Ch 17: Temperature and Heat

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 17: Temperature and Heat

Problem 16

Chapter 17, Problem 16

A geodesic dome constructed with an aluminum framework is a nearly perfect hemisphere; its diameter measures 55.0 m on a winter day at a temperature of -15°C. How much more interior space does the dome have in the summer, when the temperature is 35°C?

Verified step by step guidance

First, understand that the problem involves thermal expansion, which is the increase in volume of a material as its temperature increases. For a solid, the change in volume $ \Delta V $ can be calculated using the formula: $ \Delta V = \beta V_0 \Delta T $, where $ \beta $ is the coefficient of volume expansion, $ V_0 $ is the initial volume, and $ \Delta T $ is the change in temperature.

Calculate the initial volume $ V_0 $ of the hemisphere. The formula for the volume of a hemisphere is $ V = \frac{2}{3} \pi r^3 $. First, find the radius $ r $ by dividing the diameter by 2: $ r = \frac{55.0}{2} $ m.

Substitute the radius into the volume formula to find $ V_0 $: $ V_0 = \frac{2}{3} \pi (\frac{55.0}{2})^3 $.

Determine the change in temperature $ \Delta T $ by subtracting the initial temperature from the final temperature: $ \Delta T = 35°C - (-15°C) = 50°C $.

Use the coefficient of volume expansion for aluminum, $ \beta \approx 69 \times 10^{-6} \text{°C}^{-1} $, to calculate the change in volume $ \Delta V $ using the formula: $ \Delta V = \beta V_0 \Delta T $. This will give you the additional interior space in the summer.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Thermal Expansion

Thermal expansion refers to the tendency of matter to change in volume in response to a change in temperature. For solids like aluminum, this expansion is typically linear, meaning the material expands uniformly in all directions. The coefficient of linear expansion quantifies how much a material expands per degree of temperature change, which is crucial for calculating changes in the dome's dimensions.

Volume of a Hemisphere

The volume of a hemisphere is calculated using the formula V = (2/3)πr³, where r is the radius of the hemisphere. Understanding this formula is essential for determining the interior space of the dome. As the dome's diameter changes due to thermal expansion, the radius changes, affecting the volume calculation.

Temperature Conversion

Temperature conversion is necessary to ensure consistent units when applying formulas involving thermal expansion. In this context, temperatures are given in Celsius, which is suitable for calculating changes in physical properties. Recognizing the temperature difference between winter and summer is key to determining the extent of expansion and its impact on the dome's volume.

Recommended video:

Guided course

07:46

Unit Conversions

Related Practice

Textbook Question

A 6.00-kg piece of solid copper metal at an initial temperature T is placed with 2.00 kg of ice that is initially at -20.0°C. The ice is in an insulated container of negligible mass and no heat is exchanged with the surroundings. After thermal equilibrium is reached, there is 1.20 kg of ice and 0.80 kg of liquid water. What was the initial temperature of the piece of copper?

views

Textbook Question

Calculate the one temperature at which Fahrenheit and Kelvin thermometers agree with each other

views

Textbook Question

A steel tank is completely filled with 1.90 m³ of ethanol when both the tank and the ethanol are at 32.0°C. When the tank and its contents have cooled to 18.0°C, what additional volume of ethanol can be put into the tank?

views

Textbook Question

The pressure of a gas at the triple point of water is $1.35$ atm. If its volume remains unchanged, what will its pressure be at the temperature at which CO₂ solidifies?

Textbook Question

Steel train rails are laid in 12.0-m-long segments placed end to end. The rails are laid on a winter day when their temperature is -9.0°C. How much space must be left between adjacent rails if they are just to touch on a summer day when their temperature is 33.0°C?

views

Textbook Question

A constant-volume gas thermometer registers an absolute pressure corresponding to $325$ mm of mercury when in contact with water at the triple point. What pressure does it read when in contact with water at the normal boiling point?

views