Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Giancoli Douglas - Physics for Scientists and Engineers 5th edition

Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook

Giancoli Douglas 5th edition

Ch. 17 - Temperature, Thermal Expansion, and the Ideal Gas Law

Problem 67b

Chapter 17, Problem 67b

A precise steel tape measure has been calibrated at 14°C. At 37°C, what will be the percentage error?

Verified step by step guidance

Understand the problem: The steel tape measure expands when the temperature increases. The percentage error arises because the tape measure's length changes, leading to incorrect measurements. We need to calculate the percentage error due to thermal expansion.

Step 1: Write the formula for linear thermal expansion:

L = L_0 (1 + \(\alpha\) \(\Delta\) T)

, where

L

is the final length,

L_0

is the original length,

\(\alpha\)

is the coefficient of linear expansion for steel, and

\(\Delta\) T

is the temperature change.

Step 2: Calculate the temperature change:

\(\Delta\) T = T_{final} - T_{initial}

. Substitute

T_{final} = 37\,^{\(\circ\)}\(\text{C}\)

and

T_{initial} = 14\,^{\(\circ\)}\(\text{C}\)

to find

\(\Delta\) T

Step 3: Use the formula for percentage error:

\(\text{Percentage Error}\) = \(\frac{\Delta L}{L_0}\) \(\times\) 100

, where

\(\Delta\) L = L - L_0

. Substitute

\(\Delta\) L = L_0 \(\alpha\) \(\Delta\) T

into the formula to get

\(\text{Percentage Error}\) = \(\alpha\) \(\Delta\) T \(\times\) 100

Step 4: Substitute the known values: Use the coefficient of linear expansion for steel (

\(\alpha\) \(\approx\) 1.2 \(\times\) 10^{-5}\,\(\text{°C}\)^{-1}

) and the calculated

\(\Delta\) T

to compute the percentage error. Simplify the expression to find the result.

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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 increase in size or volume of materials as they are heated. In solids, this occurs due to the increased vibration of atoms, which causes them to occupy more space. For a steel tape measure, this means that as the temperature rises from its calibration point, the tape will expand, leading to inaccuracies in measurements.

Coefficient of Linear Expansion

The coefficient of linear expansion is a material-specific constant that quantifies how much a material expands per degree of temperature increase. For steel, this coefficient is typically around 11 x 10^-6 /°C. Understanding this concept is crucial for calculating the change in length of the tape measure as the temperature changes from the calibration temperature.

Percentage Error

Percentage error is a way to express the accuracy of a measurement by comparing the difference between the measured value and the true value relative to the true value. It is calculated using the formula: (|Measured Value - True Value| / True Value) x 100%. In this context, it helps quantify how much the expanded length of the tape measure deviates from the intended measurement due to thermal expansion.

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