In levelling between two points A and B on opposite banks of a river, the instrument was set up near A and the staff readings on A and B were 2.107 and 3.335 respectively. The instrument was then moved and set up near B and the respective staff reading on A and B were 1.934 and 3.076 respectively. What is the true difference of level of A and B?
1.185 m
Understanding Reciprocal Levelling for River Crossings
This problem involves calculating the true difference in elevation between two points, A and B, located on opposite banks of a river. This scenario requires the use of reciprocal levelling, a technique used to minimize errors caused by atmospheric refraction and the curvature of the Earth, especially over longer distances like a river span.
Levelling Readings and Setups
We are given two sets of readings taken from two different instrument positions:
- Instrument Setup 1 (Near A): The level is positioned closer to point A.
- Instrument Setup 2 (Near B): The level is moved and positioned closer to point B.
The staff readings are recorded as follows:
| Setup Location | Staff Reading on A (m) | Staff Reading on B (m) |
|---|---|---|
| Near A | 2.107 (Backsight - BS) | 3.335 (Foresight - FS) |
| Near B | 1.934 (Foresight - FS) | 3.076 (Backsight - BS) |
Calculating Apparent Differences in Level
The difference in level between two points is calculated as the Backsight (BS) reading minus the Foresight (FS) reading.
Calculation for Setup 1 (Near A):
In this setup, the reading on A (2.107 m) is the backsight (BS1), and the reading on B (3.335 m) is the foresight (FS1).
Apparent Difference of Level (B relative to A) = $BS_1 - FS_1$
Apparent Difference = $2.107 \text{ m} - 3.335 \text{ m} = -1.228 \text{ m}$
This suggests that point B is 1.228 m higher than point A.
Calculation for Setup 2 (Near B):
In this setup, the reading on B (3.076 m) is the backsight (BS2), and the reading on A (1.934 m) is the foresight (FS2).
Apparent Difference of Level (A relative to B) = $BS_2 - FS_2$
Apparent Difference = $3.076 \text{ m} - 1.934 \text{ m} = 1.142 \text{ m}$
This suggests that point A is 1.142 m higher than point B.
Determining the True Difference of Level
Reciprocal levelling helps average out the effects of curvature and refraction. The true difference in level is the average of the differences calculated from the two setups.
Let's calculate the difference as ($RL_A - RL_B$) from both setups:
- From Setup 1: $RL_A - RL_B = -(BS_1 - FS_1) = -(2.107 - 3.335) = -(-1.228) = 1.228 \text{ m}$
- From Setup 2: $RL_A - RL_B = BS_2 - FS_2 = 3.076 - 1.934 = 1.142 \text{ m}$
Now, we find the average of these two values to get the true difference of level:
True Difference of Level ($RL_A - RL_B$) = $\frac{(RL_A - RL_B)_{\text{Setup 1}} + (RL_A - RL_B)_{\text{Setup 2}}}{2}$
True Difference = $\frac{1.228 \text{ m} + 1.142 \text{ m}}{2}$
True Difference = $\frac{2.370 \text{ m}}{2}$
True Difference = $1.185 \text{ m}$
This calculation shows that point A is higher than point B by 1.185 meters.
Final Answer Explanation
The true difference of level between A and B, calculated using the reciprocal levelling method, is 1.185 meters. This value accurately represents the difference in elevation, accounting for potential systematic errors.
Important Questions from Levelling
The expression for sensitivity of the bubble tube (α) can be taken as, ______
Where n = No. of divisions, s = Net staff reading, D = Distance, R = Radius of curvature, l = Length of one division
A vertical line which is perpendicular to the level line is called:
In levelling between two points A and B on the opposite sides of a river, the level was first set up near A and the staff readings on A and B were 2.645 m and 2.30 m respectively. The level was then moved near B and set up; the respective staff readings then were 1.085 m and 1.665 m on A and B respectively. What is the true difference of level between A and B?
A level, when set up $20$ m from peg A and $70$ m from peg B, reads $0.750$ m on a staff held on A and $2.065$ m on a staff held on B, keeping the bubble at its centre while reading. If the reduced levels of A and B are $100.500$ m and $101.800$ m respectively, what is the collimation error per $100.0$ m?
In levelling back, sight is also called as ______.
