A 20 m chain was found to be 4 cm too long after chaining 1400 m. It was 8 cm too long at the end of day’s work after chaining a total distance of 2420 m. If the chain was correct before commencement of the work, find the true distance.

Key Information

• Nominal Length of Chain (L_nominal) = 20 m
• Initial Actual Length (L_start) = 20.00 m (Error = 0 cm)
• Actual Length after 1400 m (L_mid) = 20 m + 0.04 m = 20.04 m (Error = +4 cm)
• Actual Length after 2420 m (L_end) = 20 m + 0.08 m = 20.08 m (Error = +8 cm)
• Measured Distance Segment 1 = 1400 m
• Measured Distance Segment 2 = 2420 m – 1400 m = 1020 m
• Goal: Find the Total True Distance.

Step 1: Calculate True Distance for First Segment (0 m to 1400 m)

Step 2: Calculate True Distance for Second Segment (1400 m to 2420 m)

Step 3: Calculate Total True Distance

Final Result

Core Concepts:

1. Systematic Error (Variable): Unlike the previous example where the chain error was constant, here the error changes during the measurement process (the chain stretches). This is also a systematic error, but its magnitude varies.
2. Calibration Drift: The change in the instrument’s accuracy over time or due to use. In this case, the chain progressively became longer.
3. Segmented Correction: When the error is known to vary, the measurement must be broken down into segments between points where the error was known or checked.
4. Averaging Error Assumption: Within each segment, it’s common practice to assume the error changes linearly. Therefore, the average error (or average actual length) for the segment is used for correction. Average Length = (Length at Start + Length at End) / 2.
5. Correction Formula Application: The standard formula (True = (L_actual_avg / L_nominal) * Measured) is applied individually to each segment.

Real-World Applications:

• Precise Surveying over Long Durations: Especially relevant when environmental factors (like temperature causing expansion/contraction) or instrument wear can cause calibration drift during the work. Intermediate checks are crucial.
• Using Older Equipment: Older tapes or chains might be prone to stretching or deforming with use, necessitating periodic checks and segmented corrections.
• Analysis of Historical Data: Interpreting old survey field notes where intermediate instrument checks might have been recorded.
• High-Accuracy Construction Layout: For large projects, ensuring that measurement errors due to instrument drift are accounted for throughout the process.
• Geodetic Control Surveys: Where very high precision is required, understanding and correcting for all sources of error, including instrument drift, is critical.

Why It Works:
Treating the entire 2420 m measurement with a single correction factor (e.g., based only on the final error of +8 cm) would be inaccurate because the chain wasn’t that long for the entire duration. Similarly, using only the initial state (correct length) is also wrong. By dividing the work into segments (0-1400 m and 1400-2420 m) based on when the chain’s length was known, we can isolate periods where the change in length was relatively small. Assuming a linear change in length *within* each segment allows us to estimate the *average* actual length of the chain during that specific measurement period (20.02 m for the first segment, 20.06 m for the second). Applying the standard correction formula to each segment using its corresponding average length provides a much better estimate of the true distance covered in that segment. Summing these individually corrected segment distances yields the most accurate overall true distance given the available information about the changing chain length.