Which of the following formula find out tangential angle by using Rankine’s formula.

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Examoogle · Accepted Answer

Option A: $\delta = 1718.9 \frac{C}{R}$ minutes

Explanation & Concepts:
Rankine's method of tangential angles (also known as the deflection angle method) is widely used for setting out simple circular curves in surveying. The tangential angle (δ) for any chord is given by the formula $\delta = 1718.9 \frac{C}{R}$ minutes, where $C$ is the length of the chord and $R$ is the radius of the curve in the same units.

Working Principle: The method is based on the geometric principle that the angle subtended by a chord at the tangent point (tangential angle) is equal to half the angle subtended by that chord at the center of the curve. Expressing this angle in radians gives $\delta = \frac{C}{2R}$ radians. Converting radians to minutes by multiplying by $\frac{180 	imes 60}{\pi} \approx 3437.75$ yields $\delta = 3437.75 	imes \frac{C}{2R} = 1718.87 \frac{C}{R} \approx 1718.9 \frac{C}{R}$ minutes.

Key Formulas: $\delta = 1718.9 \frac{C}{R}	ext{ minutes}$ — Tangential angle for a single chord; $\Delta_n = \sum \delta_i$ — Total deflection angle for point $n$; $\delta = \frac{C}{2R}	ext{ radians}$ — Tangential angle in radians

Key Points: Rankine's method is precise and universally adopted for setting out circular curves using a theodolite and tape/chain.; The total deflection angle to any point on the curve is the sum of the individual tangential angles up to that point.; The total deflection angle for the end of the curve (Point of Tangency) is equal to half of the total deflection angle ($\Delta/2$).; The constant factor $1718.9$ is derived directly from $\frac{180 	imes 60}{2\pi} \approx 1718.87$.

Advantages: Highly accurate for both short and long curves.; Saves field work compared to offset methods.

Disadvantages: Requires a theodolite and skilled personnel.; Errors in measuring individual chords can accumulate sequentially.

Applications: Setting out simple circular curves in road and railway engineering.; Alignment of curves where high precision is required.

Additional Info: Option B ($ \delta = 1718.9 \frac{R}{C}$) inverts the relationship between chord length and radius.. Option C uses an incorrect numerical constant ($1719.9$).. Option D uses $\pi$ instead of chord length $C$ in the numerator.

Answer: A is correct — Rankine's tangential angle formula is $\delta = 1718.9 \frac{C}{R}$ minutes.

Exam Tip: Remember that $1718.9$ is derived from converting $0.5	ext{ radians}$ into minutes ($\frac{180 	imes 60}{2\pi} = 1718.87 \approx 1718.9$). This value is frequently tested in GATE and state PSC exams.

Examoogle · Answer

Option B: $\delta = 1718.9 \frac{R}{C}$ minutes

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Option C: $\delta = 1719.9 \frac{C}{R}$ minutes

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Option D: $\delta = 1718.9 \frac{\pi}{R}$ minutes

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