/*! 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 14 According to Archimedes' princip... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 14

According to Archimedes' principle, the mass of a floating object equals the mass of the fluid displaced by the object. Use this principle to solve the following problems. (a) A wooden cylinder 30.0 cm high floats vertically in a tub of water (density \(=1.00 \mathrm{g} / \mathrm{cm}^{3}\) ). The top of the cylinder is \(13.5 \mathrm{cm}\) above the surface of the liquid. What is the density of the wood? (b) The same cylinder floats vertically in a liquid of unknown density. The top of the cylinder is \(18.9 \mathrm{cm}\) above the surface of the liquid. What is the liquid density? (c) Explain why knowing the length and width of the wooden objects is unnecessary in solving Parts (a) and (b).

Short Answer

Expert verified
The density of wood is 0.55 g/cm鲁, the density of the unknown liquid is approximately 1.49 g/cm鲁, and the length and width of the wooden objects aren't necessary to solve parts (a) and (b) because only the proportion of height that is submerged is relevant in these calculations.

Step by step solution

01

Calculating the Density of the Wood

First, let's use the volume percent submerged to get the density of the wood according to the following formula: \(\text{Density of object} = \text{Density of fluid} * \frac{\text{Volume submerged}}{\text{Total volume}}\). Here, the density of fluid is the density of water, 1.00 g/cm鲁. The volume percent submerged is equal to the submerged height over the total height, (30.0 cm - 13.5 cm) / 30.0 cm = 0.55 . Substituting these values we get: \(\text{Density of wood} = 1 g/cm鲁 * 0.55 = 0.55 g/cm鲁\) .
02

Calculating the Density of the Unknown Fluid

Next, let's calculate the unknown liquid density where the wooden cylinder is floating, following the same process. Now in this case, the density of the fluid is unknown. However, we know the density of the object (wood) which we calculated to be 0.55 g/cm鲁. The volume percent submerged is (30.0 cm - 18.9 cm) / 30.0 cm = 0.37. Substituting these values into the formula transforms it to: Density of fluid = Density of object / Percent volume submerged = 0.55 g/cm鲁 / 0.37 = 1.486 g/cm鲁. Therefore, the density of the unknown liquid is approximately 1.49 g/cm鲁 when rounded off to two decimal places.
03

Reasoning Regarding Length and Width of the Cylinder

Finally, let's explain why knowing the length and width of the wooden objects is unnecessary to solve parts (a) and (b). Since the densities (whether of the wood or the liquid) are defined as the mass divided by the volume, both height and cross-sectional area of the cylinder would be in the numerator and the denominator of the fraction respectively. Thus, these factors would cancel out, leaving only the vertical height at play in determining volume submerged. Hence, the length and width of the cylinder don't matter in this case, only the proportion of the height that is submerged is important.

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 key concept in understanding why objects float or sink. It is defined as mass per unit volume and is represented by the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). This means if you have two objects of the same size, the one with the higher mass will have higher density.

For the exercise, calculating the density requires comparing how much of the wooden cylinder is submerged in water, which tells us how dense the wood is compared to water. Water has a density of 1.00 g/cm鲁. If an object has a density less than 1.00 g/cm鲁, it will float, and more of the object will stick out of the water.

In this case, the density of the wooden cylinder is calculated using the proportion of its height that is submerged. Only 55% of the wooden cylinder鈥檚 height is submerged, indicating its density is lower than that of water and is found to be 0.55 g/cm鲁 by the formula: \( \text{Density of wood} = \text{Density of water} \times \text{Volume percent submerged} \).
Buoyancy
Buoyancy explains why objects float and it is based on Archimedes' Principle, which states that a floating object displaces a amount of fluid equal to its own mass. Buoyancy is the force that pushes objects up in a fluid, countering gravity's pull downwards.

A balance exists between the weight of the object and the weight of the fluid displaced; this balance is what enables floating. If the weight of the object is less than or equal to the weight of the fluid it displaces, it floats.

In the exercise, when the wooden cylinder was placed in water and then in another liquid, its different submersion levels indicated different buoyant forces due to different densities of the liquids. The height of the cylinder submerged in water was used to determine its density and subsequently, using this density, the unknown liquid's density was determined as well.
Fluid Mechanics
Fluid mechanics is the study of fluids and how they behave when at rest and when in motion. It combines principles from physics and engineering to explore how fluids exert forces and how they flow.

This field of study includes understanding concepts like pressure, buoyant force, and density dynamics. Archimedes' Principle is a fundamental aspect of fluid mechanics, as it helps explain how objects interact with fluids around them.

In practical terms, for this exercise, fluid mechanics helps us understand how a wooden cylinder floats on water and how it affects its positioning when placed into a liquid with unknown density. By grasping the essence of fluid mechanics, one can predict, solve, and understand fluid-related problems much effectively even without knowing the exact dimensions of the objects, as only relative submerged volume matters.

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

In modeling the effect of an impurity on crystal growth, the following equation was derived: \(\frac{G-G_{\mathrm{L}}}{G_{0}-G}=\frac{1}{K_{\mathrm{L}} C^{m}}\) where \(C\) is impurity concentration, \(G_{\mathrm{L}}\) is a limiting growth rate, \(G_{0}\) is the growth rate of the crystal with no impurity present, and \(K_{\mathrm{L}}\) and \(m\) are model parameters. In a particular experiment, \(G_{0}=3.00 \times 10^{-3} \mathrm{mm} / \mathrm{min},\) and \(G_{\mathrm{L}}=1.80 \times 10^{-3} \mathrm{mm} / \mathrm{min} .\) Growth rates are measured for several impurity concentrations \(C\) (parts per million, or ppm), with the following results: $$\begin{array}{|c|c|c|c|c|c|}\hline C(\mathrm{ppm}) & 50.0 & 75.0 & 100.0 & 125.0 & 150.0 \\\\\hline G(\mathrm{mm} / \mathrm{min}) \times 10^{3} & 2.50 & 2.20 & 2.04 & 1.95 & 1.90 \\\\\hline\end{array}$$ (For example, when \(\left.C=50.0 \mathrm{ppm}, G=2.50 \times 10^{-3} \mathrm{mm} / \mathrm{min}\right)\). (a) Determine \(K_{\mathrm{L}}\) and \(m,\) giving both numerical values and units. (b) A solution is fed to a crystallizer in which the impurity concentration is 475 ppm. Estimate the expected crystal growth rate in (mm/min). Then state why you would be extremely skeptical about this result.

The following \((x, y)\) data are recorded: $$\begin{array}{|c|c|c|c|}\hline x & 0.5 & 1.4 & 84 \\\\\hline y & 2.20 & 4.30 & 6.15 \\\\\hline \end{array}$$ (a) Plot the data on logarithmic axes. (b) Determine the coefficients of a power law expression \(y=a x^{b}\) using the method of least squares. (Remember what you are really plotting \(-\) there is no way to avoid taking logarithms of the data point coordinates in this case.) (c) Draw your calculated line on the same plot as the data.

The doozy is defined as the unit of force required to accelerate a unit of mass, called the cuz, with the gravitational acceleration on the surface of the moon, which is one-sixth of the normal gravitational acceleration on earth. (a) What is the conversion factor that would be used to convert a force from the natural unit to the derived unit in this system? (Give both its numerical value and its units.) (b) What is the weight in doozies of a 3 -cuz object on the moon? What does the same object weigh in Lizard Lick, North Carolina?

A hygrometer, which measures the amount of moisture in a gas stream, is to be calibrated using the apparatus shown here: Steam and dry air are fed at known flow rates and mixed to form a gas stream with a known water content, and the hygrometer reading is recorded; the flow rate of either the water or the air is changed to produce a stream with a different water content and the new reading is recorded, and so on. The following data are taken: $$\begin{array}{cc}\hline \begin{array}{c}\text { Mass Fraction } \\\\\text { of Water, } y\end{array} & \begin{array}{c}\text { Hygrometer } \\\\\text { Reading, } R\end{array} \\\\\hline 0.011 & 5 \\\0.044 & 20 \\\0.083 & 40 \\\0.126 & 60 \\\0.170 & 80 \\ \hline\end{array}$$ (a) Draw a calibration curve and determine an equation for \(y(R)\). (b) Suppose a sample of a stack gas is inserted in the sample chamber of the hygrometer and a reading of \(R=43\) is obtained. If the mass flow rate of the stack gas is \(1200 \mathrm{kg} / \mathrm{h}\), what is the mass flow rate of water vapor in the gas?

The following expression has occurred in a problem solution: $$R=\frac{(0.6700)(264,980)(6)\left(5.386 \times 10^{4}\right)}{(3.14159)\left(0.479 \times 10^{7}\right)}$$ The factor 6 is a pure integer. Estimate the value of \(R\) without using a calculator, following the procedure outlined in Section 2.5b. Then calculate \(R\), expressing your answer in both scientific and decimal notation and making sure it has the correct number of significant figures.
See all solutions

Recommended explanations on Chemistry Textbooks

Kinetics

Read Explanation

Nuclear Chemistry

Read Explanation

Inorganic Chemistry

Read Explanation

Ionic and Molecular Compounds

Read Explanation

Organic Chemistry

Read Explanation

Chemical Reactions

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.