Which of the following is not a case in trigonometric levelling?

Question

Examoogle · Accepted Answer

Option C: Base of object is at accurate position

Explanation & Concepts:
Trigonometric levelling involves finding the elevated difference between points using measured vertical angles and horizontal distances. The standard cases depend on whether the base of the object is accessible, inaccessible with instrument stations in the same vertical plane, or inaccessible with instrument stations not in the same vertical plane.

Working Principle: The elevation of a point is determined using trigonometric ratios (specifically tangents) applied to a right-angled triangle formed by the line of sight, horizontal distance, and the vertical height of the object. When the base is inaccessible, two instrument stations are used to determine both the horizontal distance and height indirectly.

Key Formulas: $h = D 	an \alpha$ — Vertical height when the base is accessible ($D$ is horizontal distance, $\alpha$ is vertical angle); $h = \frac{b \sin \alpha_1 \sin \alpha_2}{\sin(\alpha_1 - \alpha_2)}$ — Vertical height when base is inaccessible and stations are in the same vertical plane

Key Points: Trigonometric levelling is an indirect method of levelling used primarily in hilly terrain, tall structures, or inaccessible locations.; The three standard cases are classified based on the accessibility of the object's base and the alignment of the instrument stations relative to the vertical plane containing the object.

Advantages: Allows determination of heights of inaccessible structures like towers, chimneys, and hill peaks.; Rapid method for determining elevated differences over long distances compared to spirit levelling.

Disadvantages: Requires precise measurement of vertical angles using a theodolite.; Errors in distance measurement or angular measurement directly impact the calculated elevation.

Applications: Determining the height of tall buildings, communication towers, and monuments.; Topographic surveys in mountainous and rugged terrains.

Additional Info: Option A represents Case I where horizontal distance $D$ can be directly measured using a tape or chain.. Option B represents Case II where the horizontal distance to the base cannot be measured, but both instrument stations lie in the same vertical plane with the object.. Option D represents Case III where instrument stations do not lie in the same vertical plane as the object, requiring triangulation techniques.. Option C ('Base of object is at accurate position') is not a classification or case in trigonometric levelling terminology.

Answer: C is correct — 'Base of object is at accurate position' is not a recognized case in trigonometric levelling.

Exam Tip: For competitive exams, remember that if instrument stations are at different levels, correction for difference in line of sight heights ($h_1, h_2$) must be applied when calculating total height.

Examoogle · Answer

Option A: Base of object is accessible

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Option B: Base of object is inaccessible

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Option D: Base of object is inaccessible, station is not in vertical plane

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