1.20.2. Finding height of an object which is inaccessible

To find the height of the tower PP’ above a BM, select two stations A and B suitably on a fairly level ground so that these points lie in a vertical plane with the tower, and measure the distance AB with a tape (Figure 1.50).

To measure the height of a tower that cannot be accessed directly, we first choose two points (stations A and B) on level ground that are aligned vertically with the tower. The distance between these two points (AB) is measured using a tape measure. This setup is crucial as it provides a stable reference point to work with while ensuring the tower is directly in line with our measurements.

Imagine you're trying to measure the height of a tall building from where you can’t get too close. You find two spots some distance away where you can still see the building clearly. By measuring the distance between those two spots, you can indirectly deduce how tall the building is.

Set up the theodolite at station A to take a staff reading kept on the BM. Read the vertical angle α. Shift the theodolite at B point and take similar observations as taken at A point. Read the vertical angle β.

Once you have established your stations A and B, the next step is to use a theodolite, an instrument for measuring angles, to get precise readings. At station A, you will take a reading from a horizontal rod (staff) placed at a known reference point (BM). The vertical angle α is measured to the top of the tower from this position. Similarly, you will shift the theodolite to station B and repeat the process to measure the vertical angle β to the same tower.

Think of this like using binoculars to get a better view of an object far away. At the first point, you focus on the top of the building, noting the angle you've turned the binoculars to. Then you move slightly and do the same—this helps create a more complete picture of how high the building is compared to where you're standing.

Let b = horizontal distance between A and B. D = distance of the object from A point. h = height of the tower P above instrument axis at A’. h = staff reading at the BM when the instrument is at A. a h = staff reading at the BM when the instrument is at B. b h = the level difference between A and B of the instrument axes = h ~h . d a b

Here, we define some important terms used in our calculations: 'b' is the distance measured between stations A and B, and 'D' is the distance from point A to the base of the tower. The 'height of the tower' above the point where the instrument is set (A) is denoted by 'h'. It's important to track the staff readings (measurements from the reference point) taken from both stations A and B. Lastly, we consider the difference in height between the two instrument axes to ensure all calculations account for any elevation change between the two points.

Imagine you're using two different ladder heights to measure the height of a tree. You need to note how tall each ladder is (our reference points), the distance between them, and the height at which you can see the top of the tree from each position. This helps ensure that your final measurement is accurate.

When the instrument at farther station B is higher than that at the near station A. h = D tan α h – h d = (D + b) tan β Putting the value of h from (i) and (ii), D tan α - h d = (D + b) tan β or D (tan α – tan β) = h d + b tan β

In scenarios where station B is higher than station A, we can apply trigonometric principles to find the height of the tower. The height 'h' can be derived using the tangent of the angle measured (α) and the distance 'D'. Similar calculations with angle β help us establish relationships between the heights calculated from both stations. By substituting and rearranging these equations, we can isolate D, which is vital for determining the overall height of the tower.

Imagine trying to solve for how high a roller coaster peak is from different spots around it. You use your vantage point (the angle you see the top from) and the distance you are from it. By piecing together your angles and distances, you can triangulate and find the exact height.

Height of the tower above the BM, H = h + h d

Finally, we combine the calculated heights from the previous steps to get the total height of the tower above the known benchmark (BM). The height 'H' includes both the height derived from station measurements (h) and the difference in height between the two instrument readings (h d). This step is crucial as it gives us the final measurement needed for practical applications.

Consider this as compiling all the notes you've taken from both places to write a final report about the height of the building or the tree. You summarize all your findings to get the complete picture of how tall it actually is.

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