Problem Statement
The distance between two points, measured with a 20 m chain, was recorded as 327 m. It was afterwards found that the chain was 3 cm too long. What was the true distance between the points?
Step-by-Step Solution
Key Information & Setup
- Nominal length of chain (L) = 20 m
- Error in chain length (e) = 3 cm = 0.03 m (too long)
- Actual chain length (L’) = L + e = 20 + 0.03 = 20.03 m
- Measured distance = 327 m
- Goal: Find the true distance between the points
Step 1: Understanding The Error
When a chain is longer than its nominal length, the recorded number of chains will be fewer than reality. This means the reported distance will be shorter than the true distance.
For example, if we measure exactly 1 chain length with our elongated chain:
- We report: “1 chain = 20 m”
- But actually measured: 20.03 m
So for any measurement, the ratio of true distance to recorded distance is:
True Distance / Recorded Distance = Actual Chain Length / Nominal Chain Length
True Distance / Recorded Distance = L’ / L = 20.03 / 20 = 1.0015
Step 2: Calculate The True Distance
We can now calculate the true distance:
True Distance = Recorded Distance × (L’ / L)
True Distance = 327 × (20.03 / 20)
True Distance = 327 × 1.0015
True Distance = 327.49 m
Alternative Approach Using Correction Formula
We can also solve this using the standard chain correction formula:
True Distance = Recorded Distance + (Recorded Distance × Error / Nominal Length)
True Distance = 327 + (327 × 0.03 / 20)
True Distance = 327 + (327 × 0.0015)
True Distance = 327 + 0.49
True Distance = 327.49 m
Final Result
Explanation of Chain Error Corrections
Understanding Chain Errors in Surveying:
- Chain Too Long: When a chain is longer than its nominal length, fewer chain lengths are needed to cover the same distance. This results in a reported measurement that is shorter than the true distance. To correct this, we must multiply the reported distance by the ratio (actual length / nominal length), which will be greater than 1.
- Chain Too Short: Conversely, when a chain is shorter than its nominal length, more chain lengths are needed to cover the same distance. This results in a reported measurement that is longer than the true distance. To correct this, we multiply the reported distance by the ratio (actual length / nominal length), which will be less than 1.
General Formula:
True Distance = Recorded Distance × (Actual Chain Length / Nominal Chain Length)
In traditional surveying with chains, this type of error correction was critical for ensuring accurate measurements, especially over long distances where small errors in the measuring instrument could lead to significant cumulative errors in the final results.