/*! 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 65 In seawater, a life preserver wi... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 65

In seawater, a life preserver with a volume of 0.0400 \(\mathrm{m}^{3}\) will support a 75.0 \(\mathrm{kg}\) person (average density 980 \(\mathrm{kg} / \mathrm{m}^{3} )\) with 20\(\%\) of the person's volume above water when the life preserver is fully submerged. What is the density of the material composing the life preserver?

Short Answer

Expert verified
The density of the life preserver material is 719.5 kg/m鲁.

Step by step solution

01

Calculate the Volume of the Person

The volume of the person can be found using their mass and density. The formula to find volume is \( V = \frac{m}{\rho} \). So, the volume of the person \( V_{person} = \frac{75.0}{980} = 0.0765 \, \mathrm{m}^3 \).
02

Determine Submerged Volume of the Person

Since 20% of the person's volume is above water, the submerged volume is 80% of the total volume. Thus, \( V_{submerged, person} = 0.8 \times 0.0765 = 0.0612 \, \mathrm{m}^3 \).
03

Apply Buoyancy Principle

For the system to float, the buoyant force must equal the gravitational force on the person plus the life preserver. The buoyant force is due to the total submerged volume of the life preserver and person \( V_{submerged, total} = 0.0400 + 0.0612 = 0.1012 \, \mathrm{m}^3 \).
04

Calculate Mass of Displaced Seawater

The density of seawater is about 1025 \( \mathrm{kg/m^3} \). The mass of the displaced seawater is \( m_{seawater} = 1025 \times 0.1012 = 103.78 \, \mathrm{kg} \).
05

Determine Mass of Life Preserver

Since the buoyant force equals the weight of the person plus the life preserver, we find the mass of the life preserver by subtracting the mass of the person from the displaced water's mass: \( m_{preserver} = 103.78 - 75 = 28.78 \, \mathrm{kg} \).
06

Calculate Density of Life Preserver Material

Density is mass divided by volume. The density of the life preserver \( \rho_{preserver} = \frac{m_{preserver}}{V_{preserver}} = \frac{28.78}{0.0400} = 719.5 \, \mathrm{kg/m^3} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Density calculations
Density is a crucial concept in physics and helps us understand how mass is distributed in a given volume. To calculate density (\( \rho \)), you need to use the formula:
  • \( \rho = \frac{m}{V} \)
where \( m \) is the mass and \( V \) is the volume.
In our example, a life preserver must have a calculated density to ensure it assists a person in floating. By finding the mass of the life preserver and dividing it by its volume, we can calculate its density.
Understanding density also helps explain why certain objects float or sink in fluids.
In simpler terms, if an object's density is less than the fluid it is in, it will float. Here, the life preserver must have a lower density than seawater for it to support the person on water successfully.
Buoyant force
Buoyant force is the upward push a fluid exerts on an object submerged in it. This is the reason objects feel lighter in water.
The buoyant force can be understood by looking at how much fluid an object displaces.
The key to buoyant force is that it equals the weight of the fluid that the object displaces, which is fundamental in determining if an object will float or sink. When you place the life preserver and the person in water, they displace a certain volume of seawater. The weight of this displaced seawater equals the buoyant force applied by the water.
So, when the total weight of the person and the life preserver equals the buoyant force, the system floats.
This balance leads to 80% of the person's volume being submerged while the rest remains above the water, maintaining buoyancy.
Archimedes' Principle
Archimedes' Principle is a vital principle in fluid mechanics that helps us understand buoyancy. It states that an object wholly or partially submerged in a fluid experiences an upward force known as the buoyant force. This force is equal to the weight of the fluid displaced by the object.
For example, when you immerse a life preserver and a person in water, they push aside a certain volume of that water.
According to Archimedes' Principle:
  • The weight of the displaced water is equal to the buoyant force acting on the submerged object.
In our case, the displacement of 0.1012 \( \mathrm{m}^3 \) of seawater results in a buoyant force that precisely balances the combined weight of the person and the life preserver.
By understanding this principle, we can predict and analyze the floating and sinking behavior of various objects when placed in a fluid.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

(a) Calculate the buoyant force of air (density 1.20 \(\mathrm{kg} / \mathrm{m}^{3} )\) on a spherical party balloon that has a radius of 15.0 \(\mathrm{cm}\) . (b) If the rubber of the balloon itself has a mass of 2.00 \(\mathrm{g}\) and the balloon is filled with helium (density 0.166 \(\mathrm{kg} / \mathrm{m}^{3}\) ), calculate the net upward force (the "lift") that acts on it in air.

Fish navigation. (a) As you can tell by watching them in an aquarium, fish are able to remain at any depth in water with no effort. What does this ability tell you about their density? (b) Fish are able to inflate themselves using a sac (called the swim bladder) located under their spinal column. These sacs can be filled with an oxygen-nitrogen mixture that comes from the blood. If a 2.75 \(\mathrm{kg}\) fish in fresh water inflates itself and increases its volume by \(10 \%,\) find the net force that the water exerts on it. (c) What is the net external force on it? Does the fish go up or down when it inflates itself?

An electrical short cuts off all power to a submersible diving vehicle when it is 30 m below the surface of the ocean. The crew must push out a hatch of area 0.75 \(\mathrm{m}^{2}\) and weight 300 \(\mathrm{N}\) on the bottom to escape. If the pressure inside is 1.0 atm, what downward force must the crew exert on the hatch to open it?

You are designing a machine for a space exploration vehicle. It contains an enclosed column of oil that is 1.50 m tall, and you need the pressure difference between the top and the bottom of this column to be 0.125 atm. (a) What must be the density of the oil? (b) If the vehicle is taken to Mars, where the acceleration due to gravity is \(0.379 g,\) what will be the pressure difference (in earth atmospheres) between the top and bottom of the oil column?

A shower head has 20 circular openings, each with radius 1.0 \(\mathrm{mm} .\) The shower head is connected to a pipe with radius 0.80 \(\mathrm{cm} .\) If the speed of water in the pipe is \(3.0 \mathrm{m} / \mathrm{s},\) what is its speed as it exits the shower-head openings?
See all solutions

Recommended explanations on Physics Textbooks

Electrostatics

Read Explanation

Thermodynamics

Read Explanation

Measurements

Read Explanation

Magnetism and Electromagnetic Induction

Read Explanation

Astrophysics

Read Explanation

Energy Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.