How to find out mid ordinate length for curve ?

Question

Examoogle · Accepted Answer

Option C: Both A and B

Explanation & Concepts:
The mid-ordinate (also known as the versine of a curve) is the distance from the midpoint of a chord to the midpoint of the circular arc. It can be derived geometrically either using the chord length ($L$) and radius ($R$) or using the deflection angle ($\Delta$) of the curve.

Working Principle: Geometrically, for a circular curve, the distance from the center to the chord is $\sqrt{R^2 - (L/2)^2}$. By subtracting this distance from the total radius $R$, we obtain the mid-ordinate $O_o$. Alternatively, using the triangle formed by the radius and the angle $\Delta/2$, the offset can be expressed via trigonometric projection as $R(1 - \cos(\Delta/2))$.

Key Formulas: $O_o = R - \sqrt{R^2 - (L/2)^2}$ — Chord-based method; $M = R(1 - \cos(\frac{\Delta}{2}))$ — Angle-based method

Key Points: The mid-ordinate is the maximum perpendicular distance from the chord to the curve.; It is primarily used in field engineering for setting out circular curves using offsets from the long chord.; The value of $O_o$ depends inversely on the radius of the curve; flatter curves have smaller mid-ordinates.

Advantages: Provides a simple way to define curvature without complex transit equipment.; Useful for checking the alignment of existing railway or road curves.

Disadvantages: Accuracy decreases significantly for very long chords relative to the radius.; Requires precise measurement of the chord midpoint.

Applications: Highway and Railway curve setting out.; Verification of circular arc geometry in surveying.

Additional Info: Option A uses the Pythagorean theorem applied to the triangle formed by the radius, the chord half-length, and the mid-ordinate.. Option B uses simple trigonometric resolution of the radius vector relative to the tangent-intercept point.. Both methods are mathematically equivalent expressions for the same physical property.

Answer: C is correct — Both formulas correctly represent the geometrical determination of the mid-ordinate of a circular curve.

Exam Tip: Always verify if the question asks for the long chord offset (mid-ordinate) or a specific tangent offset, as the formulas differ.

Examoogle · Answer

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

Examoogle · Answer

Option B: $M = R(1 - \cos(\frac{\Delta}{2}))$

Examoogle · Answer

Option D: None of these

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