Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 17: Problem 21

A solid cylinder and a cylindrical shell, of identical radius and length and made of the same material, experience the same temperature increase \(\Delta T .\) Which of the two will expand to a larger outer radius?

Short Answer

Expert verified
Answer: Both the solid cylinder and the cylindrical shell will expand to the same larger outer radius when they experience the same temperature increase.

Step by step solution

01

Calculate Volume Change

Firstly, let's calculate the volume change of each cylinder with respect to the temperature difference. Since both cylinders are made from the same material, they will both have the same coefficient of volumetric expansion alpha (\(\alpha\)), which relates volume change and temperature change as follows: \(\Delta V = V_{0} \alpha \Delta T\) Here, \(V_{0}\) is the initial volume of the cylinder, \(\Delta V\) is the change in volume, and \(\Delta T\) is the temperature difference.
02

Find the Volume Relationship of Both Cylinders

We know that the solid cylinder and the cylindrical shell have the same radius and length, so let's define the volume of the solid cylinder as \(V_{s}\) and the volume of the cylindrical shell as \(V_{sh}\). As the cylindrical shell can be imagined as the result of subtracting a smaller internal cylinder from the solid cylinder with the same height, we can write: \(V_{sh} = V_{s} - V_{i}\), where \(V_{i}\) is the volume of the internal cylinder.
03

Calculate the Linear Expansion

The linear expansion, which relates the initial size of any dimension to the size after a temperature change, can be described as: \(\Delta L = L_{0} \beta \Delta T\) Here, \(L_0\) is the initial length of the object, \(\Delta L\) is the change in length, and \(\beta\) is the linear expansion coefficient.
04

Relate Volumetric and Linear Expansion

The relationship between the volumetric and linear expansion coefficients is: \(\alpha = 3\beta\)
05

Calculate the Change in Outer Radius

Now, we will calculate the change in outer radius for both cylinders. Let's denote the initial radius of the solid cylinder and the cylindrical shell as \(r_{s}\) and denote the change in the outer radius as \(\Delta r_{s}\). Then: \(\Delta r_{s} = r_{s} \beta \Delta T\) For the cylindrical shell, let's denote the initial outer radius as \(r_{sh}\) and the initial inner radius as \(r_{i}\). Then the change in the outer radius for the cylindrical shell can be written as: \(\Delta r_{sh} = r_{sh} \beta \Delta T\)
06

Compare the Change in Outer Radius

Since both the solid cylinder and the cylindrical shell have the same initial dimensions and coefficients of linear expansion, we can observe the change in the outer radius directly: \(\Delta r_{s} = r_{s} \beta \Delta T = r_{sh} \beta \Delta T = \Delta r_{sh}\) Since the change in the outer radius \(\Delta r_{s}\) is equal to the change in the outer radius \(\Delta r_{sh}\), we can conclude that both the solid cylinder and the cylindrical shell will expand to the same larger outer radius when they experience the same temperature increase \(\Delta T\).

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.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

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

\(\cdot 17.41\) A clock based on a simple pendulum is situated outdoors in Anchorage, Alaska. The pendulum consists of a mass of 1.00 kg that is hanging from a thin brass rod that is \(2.000 \mathrm{~m}\) long. The clock is calibrated perfectly during a summer day with an average temperature of \(25.0^{\circ} \mathrm{C}\). During the winter, when the average temperature over one 24 -h period is \(-20.0^{\circ} \mathrm{C}\), find the time elapsed for that period according to the simple pendulum clock.

A 25.01 -mm-diameter brass ball sits at room temperature on a 25.00 - mm- diameter hole made in an aluminum plate. The ball and plate are heated uniformly in a furnace, so both are at the same temperature at all times. At what temperature will the ball fall through the plate?

On a hot summer day, a cubical swimming pool is filled to within \(1.0 \mathrm{~cm}\) of the top with water at \(21{ }^{\circ} \mathrm{C} .\) When the water warms to \(37^{\circ} \mathrm{C}\), the pool overflows. What is the depth of the pool?

You are designing a precision mercury thermometer based on the thermal expansion of mercury \(\left(\beta=1.8 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\right)\) which causes the mercury to expand up a thin capillary as the temperature increases. The equation for the change in volume of the mercury as a function of temperature is \(\Delta V=\beta V_{0} \Delta T\) where \(V_{0}\) is the initial volume of the mercury and \(\Delta V\) is the change in volume due to a change in temperature, \(\Delta T .\) In response to a temperature change of \(1.0^{\circ} \mathrm{C}\), the column of mercury in your precision thermometer should move a distance \(D=1.0 \mathrm{~cm}\) up a cylindrical capillary of radius \(r=0.10 \mathrm{~mm} .\) Determine the initial volume of mercury that allows this change. Then find the radius of a spherical bulb that contains this volume of mercury.

In a pickup basketball game, your friend cracked one of his teeth in a collision with another player while attempting to make a basket. To correct the problem, his dentist placed a steel band of initial internal diameter \(4.4 \mathrm{~mm},\) and a crosssectional area of width \(3.5 \mathrm{~mm},\) and thickness \(0.45 \mathrm{~mm}\) on the tooth. Before placing the band on the tooth, he heated the band to \(70 .{ }^{\circ} \mathrm{C}\). What will be the tension in the band once it cools down to the temperature in your friend's mouth \(\left(37^{\circ} \mathrm{C}\right) ?\)
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Physics of Motion

Read Explanation

Electromagnetism

Read Explanation

Energy Physics

Read Explanation

Translational Dynamics

Read Explanation

Physical Quantities And Units

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.

Sign-up for free