Problem Statement
The length of a survey line was measured using a chain with a nominal length of 30 m and found to be 631.5 m. Upon comparison with a standard, the chain was found to be actually 0.10 m too long. Find the true length of the survey line.
Step-by-Step Solution
Key Information
- Measured Length of Survey Line = 631.5 m
- Nominal Length of Chain (L) = 30 m
- Error in Chain = +0.10 m (too long)
- Actual Length of Chain (L’) = L + Error = 30 m + 0.10 m = 30.10 m
- Goal: Find the True Length of the survey line.
Step 1: Understand the Correction Principle
When the measuring tool (chain) is longer than its stated nominal length, it covers more ground than recorded for each measurement increment. This means the measured length is shorter than the true length.
To find the true length, we need to apply a correction factor based on the ratio of the actual chain length (L’) to the nominal chain length (L).
The correction formula is:
True Length = (Actual Chain Length / Nominal Chain Length) × Measured Length
True Length = (L’ / L) × Measured Length
Step 2: Apply the Formula and Calculate
Substitute the known values into the formula:
L’ = 30.10 m
L = 30 m
Measured Length = 631.5 m
True Length = (30.10 m / 30 m) × 631.5 m
True Length = (1.00333…) × 631.5 m
True Length ≈ 633.605 m
Final Result
Conceptual Explanation & Applications
Core Concepts:
- Linear Measurement: Determining the distance between two points along a straight or curved line using tools like chains, tapes, or electronic distance measurers (EDMs).
- Measurement Standard & Calibration: A standard is an accepted reference (e.g., a standard meter). Calibration is the process of comparing a measuring instrument against a standard to determine its accuracy and any deviation (error).
- Systematic Error: An error that is consistent and repeatable, often due to instrument inaccuracy (like a chain being too long or too short). This type of error can usually be corrected if the magnitude of the error is known.
- Correction Factor: A numerical value used to adjust a measured quantity to account for known errors. In this case, the factor is the ratio of the actual instrument length to its nominal length (L’/L).
- Proportional Reasoning: The correction assumes the error is consistent along the entire length of the chain and applies proportionally to the total measured distance.
Real-World Applications:
- Land Surveying: Essential for accurate boundary determination, topographic mapping, and setting out construction works. Chains and tapes require regular calibration.
- Construction & Civil Engineering: Precise layout of buildings, roads, bridges relies on accurate distance measurements. Correction for tape errors is vital for quality control.
- Manufacturing: Ensuring measuring tools used in production meet accuracy standards.
- Scientific Experiments: Correcting measurements made with instruments that have known calibration errors.
- Calibration Services: Laboratories specialize in comparing measuring instruments to standards and providing correction data.
Why It Works:
The measurement process involves laying the chain end-to-end multiple times. The measured length (631.5 m) implicitly represents the number of times the *nominal* length (30 m) was thought to be laid out. However, each time the 30 m chain was laid, a true distance of 30.10 m was covered. The ratio L’/L = 30.10 / 30 calculates the correction factor for each nominal meter measured; it tells us that for every 1 meter *recorded*, 1.00333… meters were actually covered. Multiplying the total measured length (631.5 m) by this correction factor scales the entire measurement to reflect the true distance covered by the slightly longer chain. If the chain were too short (L’ < L), the factor would be less than 1, resulting in a true length shorter than the measured length.
