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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: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 17 - Temperature, Thermal Expansion, and the Ideal Gas LawProblema 67b
Capitolo 17, Problema 67b

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

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1
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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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.
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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.
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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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