With what accuracy should an offset be measured if the angular error in laying off the perpendicular direction is 5°, so that the maximum displacement of the point on paper is the same due to these two sources of error?

Surveying Offset Accuracy Calculation

Surveying Problem: Required Offset Measurement Accuracy

Problem: With what accuracy should an offset be measured if the angular error in laying off the perpendicular direction is 5°, so that the maximum displacement of the point on paper is the same due to these two sources of error?

Setup Variables:

Let ‘l’ be the length of the offset (in meters).
Angular error (α) = 5°.
Required linear measurement accuracy = 1 in ‘n’.

Solution:

Equating Displacement Errors

Displacement due to angular error ≈ l * sin(α)

= l * sin(5°)

Maximum displacement due to linear error = l / n

Set the displacements equal:

l * sin(5°) = l / n

sin(5°) = 1 / n (Dividing both sides by ‘l’)

n = 1 / sin(5°)

n = cosec(5°)

n ≈ 11.473

Required accuracy ≈ 1 in 11.5

Explanation of Principles

This problem deals with balancing different sources of error in offset surveying to achieve a consistent level of precision in the final plotted point.

1. Sources of Error & Displacement: When plotting a point using an offset from a chain line, errors can arise from several sources. This problem considers two primary ones:

Angular Error (α): An error in setting out the intended angle (usually 90°) from the chain line. For a small angular error ‘α’, the resulting displacement on the ground, primarily parallel to the chain line, is approximately l * sin(α), where ‘l’ is the offset length.
Linear Measurement Error: An error in measuring the length ‘l’ of the offset itself. If the required accuracy is specified as ‘1 in n’, it signifies that the maximum potential error in measuring the length ‘l’ is l / n. This maximum error directly translates to a potential displacement of the plotted point by l / n along the direction of the offset measurement.

2. Equating Error Magnitudes: The core requirement is to find the linear accuracy ‘n’ such that the maximum potential displacement caused by the linear measurement error is *equal* to the displacement caused by the given 5° angular error. By setting the magnitudes of these two displacements equal, we determine the measurement precision needed to match the effect of the angular uncertainty.

Displacement (from Angular Error) = Max Displacement (from Linear Error)
l * sin(α) = l / n

It’s important to note that the offset length ‘l’ cancels out. This means the required relative accuracy ‘n’ is independent of the actual length of the offset being measured; it depends only on the specified angular error. The scale of plotting also cancels out because it would affect both displacement calculations equally when converting from ground to paper.

How it was Solved:

Identify the given angular error (α = 5°).
Define the required linear accuracy notation as ‘1 in n’.
Write the expression for the displacement component caused by the angular error: l * sin(5°).
Write the expression for the maximum displacement caused by the linear measurement error: l / n.
Equate these two expressions based on the problem statement: l * sin(5°) = l / n.
Simplify the equation by dividing both sides by ‘l’: sin(5°) = 1 / n.
Solve for ‘n’ by rearranging the equation: n = 1 / sin(5°).
Recognize that 1 / sin(θ) is the cosecant function: n = cosec(5°).
Calculate the numerical value: cosec(5°) ≈ 11.473.
State the required accuracy using the ‘1 in n’ format, rounding appropriately: 1 in 11.5. This indicates that the offset length must be measured with an accuracy of at least 1 part in 11.5 to ensure its maximum error contribution does not exceed the displacement caused by the 5° angular error.