/*! 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 42 When you apply the brakes on you... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 42

When you apply the brakes on your car, the kinetic energy of your vehicle is transformed into thermal energy in your brake disks. During a mountain descent, a 28.00 -cm-diameter iron brake disk heats up from \(30^{\circ} \mathrm{C}\) to \(180^{\circ} \mathrm{C} .\) What is the diameter of the disk after it heats up?

Short Answer

Expert verified
The new diameter of the heated disk is approximately 28.0504 cm.

Step by step solution

01

Identify the given variables

The diameter of the brake disk is initially 28.00 cm, the initial temperature is 30掳C, the final temperature is 180掳C, and the coefficient of linear expansion for iron, \(\alpha\), is \(12 \times 10^{-6}/\mathrm{C}\). Therefore, we have: \(L_0 = 28.00\) cm, \(\Delta T = 180掳C - 30掳C = 150掳C\), and \(\alpha = 12 \times 10^{-6}/\mathrm{C}\).
02

Substitute values into the formula

Next, we substitute the known values into the formula for linear thermal expansion \(\Delta L = L_0 \alpha \Delta T\). Doing so gives us \(\Delta L = 28.00 \, \text{cm} \times 12 \times 10^{-6}/\mathrm{C} \times 150掳C\).
03

Calculate the change in length

After performing the multiplication, we find that \(\Delta L\), the change in the diameter, is approximately 0.0504 cm.
04

Find the final diameter

Now that we have the change in diameter, we can easily find the new diameter of the disk: \(D_{new} = L_0 + \Delta L = 28.00 \, cm + 0.0504 \, cm = 28.0504 \, cm\). Since the brake disk is round, the diameter change will be uniform all around the disk.

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.

Thermal Energy
Thermal energy can be thought of as the energy that is associated with the temperature of an object. It is a form of kinetic energy because it arises from the motion of particles within a substance. The faster these particles move, the higher the temperature and, consequently, the greater the thermal energy of the object.

In practical terms, when you apply the brakes on your car, the kinetic energy of the moving car is transformed into heat due to the friction between the brake pads and the brake disks. This process increases the thermal energy of the brake disks, leading to an increase in temperature. As a result, various properties of the brake disk, such as its size, can change due to thermal expansion.
Coefficient of Linear Expansion
The coefficient of linear expansion, represented by the symbol \( \alpha \), quantifies how much a material's linear dimension changes with temperature. It is specific to each material and is typically expressed as an increase in length per degree of temperature increase, often in units such as \(1/掳C\) or \(1/K\).

To continue with the brake disk example, iron has a coefficient of linear expansion, \( \alpha \), of \(12 \times 10^{-6}/掳C\). This value tells us how much a given length of iron will expand or contract for each degree of temperature change. The calculation for linear expansion \( \Delta L = L_0 \alpha \Delta T \) uses this coefficient to determine the new size of an object after a temperature change.
Temperature Change in Physics
Temperature change in physics refers to the variation in temperature of a substance or object between two states or over time. This concept is critical because it's a driving force behind many physical phenomena, including the expansion or contraction of materials, phase changes, and the generation of thermal energy.

The temperature change is denoted as \( \Delta T \) and is calculated as the final temperature minus the initial temperature. In our example, the 28.00 cm-diameter iron brake disk heats up from \(30^\circ C\) to \(180^\circ C\), resulting in a temperature change, \( \Delta T \) of \(150^\circ C\). This temperature change, when plugged into the linear expansion formula along with the coefficient of linear expansion, allows us to predict how much the iron brake disk will expand.

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 compressed-air cylinder is known to fail if the pressure exceeds 110 atm. A cylinder that was filled to 25 atm at \(20^{\circ} \mathrm{C}\) is stored in a warehouse. Unfortunately, the warehouse catches fire and the temperature reaches \(950^{\circ} \mathrm{C} .\) Does the cylinder explode?

Electronics and inhabitants of the International Space Station generate a significant amount of thermal energy that the station must get rid of. The only way that the station can exhaust thermal energy is by radiation, which it does using thin, 1.8 -m-by3.6-m panels that have a working temperature of about \(6^{\circ} \mathrm{C}\). How much power is radiated from each panel? Assume that the panels are in the shade so that the absorbed radiation will be negligible. Assume that the emissivity of the panels is \(1.0 .\) Hint: Don't forget that the panels have two sides!

When air is inhaled, it quickly becomes saturated with water vapor as it passes through the moist upper airways. When a person breathes dry air, about 25 mg of water are exhaled with each breath. At 12 breaths/min, what is the rate of energy loss due to evaporation? Express your answer in both watts and Calories per day. At body temperature, the heat of vaporization of water is \(L_{\mathrm{v}}=24 \times 10^{5} \mathrm{J} / \mathrm{kg}\).

An often-overlooked consequence of power generation is "thermal pollution" from excess heat deposited in the environment. The heating can be considerable. Here are some typical numbers: A nuclear power plant generates \(1.0 \mathrm{GW}\) of electric power with an operating efficiency of \(35 \% .\) The reactor has a single-pass cooling system; water from a nearby river is brought in, warmed, and returned to the river. When operating at full power, the plant takes in \(90 \mathrm{m}^{3}\) of water per second, comparable to the flow of a small river. If the river water comes in at \(10^{\circ} \mathrm{C},\) at what temperature does it emerge?

A 5.0-m-diameter garden pond holds \(5.9 \times 10^{3} \mathrm{kg}\) of water. \(400 \mathrm{W} / \mathrm{m}^{2}\) of solar energy strikes the pond's surface, meaning each square meter of the pond's surface is absorbing \(400 \mathrm{W}\). How many hours will it take for the pond to warm from \(15^{\circ} \mathrm{C}\) to \(25^{\circ} \mathrm{C} ?\)
See all solutions

Recommended explanations on Physics Textbooks

Radiation

Read Explanation

Circular Motion and Gravitation

Read Explanation

Linear Momentum

Read Explanation

Kinematics Physics

Read Explanation

Particle Model of Matter

Read Explanation

Conservation of Energy and Momentum

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.