weight and decreases with fluid density.
  • Volume displaced is a measure of how much fluid is pushed aside.
As the boat switches from fresh to salt water, the volume of displaced fluid changes due to differing fluid densities, while the weight of displaced water (50.1 kN) remains the same.
Density Comparison
Density comparison is crucial in determining the volume of fluid displaced by a floating object. It is defined as mass per unit volume and affects how much fluid needs to be displaced for an object to float.
  • Higher density means less volume needs to be displaced.
  • Lower density requires more volume to be displaced for the same weight.
In the exercise, fresh water (density = 1000 kg/m鲁) and salt water (density = 1100 kg/m鲁) show different behaviors. Salt water's higher density means the boat displaces less volume (approximately 4.64 m鲁) compared to fresh water (approximately 5.11 m鲁) to achieve buoyancy. Thus, the change in volume displaced (0.47 m鲁 difference) highlights how density alters the fluid's displacement by a floating object.

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