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The coefficient of linear expansion, denoted as \( \alpha \), is a measure that describes how the size of an object changes with a change in temperature. It is crucial in understanding how materials expand or contract when subjected to temperature variations. In the context of linear expansion, this coefficient indicates how much a unit length of a material will expand for a one-degree Celsius change in temperature.
In our example, the steel tape has a coefficient of linear expansion \( \alpha = 1.1 \times 10^{-5} \ { }^{\circ} \mathrm{C}^{-1} \). This value means that for every degree Celsius change, a 1-meter length of the steel tape would expand or contract by \( 1.1 \times 10^{-5} \) meters. This concept is helpful when calibrating measuring instruments in varying environmental conditions.

• Smaller coefficient values indicate less expansion or contraction with temperature changes.
• Material properties, like atomic structure and bonding, influence the coefficient.

Understanding the coefficient allows engineers and designers to predict how materials will behave in different temperatures and adjust their designs accordingly.

Temperature change, denoted as \( \Delta T \), refers to the difference in temperature that an object experiences. It plays a key role in determining how much an object made from a given material will expand or contract.
For the steel tape, the initial temperature is \( 20^{\circ} \mathrm{C} \), and it was later exposed to \( -15^{\circ} \mathrm{C} \). To calculate \( \Delta T \), we subtract the final temperature from the initial temperature:
\[ \Delta T = -15^{\circ} \mathrm{C} - 20^{\circ} \mathrm{C} = -35^{\circ} \mathrm{C} \]
This negative temperature change indicates that the steel tape is experiencing a decrease in temperature, leading it to contract rather than expand.

• A positive \( \Delta T \) would cause expansion if the material can stretch with heat.
• Neutral or no \( \Delta T \) implies that there would be no size alteration in the material.

Knowing the temperature change is essential for predicting any dimensional shifts in materials due to environmental changes.

Percent error is a way to express how approximate a measured or estimated value is when compared to an expected or true value. In our exercise, we calculated the percent error to determine the impact of temperature changes on the steel tape's measurements.
We use the formula:
\[ \frac{\Delta L}{L_0} = \alpha \cdot \Delta T \]
to find the ratio of change in length \( \Delta L \) to the original length \( L_0 \). The product of the coefficient of linear expansion and the temperature change gives us this ratio, \( -3.85 \times 10^{-4} \) in this situation. Converting this ratio into a percentage helps to visualize the error:
\[ \text{Percent Error} = \left( \frac{\Delta L}{L_0} \right) \times 100 = -0.039\% \]
This percentage shows that, due to contraction, the tape is measuring 0.039% shorter than it should. It's crucial because it allows users to compensate for or correct measurement inaccuracies caused by temperature fluctuations.

• Negative percent error suggests underestimation.
• Accuracy of the calculation depends on precision in measuring \( \alpha \) and \( \Delta T \).

Understanding percent error calculation aids in making reliable measurements across different application scenarios.