Slope correction for a 100-link chain on a small angle α (radians) For a measured chain length of 100 links laid along a uniform slope of α radians, what is the approximate subtractive slope correction per chain length to obtain the horizontal distance (use small-angle approximation)?

Introduction / Context:
When a line is measured along a slope, the recorded length exceeds the required horizontal distance. A slope correction must therefore be subtracted. For small slopes, a convenient quadratic approximation in the slope angle provides fast and accurate corrections in the field for a 100-link chain (20.1168 m for a Gunter’s chain or 20 m for a metric chain, depending on convention).

Given Data / Assumptions:

• Measured along-slope chain length per segment = 100 links.
• Uniform slope angle = α radians, with α small.
• Horizontal distance L_h = L * cos α; Correction C_s = L_h − L (negative).

Concept / Approach:

Use the identity cos α ≈ 1 − α^2/2 for small α. Then L_h = L * (1 − α^2/2), so the subtractive slope correction magnitude is |C_s| = L * (1 − cos α) ≈ L * (α^2/2). For L = 100 links, the magnitude becomes 100 * α^2 / 2 = 50 * α^2 links. The correction is subtractive from the measured sloping length to recover the horizontal length.

Step-by-Step Solution:

Verification / Alternative check:

Exact correction is L * (cos α − 1). For small α (e.g., ≤ 5°), the quadratic approximation differs negligibly from the exact value for most chaining work.

Why Other Options Are Wrong:

Linear (α) or cubic (α^3) dependence is incorrect; 100/α is dimensionally wrong; zero ignores real slope effects.

Common Pitfalls:

Using degrees in place of radians in the series; forgetting that the correction is subtractive; mixing links and metres without consistent conversion.

Final Answer:

50 * α^2 links (i.e., 100 * α^2 / 2)