The observed reading on a staff held at point A is 3.55 m, if the staff is found to be 0.10 m off the vertical through its bottom, the correct staff reading on point A is:
3.5485 m
Understanding Staff Reading Correction
When performing leveling in surveying, the staff must be held perfectly vertical to obtain the correct reading. If the staff is tilted, the observed reading will be along the slanted line, which is longer than the true vertical height. This results in an incorrect reading that needs to be corrected.
Why is Correction Needed for a Tilted Staff?
- A tilted staff creates a longer sight line from the instrument to the staff compared to a vertical staff at the same point.
- The observed reading is the distance along this slanted line.
- The true height difference required for leveling is the vertical distance from the line of sight to the point on the ground (point A in this case).
- This geometry forms a right-angled triangle where the observed reading is the hypotenuse, the true vertical reading is one side, and the horizontal offset at the top of the staff is the other side.
Applying the Pythagorean Theorem
We are given the observed reading on the staff held at point A is 3.55 m. The staff is tilted such that its top is 0.10 m horizontally away from the true vertical line passing through its bottom (point A). Let's denote:
- \(R_{observed}\) = Observed staff reading = 3.55 m
- \(d\) = Horizontal offset of the staff top from vertical = 0.10 m
- \(R_{correct}\) = Correct (true vertical) staff reading
According to the Pythagorean theorem, in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the observed reading is the hypotenuse.
The relationship is:
\((R_{observed})^2 = (R_{correct})^2 + d^2\)
We need to find \(R_{correct}\), so we rearrange the formula:
\((R_{correct})^2 = (R_{observed})^2 - d^2\)
\(R_{correct} = \sqrt{(R_{observed})^2 - d^2}\)
Calculation of the Correct Staff Reading
Substitute the given values into the formula:
\(R_{correct} = \sqrt{(3.55 \, \text{m})^2 - (0.10 \, \text{m})^2}\)
First, calculate the squares:
\((3.55)^2 = 12.6025\)
\((0.10)^2 = 0.0100\)
Now, subtract \(d^2\) from \((R_{observed})^2\):
\((R_{correct})^2 = 12.6025 - 0.0100 = 12.5925\)
Finally, take the square root to find \(R_{correct}\):
\(R_{correct} = \sqrt{12.5925} \approx 3.548591\)
Comparing with Options
The calculated correct staff reading is approximately 3.54859 m. Let's compare this with the given options:
- Option 1: 3.5514 m
- Option 2: 3.65 m
- Option 3: 3.5485 m
- Option 4: 3.55 m
The value 3.5485 m is the closest option to our calculated value of 3.548591 m.
Therefore, the correct staff reading on point A is 3.5485 m.
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 ______.
