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Ch 17: Temperature and Heat
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 17: Temperature and HeatProblem 21a
Chapter 17, Problem 21a

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?

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First, identify the concept involved: thermal expansion. When materials are heated, they expand. The amount of expansion can be calculated using the formula for linear expansion: ΔL = LαΔT, where ΔL is the change in length, L is the original length, α is the coefficient of linear expansion, and ΔT is the change in temperature.
Determine the change in temperature ΔT by subtracting the initial temperature from the final temperature: ΔT = 33.0°C - (-9.0°C). This will give you the temperature increase that the rails experience from winter to summer.
Find the coefficient of linear expansion α for steel. This is a known value and can be found in physics reference materials. For steel, α is typically around 12 imes 10^{-6} \(\text{°C}\)^{-1}.
Calculate the change in length ΔL using the formula: ΔL = 12.0 \(\text{ m}\) imes 12 imes 10^{-6} \(\text{°C}\)^{-1} imes \(\text{ΔT}\). Substitute the values for L, α, and ΔT into the equation.
The result from the calculation will give you the amount of space that must be left between adjacent rails to accommodate the expansion and ensure they just touch on a summer day.

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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 shape, area, and volume in response to a change in temperature. For solids like steel, this expansion is linear and can be calculated using the formula ΔL = αLΔT, where ΔL is the change in length, α is the coefficient of linear expansion, L is the original length, and ΔT is the change in temperature.
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Coefficient of Linear Expansion

The coefficient of linear expansion (α) is a material-specific constant that quantifies how much a material expands per degree change in temperature. For steel, this value is typically around 11 x 10^-6 /°C. This coefficient is crucial for calculating the change in length of the steel rails as the temperature varies from winter to summer.
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Temperature Change Calculation

Temperature change calculation involves determining the difference between the initial and final temperatures. In this scenario, the temperature changes from -9.0°C in winter to 33.0°C in summer, resulting in a ΔT of 42.0°C. This change is used in the thermal expansion formula to calculate how much the steel rails will expand.
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Related Practice
Textbook Question

In very cold weather a significant mechanism for heat loss by the human body is energy expended in warming the air taken into the lungs with each breath. On a cold winter day when the temperature is -20°C, what amount of heat is needed to warm to body temperature (37°C) the 0.50 L of air exchanged with each breath? Assume that the specific heat of air is 1020 J/kg K and that 1.0 L of air has mass 1.3 × 10-3 kg.

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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?

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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?

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Textbook Question

As a new mechanical engineer for Engines Inc., you have been assigned to design brass pistons to slide inside steel cylinders. The engines in which these pistons will be used will operate between 20.0°C and 150.0°C. Assume that the coefficients of expansion are constant over this temperature range. If the piston just fits inside the chamber at 20.0°C, will the engines be able to run at higher temperatures? Explain.

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Textbook Question

A brass rod is 185 cm long and 1.60 cm in diameter. What force must be applied to each end of the rod to prevent it from contracting when it is cooled from 120.0°C to 10.0°C?

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Textbook Question

In an effort to stay awake for an all-night study session, a student makes a cup of coffee by first placing a 200-W electric immersion heater in 0.320 kg of water. How much time is required? Assume that all of the heater's power goes into heating the water.

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