Which of the following formula for the radial offset can be given as

Question

Examoogle · Accepted Answer

Option C: $O_x = \sqrt{R^2 + x^2} - R$

Explanation & Concepts:
The radial offset ($O_x$) is a method used in surveying to locate points on a circular curve from the tangent. It represents the perpendicular distance from the tangent line to the curve at a distance 'x' from the point of tangency along the tangent.

Working Principle: In a circular curve, consider a triangle formed by the radius to the point of tangency (T), the distance along the tangent (x), and the distance from the center to a point on the curve. By the Pythagorean theorem, the distance from the center of the curve to the point on the tangent is $\sqrt{R^2 + x^2}$. Subtracting the radius 'R' from this hypotenuse gives the radial offset $O_x$.

Key Formulas: $O_x = \sqrt{R^2 + x^2} - R$ — Radial offset distance; $R = \frac{1719}{D}$ — Relationship between radius and degree of curve (in degrees)

Key Points: The radial offset is measured perpendicular to the tangent.; It is primarily used for setting out curves of large radii.; The offset increases as the distance 'x' from the point of tangency increases.

Advantages: Simple to compute for field staff.; Efficient for small sections of curves.

Disadvantages: Not suitable for very long curves as errors propagate.; Requires precise right-angle measurements.

Applications: Railway track alignment.; Highway geometry design.; Setting out circular curves in construction surveying.

Additional Info: The formula assumes the offset is measured perpendicular to the tangent line at distance x.. Option A and B represent incorrect geometric derivations often confused with chord offsets.. Option D is dimensionally inconsistent as it would yield a value larger than R.

Answer: C is correct — the radial offset $O_x$ is mathematically derived from the right-angled triangle formed by the radius and the tangent distance using the hypotenuse $\sqrt{R^2 + x^2}$.

Exam Tip: Always verify if the offset is 'radial' (measured from the center to the tangent) or 'perpendicular' to the chord, as formulas differ significantly between these methods.

Examoogle · Answer

Option A: $O_x = \sqrt{R^2 - x^2} - R$

Examoogle · Answer

Option B: $O_x = R - \sqrt{R^2 - x^2}$

Examoogle · Answer

Option D: $O_x = R + \sqrt{R^2 - x^2}$

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