How to finding R.L of staff station in tangential tacheometry for one angle elevation and other depression ?

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Examoogle · Accepted Answer

The correct answer is Option B: R.L of axis $- V + r$. In tangential tacheometry, target vanes/staff targets are fixed at a known distance $h$ apart on the staff. When one sight is an angle of elevation ($\alpha_1$) and the other is an angle of depression ($\alpha_2$), the line of sight to the lower target lies below the horizontal axis, and the upper target lies above it. Thus, the vertical position of the lower target relative to the instrument axis is obtained by subtracting $V$ ($V_2$), and adding the height of the lower staff hair $r$ gives the R.L of the staff station.

Step-by-Step Solution & Concepts:
In tangential tacheometry, target vanes/staff targets are fixed at a known distance $h$ apart on the staff. When one sight is an angle of elevation ($\alpha_1$) and the other is an angle of depression ($\alpha_2$), the line of sight to the lower target lies below the horizontal axis, and the upper target lies above it. Thus, the vertical position of the lower target relative to the instrument axis is obtained by subtracting $V$ ($V_2$), and adding the height of the lower staff hair $r$ gives the R.L of the staff station.

Working Principle: Let $S_1$ and $S_2$ be two staff targets with spacing $h$. With one angle of elevation $\alpha_1$ and one angle of depression $\alpha_2$, the horizontal distance $D = \frac{h}{	an \alpha_1 + 	an \alpha_2}$. The vertical displacement to the lower target is $V_2 = D 	an \alpha_2$. Therefore, R.L of Staff Station = R.L of Instrument Axis $- V_2 + r$, where $r$ is the height of the lower target vane above the staff foot.

Key Formulas: $D = \frac{s}{	an \alpha_1 + 	an \alpha_2}$ — Horizontal distance in tangential tacheometry (one elevation, one depression); $V = D 	an \alpha_2$ — Vertical distance to the lower target vane; $	ext{R.L of Staff Station} = 	ext{R.L of Axis} - V + r$ — Formula for reduced level of staff station

Key Points: Tangential tacheometry is used when the stadium hairs are not available or stadia method cannot be applied.; It requires two sights to two targets fixed on the staff at a known distance $s$ apart.; When target 1 is above horizontal and target 2 is below horizontal, vertical displacement down to target 2 is $V = D 	an \alpha_2$.; To reach the base of the staff from the instrument axis, subtract $V$ and add the target height $r$ above the ground.

Advantages: Can be performed using an ordinary transit theodolite without stadia diaphragm.; Useful when staff is very far and stadia hair readings are difficult to take.

Disadvantages: Requires reading two vertical angles, increasing time and chance of observation error.; Refraction and instrument movement between the two angular measurements can introduce errors.

Applications: Tacheometric surveying in rugged terrain where stadia wires are absent.; Determining elevation and distance of distant staff stations.

Additional Info: Option A (R.L of axis + V - r) applies when both angles are angles of elevation.. Option C (R.L of axis - V - r) applies when both angles are angles of depression.

Answer: B is correct — R.L of staff station is obtained by subtracting vertical height $V$ to the lower target and adding $r$ (the height of lower vane above the ground).

Exam Tip: Always trace the sign of vertical distances from the instrument axis line: going UP is $+V$, going DOWN is $-V$. Then apply $+r$ or $-r$ to reach the foot of the staff.

Examoogle · Answer

Option A: R.L of axis $+ V - r$

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Option C: R.L of axis $- V - r$

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Option D: R.L of axis $+ V + r$

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