Understanding Chain Surveying Corrections

In surveying, accurate measurements are crucial. Often, the measuring instrument, like a chain, might not be its exact nominal length. This discrepancy leads to errors in the recorded measurements, which must be corrected to find the true length of the line being measured.

Problem Description

We are given a scenario where a distance was measured using a 20 m chain. The recorded distance is 327 m. However, it was discovered that the chain was actually 3 cm too long.

We need to determine the true length of the line measured.

Analyzing the Given Data

• Nominal length of the chain = 20 m
• Recorded (measured) distance = 327 m
• Error in chain length = +3 cm (chain is too long)

Calculating the Actual Length of the Chain

The nominal length is 20 m. The chain is 3 cm too long. First, convert the error to meters.

3 cm = $\frac{3}{100}$ m = 0.03 m

The actual length of the chain is the nominal length plus the error:

Actual Length of Chain = Nominal Length + Error

Actual Length of Chain = 20 m + 0.03 m = 20.03 m

Formula for True Length Calculation

When a measurement is taken with a chain that is not the correct length, the true length of the line can be calculated using the following formula:

$\text{True Length} = \left(\frac{\text{Actual Length of Chain}}{\text{Nominal Length of Chain}}\right) \times \text{Measured Length}$

This formula accounts for the difference between the standard length and the actual length of the measuring instrument over the entire measured distance.

Step-by-Step Calculation of True Length

Using the formula and the values we have:

• Actual Length of Chain = 20.03 m
• Nominal Length of Chain = 20 m
• Measured Length = 327 m

Substitute these values into the formula:

$\text{True Length} = \left(\frac{20.03 \text{ m}}{20 \text{ m}}\right) \times 327 \text{ m}$

Calculate the ratio of the actual length to the nominal length:

Ratio = $\frac{20.03}{20} = 1.0015$

Now, multiply this ratio by the measured length:

True Length = $1.0015 \times 327 \text{ m}$

True Length = $327.4905 \text{ m}$

Result Analysis

The calculated true length is 327.4905 m. When rounded to two decimal places, this is 327.49 m.

Let's compare this result with the given options:

Our calculated true length of 327.49 m matches Option 2.

It makes sense that the true length is greater than the measured length. Since the chain was too long, each application of the chain covered slightly more distance than the nominal 20 m. Therefore, fewer applications of the faulty chain were needed to cover the distance compared to a standard 20 m chain. This results in a measured distance that is less than the actual or true distance.

Revision Table: Chain Correction Formula

Additional Information on Surveying Corrections

Chain surveying involves various sources of error that surveyors must account for. Errors can be systematic or accidental.

• Systematic Errors: These errors follow a definite pattern and can be calculated and corrected. The error due to incorrect chain length is a systematic error. Other examples include errors due to temperature changes affecting chain length, sag in the chain, and improper alignment.
• Accidental Errors: These errors are random and unpredictable. They tend to balance out over a large number of measurements. Examples include errors in marking the chain ends, errors in reading the tape, or slight variations in pulling tension.

Correcting for incorrect chain length is a fundamental calculation in surveying to ensure the accuracy of measurements and subsequent maps or plans.

When the chain is too long, the measured distance is less than the true distance, so the correction increases the measured length.

When the chain is too short, the measured distance is greater than the true distance, so the correction decreases the measured length.

The formula used, $\text{True Length} = \left(\frac{\text{Actual Length}}{\text{Nominal Length}}\right) \times \text{Measured Length}$, correctly applies this principle by scaling the measured length by the ratio of the actual chain length to its intended length.