A surveyor has a steel measuring tape that is calibrated to be 100.000 m long (i.e., accurate to ±1 mm) at 20°C. If she measures the distance between two stakes to be 65.175 m on a 3°C day, does she need to add or subtract a correction factor to get the true distance? How large, in mm, is the correction factor?

Verified step by step guidance

1. Identify the coefficient of linear expansion for steel, which is typically around $11 \times 10^{-6} \, \text{°C}^{-1}$.
2. Calculate the change in temperature from the calibration temperature to the temperature on the day of measurement. This is done by subtracting the day's temperature from the calibration temperature: $\Delta T = 20°C - 3°C$.
3. Use the formula for linear expansion to find the change in length of the tape. The formula is $\Delta L = L_0 \alpha \Delta T$, where $L_0$ is the original length, $\alpha$ is the coefficient of linear expansion, and $\Delta T$ is the change in temperature.
4. Determine if the tape length has increased or decreased. Since the temperature is lower than the calibration temperature, the tape contracts, and the measured length is less than the true distance.
5. Calculate the correction factor in millimeters to find the true distance. Add this correction factor to the measured length to get the true distance between the stakes.

Key Concepts

Thermal Expansion

Thermal expansion refers to the tendency of materials to change in size or volume in response to changes in temperature. For metals like steel, this means that as the temperature decreases, the material contracts, leading to a shorter length. Understanding this concept is crucial for determining how much the measuring tape's length will change when the temperature drops from 20°C to 3°C.

Calibration and Measurement Accuracy

Calibration is the process of adjusting the precision of a measuring instrument to ensure accurate readings. The steel measuring tape is calibrated to be accurate at 20°C, meaning its length is defined at this temperature. When measuring at a different temperature, it is essential to account for any discrepancies that arise due to thermal expansion or contraction, which can affect the accuracy of the measurement.

Correction Factor

A correction factor is a numerical adjustment applied to a measurement to account for systematic errors or deviations from the true value. In this scenario, the correction factor will be calculated based on the change in temperature and the coefficient of linear expansion for steel. This factor will indicate whether the measured distance should be increased or decreased to reflect the true distance between the stakes.