/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 46 A lost shipping container is fou... [FREE SOLUTION] | 91影视

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Chapter 11: Problem 46

A lost shipping container is found resting on the ocean floor and completely submerged. The container is 6.1 m long, 2.4 m wide, and 2.6 m high. Salvage experts attach a spherical balloon to the top of the container and inflate it with air pumped down from the surface. When the balloon鈥檚 radius is 1.5 m, the shipping container just begins to rise toward the surface. What is the mass of the container? Ignore the mass of the balloon and the air within it. Do not neglect the buoyant force exerted on the shipping container by the water. The density of seawater is 1025 \(\mathrm{kg} / \mathrm{m}^{3}\)

Short Answer

Expert verified
The mass of the container is approximately 18,418 kg.

Step by step solution

01

Calculate the volume of the container

To find the volume of the rectangular prism-shaped container, use the formula for the volume of a rectangle: \( V = \text{length} \times \text{width} \times \text{height} \). Substituting in the values given, we have \( V = 6.1 \times 2.4 \times 2.6 \). Calculate this to get the volume of the container in cubic meters.
02

Determine the volume of the balloon

The balloon is spherical in shape, so we use the formula for the volume of a sphere: \( V = \frac{4}{3} \pi r^{3} \), where \( r = 1.5 \) m is the radius of the balloon. Calculate the volume of the balloon using this formula.
03

Calculate the buoyant force

The buoyant force is equal to the weight of the water displaced by the submerged parts (container and balloon). The volume of water displaced is equal to the combined volume of the container and the balloon. Calculate the total displaced volume and multiply by the density of seawater \( 1025 \, \mathrm{kg/m^3} \) and gravitational acceleration \( 9.81 \, \mathrm{m/s^2} \) to get the buoyant force.
04

Set buoyant force equal to weight of container

At the point when the container begins to rise, the upward buoyant force is equal to the downward gravitational force (weight) of the container. Use the equation: \( F_{\text{buoyant}} = \text{mass of container} \times g \), where \( F_{\text{buoyant}} \) is the force you calculated in the previous step. Solve for the mass of the container.

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Key Concepts

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

Buoyant Force
The buoyant force is a key concept in understanding why objects float or sink in a fluid. It's an upward force exerted by the fluid that opposes the weight of an object submerged in it. This force is determined by how much fluid is displaced by the object. In the context of our shipping container scenario, when the container is completely submerged, it displaces a certain amount of seawater. Hence, the buoyant force is exactly equal to the weight of this displaced seawater.
  • Think of buoyant force as nature's way of balancing things out. It's the reason we feel lighter in water.
  • It results from the pressure difference on the bottom and top of the object submerged in the fluid.
  • Archimedes' Principle is the rule here: the buoyant force on an object is equal to the weight of the fluid displaced by the object.
The buoyant force doesn't just play a role when the container starts rising. It's present from the moment the container is submerged, constantly working against gravity.
Volume Calculation
Volume calculation is essential to determine how much space an object occupies. Knowing the volume helps us calculate how much fluid is displaced, which is directly linked to the buoyant force. Here, we calculate the volume of both the shipping container and the attached spherical balloon.
  • For the rectangular container, the volume is calculated as the product of its length, width, and height. This is straightforward and involves simple multiplication.
  • For the spherical balloon, the volume formula differs. It involves raising the radius to the third power and multiplying by \(\frac{4}{3} \, \pi\).
  • This step is crucial because, without knowing the correct total volume displaced, we wouldn't be able to accurately calculate the buoyant force.
Once we have the volumes, they help us easily transition to the next stages, including calculating the displaced seawater and subsequently the buoyant force.
Density of Seawater
The density of seawater is a constant in this problem, given as 1025 \(\mathrm{kg/m^3}\). Density is defined as mass per unit volume and is crucial in calculating buoyant forces.
  • Seawater density is slightly higher than that of pure water due to the salts dissolved in it.
  • This higher density means that seawater provides a greater buoyant force for objects submerged in it than freshwater.
  • The buoyant force calculation relies on the density because it sets the weight of the water displaced per unit volume of the object submerged.
A constant density assumption simplifies our calculations, allowing us to focus on other variable elements like volume and gravitational effects.
Rectangular Prism
A rectangular prism is a 3D shape with six rectangular faces, and it forms the basis for calculating the volume of the container. Picture a box or a brick; those are everyday examples of rectangular prisms.
  • To calculate the volume of a rectangular prism, multiply the length, width, and height: \(V = \text{length} \times \text{width} \times \text{height}\).
  • The dimensions given are specific to our problem: 6.1 m in length, 2.4 m in width, and 2.6 m in height.
  • After inserting these numbers into the formula, you'll obtain the total volume of the shipping container.
This total volume is important for figuring out how much seawater the container displaces when submerged, directly affecting the buoyant force it experiences.
Spherical Volume
Calculating the volume of a sphere is slightly more complex than a rectangular prism because of its shape. In our problem, the spherical shape applies to the balloon attached to the container.
  • The formula for the volume of a sphere is \(V = \frac{4}{3} \pi r^3\), where \(r\) is the radius.
  • Using the radius given in the problem, 1.5 m, plug it into the formula to find the total volume of the balloon.
  • This spherical volume is necessary because it adds to the total volume of seawater displaced, amplifying the buoyant force alongside the container.
Understanding the sphere's volume concept helps grasp why and how the balloon assists the container in rising by increasing the total displaced water volume.

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Most popular questions from this chapter

The vertical surface of a reservoir dam that is in contact with the water is 120 m wide and 12 m high. The air pressure is one atmosphere. Find the magnitude of the total force acting on this surface in a completely filled reservoir. (Hint: The pressure varies linearly with depth, so you must use an average pressure.)

A paperweight, when weighed in air, has a weight of \(W=6.9 \mathrm{N}\) . When completely immersed in water, however, it has a weight of \(W_{\text { in water }}=4.3 \mathrm{N}\) . Find the volume of the paperweight.

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