Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 17: Problem 36

Even though steel has a relatively low linear expansion coefficient \(\left(\alpha_{\text {steel }}=13 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right),\) the expansion of steel railroad tracks can potentially create significant problems on very hot summer days. To accommodate for the thermal expansion, a gap is left between consecutive sections of the track. If each section is \(25.0 \mathrm{~m}\) long at \(20.0{ }^{\circ} \mathrm{C}\) and the gap between sections is \(10.0 \mathrm{~mm}\) wide, what is the highest temperature the tracks can take before the expansion creates compressive forces between sections?

Short Answer

Expert verified
Answer: The highest temperature the steel railroad tracks can take before the expansion creates compressive forces between sections is \(50.77 \, ^{\circ}C\).

Step by step solution

01

List the given information and what we want to find

- Initial length of each section, \(L_0 = 25.0 \, m\) - Gap between sections, Gap \(= 10.0 \, mm = 0.010 \, m\) - Linear expansion coefficient of steel, \(\alpha_{\text{steel}}=13 \cdot 10^{-6} \, K^{-1}\) - Initial temperature \(T_i = 20.0 \, ^{\circ}C = 293.15 \, K\) - Final temperature \(T_f\): unknown (what we want to find)
02

Calculate the change in temperature (\(\Delta T\))

We first need to find the change in temperature \(\Delta T\). To do this, we will use the linear expansion formula, rearranged to solve for \(\Delta T\): $$\Delta T = \frac{\Delta L}{L_0 \alpha}$$ We know that the change in length \(\Delta L\) must be equal to the gap between the sections for compressive forces to arise: $$\Delta L = \text{Gap} = 0.010 \, m$$ Now we can plug the known values into the formula and solve for \(\Delta T\): $$\Delta T = \frac{0.010 \, m}{25.0 \, m \cdot 13 \cdot 10^{-6} \, K^{-1}} = 30.77 \, K$$
03

Calculate the highest temperature (\(T_f\))

Now that we have found the change in temperature, we can calculate the highest temperature the tracks can take before compressive forces arise. We can do this by adding the change in temperature to the initial temperature: $$T_f = T_i + \Delta T = 293.15 \, K + 30.77 \, K = 323.92 \, K$$ Since the given initial temperature was in Celsius, we will also convert our answer back to Celsius: $$T_f = 323.92 \, K - 273.15 = 50.77 \, ^{\circ}C$$
04

State the final answer

The highest temperature the steel railroad tracks can take before the expansion creates compressive forces between sections is \(50.77 \, ^{\circ}C\).

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

For food storage, what is the advantage of placing a metal lid on a glass jar? (Hint: Why does running the metal lid under hot water for a minute help you open such a jar?)

A medical device used for handling tissue samples has two metal screws, one \(20.0 \mathrm{~cm}\) long and made from brass \(\left(\alpha_{\mathrm{b}}=18.9 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)\) and the other \(30.0 \mathrm{~cm}\) long and made from aluminum \(\left(\alpha_{\mathrm{a}}=23.0 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)\). A gap of \(1.00 \mathrm{~mm}\) exists between the ends of the screws at \(22.0^{\circ} \mathrm{C}\). At what temperature will the two screws touch?

A plastic-epoxy sheet has uniform holes of radius \(1.99 \mathrm{~cm}\). The holes are intended to allow solid ball bear- ings with an outer radius of \(2.00 \mathrm{~cm}\) to just go through. Over what temperature rise must the plastic-epoxy sheet be heated so that the ball bearings will go through the holes? The linear expansion coefficient of plastic-epoxy is about \(1.3 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\).

At what temperature do the Celsius and Fahrenheit temperature scales have the same numeric value? a) -40 degrees b) 0 degrees c) 40 degrees d) 100 degrees

Two solid objects are made of different materials. Their volumes and volume expansion coefficients are \(V_{1}\) and \(V_{2}\) and \(\beta_{1}\) and \(\beta_{2}\), respectively. It is observed that during a temperature change of \(\Delta T\), the volume of each object changes by the same amount. If \(V_{1}=2 V_{2}\) what is the ratio of the volume expansion coefficients?
See all solutions

Recommended explanations on Physics Textbooks

Modern Physics

Read Explanation

Solid State Physics

Read Explanation

Translational Dynamics

Read Explanation

Magnetism

Read Explanation

Atoms and Radioactivity

Read Explanation

Turning Points in 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.

Sign-up for free