Which of the following indicates the formula for setting a long chord by using ordinate?

Question

Examoogle · Accepted Answer

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

Explanation & Concepts:
In curve setting, the 'ordinate from the long chord' method is used to determine the perpendicular distance $O_x$ from a long chord to the curve at a given distance $x$ from the midpoint of the chord. The formula is derived using the Pythagorean theorem by considering a triangle formed by the radius, the ordinate, and the distance from the mid-point.

Working Principle: Let $R$ be the radius of the curve and $O_o$ be the mid-ordinate (the distance at the midpoint of the long chord). At a distance $x$ from the center of the chord, the perpendicular ordinate $O_x$ is calculated as the vertical distance between the circle's arc and the chord line, which geometrically simplifies to $O_x = \sqrt{R^2 - x^2} - (R - O_o)$.

Key Formulas: $O_x = \sqrt{R^2 - x^2} - (R - O_o)$ — Ordinate at distance $x$ from midpoint; $O_o = R - \sqrt{R^2 - (L/2)^2}$ — Mid-ordinate of the long chord of length $L$

Key Points: Used for setting out simple circular curves in the field.; The method is most accurate when the length of the chord is relatively small.; It requires precise measurement of the distance $x$ along the long chord from its center.

Advantages: Simple geometric derivation.; Does not require complex angular measurements like the Theodolite method.

Disadvantages: Less accurate for very long curves.; Cumulative errors can occur if measurements are not precise.

Applications: Setting out small circular curves in highway and railway engineering.; Verification of circular arc paths in land surveying.

Additional Info: The term $(R - O_o)$ represents the perpendicular distance from the center of the circle to the long chord.. Option B is incorrect due to the '+' sign inside the root, which represents the hypotenuse calculation for a point outside the circle. Options C and D are algebraically incorrect derivations.

Answer: A is correct — The formula correctly represents the vertical distance from the chord to the arc, calculated by subtracting the constant offset $(R - O_o)$ from the varying vertical projection of the radius $\sqrt{R^2 - x^2}$.

Exam Tip: Always remember that in curve geometry, $R^2 = x^2 + y^2$ where $y$ is the perpendicular distance; shifting the origin to the chord midpoint introduces the $(R - O_o)$ term.

Examoogle · Answer

Option B: $O_x = \sqrt{R^2 + x^2} - (R - O_o)$

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

Examoogle · Answer

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

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