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Chapter 14: Problem 34

A boat floating in fresh water displaces water weighing \(35.6\) 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) 35.6 kN; (b) 0.330 m\(^3\).

Step by step solution

01

Understand the Archimedes' Principle

According to Archimedes' Principle, the buoyant force on an object submerged in a fluid is equal to the weight of the fluid that the object displaces. For a boat floating in fresh water, the weight of the displaced fresh water is equal to the weight of the boat, which is given as 35.6 kN.
02

Determine Weight of Salt Water Displaced

Since the boat floats in both fresh and salt water, the buoyant force must be equal to the weight of the boat in both cases. Therefore, the weight of the salt water displaced must also be 35.6 kN because the weight of the boat does not change.
03

Calculate Volume of Fresh Water Displaced

Using the formula for buoyant force, \( F_b = \rho V g \), where \( F_b \) is the buoyant force (35.6 kN), \( \rho \) is the density of water (\(1000 \ kg/m^3\) for fresh water), and \( g = 9.8 \, m/s^2 \). Solve for volume \( V \):\[ 35.6 \times 10^3 \, N = 1000 \, kg/m^3 \times V \times 9.8 \, m/s^2 \]\[ V_{fresh} = \frac{35.6 \times 10^3}{1000 \times 9.8} \approx 3.631 \ m^3 \]
04

Calculate Volume of Salt Water Displaced

Using the same formula for buoyant force in salt water with density \( \rho = 1.10 \times 10^3 \ kg/m^3 \):\[ 35.6 \times 10^3 \, N = 1.10 \times 10^3 \, kg/m^3 \times V \times 9.8 \, m/s^2 \]\[ V_{salt} = \frac{35.6 \times 10^3}{1.10 \times 10^3 \times 9.8} \approx 3.301 \ m^3 \]
05

Calculate Difference in Volume Displaced

Subtract the volume of salt water displaced from the volume of fresh water displaced:\[ \Delta V = V_{fresh} - V_{salt} = 3.631 \, m^3 - 3.301 \, m^3 = 0.330 \, m^3 \]
06

Conclusion

(a) The weight of water displaced in salt water is the same as in fresh water, 35.6 kN. (b) The difference in the volume of fresh and salt water displaced is 0.330 m\(^3\).

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Key Concepts

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

Buoyant Force
When it comes to floating objects like boats, the concept of buoyant force is crucial. The buoyant force is an upward force exerted by a fluid on any object placed in it. According to Archimedes' Principle, the buoyant force on an object is equal to the weight of the fluid displaced by the object.
For example, when a boat floats in water, it pushes water out of the way. The weight of this displaced water is what pushes back against the boat, keeping it afloat. This is what we refer to as the buoyant force.
  • It acts in the opposite direction to gravitational force.
  • It enables objects to float or appear lighter in water.
  • Is determined by the volume of fluid displaced and not the object's own weight or shape.
Understanding buoyant force helps us grasp how different fluids, like fresh water versus salt water, affect the floating capability of objects.
Even when a boat moves from fresh to salt water, though the density of the fluid changes, the buoyant force equals the boat's weight because it must stay afloat.
Density of Water
Density is a measure of how much mass is contained in a given volume. Water's density is an important factor that influences buoyant force.
Fresh water has a density of approximately 1000 kg/m鲁, while salt water is denser at about 1100 kg/m鲁. So, how does this difference impact buoyancy you might wonder?
When an object floats, it displaces a volume of water equal to its own weight. Because salt water is denser, the same object will displace less of it compared to fresh water.
  • Higher density means less volume is displaced for the same weight of the object.
  • The properties of the medium (fresh vs. salt water) determine the volume of fluid displaced.
  • Even though the boat's weight remains constant, the density of water affects how much water needs to be displaced for the boat to float.
This is why when you move a boat from fresh water to salt water, though its weight and the buoyant force remain the same, the volume of water displaced changes.
Volume of Water Displacement
The volume of water an object displaces is a key concept in understanding buoyancy. The volume displaced is essential because it directly relates to the buoyant force acting on an object.
When a boat floats in water, whether fresh or salt, the volume of water displaced will vary depending on the water's density. Using our boat as an example, the density of fresh water allows it to displace a larger volume compared to denser salt water.
  • Volume displaced dictates the extent of the buoyant force experienced.
  • For fresh water with a lower density, more volume is needed to equal the boat's weight, as calculated approximately 3.631 m鲁 in the exercise.
  • For salt water, only around 3.301 m鲁 needs to be displaced due to its higher density.
This difference in displacement volume is crucial for calculations related to flotation and stability of objects in varying water bodies.
Recognizing how water's density impacts volume displaced helps clarify why objects might sit differently in fresh versus salt water, even when the buoyant force remains unchanged.

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Most popular questions from this chapter

When a pilot takes a tight turn at high speed in a modern fighter airplane, the blood pressure at the brain level decreases, blood no longer perfuses the brain, and the blood in the brain drains. If the heart maintains the (hydrostatic) gauge pressure in the aorta at 120 torr (or mm Hg) when the pilot undergoes a horizontal centripetal acceleration of \(4 g\), what is the blood pressure (in torr) at the brain, \(30 \mathrm{~cm}\) radially inward from the heart? The perfusion in the brain is small enough that the vision switches to black and white and narrows to "tunnel vision" and the pilot can undergo \(g\) -LOC ("g-induced loss of consciousness"). Blood density is \(1.06 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\)

Water is moving with a speed of \(5.0 \mathrm{~m} / \mathrm{s}\) through a pipe with a cross-sectional area of \(4.0 \mathrm{~cm}^{2}\). The water graduall descends \(10 \mathrm{~m}\) as the pipe cross-sectional area increases to \(8.0 \mathrm{~cm}^{2} .\) (a) What is the speed at the lower level? (b) If the pressure at the upper level is \(1.5 \times 10^{5} \mathrm{~Pa}\), what is the pressure at the lower level?

A flotation device is in the shape of a right cylinder, with a height of \(0.500 \mathrm{~m}\) and a face area of \(4.00 \mathrm{~m}^{2}\) on top and bottom, and its density is \(0.400\) times that of fresh water. It is initially held fully submerged in fresh water, with its top face at the water surface. Then it is allowed to ascend gradually until it begins to float. How much work does the buoyant force do on the device during the ascent?

Suppose that two tanks, 1 and 2, each with a large opening at the top, contain different liquids. A small hole is made in the side of each tank at the same depth \(h\) below the liquid surface, but the hole in tank 1 has half the cross- sectional area of the hole in tank \(2 .\) (a) What is the ratio \(\rho_{1} / \rho_{2}\) of the densities of the liquids if the mass flow rate is the same for the two holes? (b) What is the ratio \(R_{V 1} / R_{V 2}\) of the volume flow rates from the two tanks? (c) At one instant, the liquid in tank 1 is \(12.0 \mathrm{~cm}\) above the hole. If the tanks are to have equal volume flow rates, what height above the hole must the liquid in tank 2 be just then?

The volume of air space in the passenger compartment of an \(1800 \mathrm{~kg}\) car is \(5.00 \mathrm{~m}^{3}\). The volume of the motor and front wheels is \(0.750 \mathrm{~m}^{3}\), and the volume of the rear wheels, gas tank, and trunk is \(0.800 \mathrm{~m}^{3}\); water cannot enter these two regions. The car rolls into a lake. (a) At first, no water enters the passenger compartment. How much of the car, in cubic meters, is below the water surface with the car floating (Fig. 14-43)? (b) As water slowly enters, the car sinks. How many cubic meters of water are in the car as it disappears below the water surface? (The car, with a heavy load in the trunk, remains horizontal.)
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