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Interactive Audio Lesson
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Understanding Instruments and Measurement
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Today we will learn about how to measure the height of an object that we cannot reach directly, such as a tower. What instrument do you think we will use for this task?
Is it a theodolite?
Exactly! A theodolite is crucial for measuring angles. It allows us to determine vertical angles to objects at distances. Can anyone tell me why measuring angles is essential?
Because angles help us figure out how high something is from where we are standing!
Correct! We can use the angle along with the distance to calculate height. Remember that understanding this concept is vital, so let’s summarize: a theodolite measures angles, and we use these angles along with distances to find heights.
First Measurement Method Using Same Vertical Plane
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Now let’s dive into our first method: measuring height when we have two stations in the same vertical plane. Please describe the steps we would take.
First, we need to measure the distance between the two stations.
Then, we can set up the theodolite at station A and get the angle to the tower’s top.
Great! Now, what about at station B?
We do the same thing! We read the vertical angle to the tower and also take the staff reading at the BM.
Fantastic! Now, how would we use those angles to find the height?
We use the formula: h = D tan α and adjust it depending on the instrument heights!
Exactly! Always remember to adjust your heights accordingly. Let’s summarize: we measure angles from two aligned positions to find the height using trigonometric functions.
Second Method When Stations Are Not Aligned
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Now let's discuss what we do when our stations aren’t aligned vertically with the tower. How would we start this measurement?
We would measure the horizontal distance between the two stations first.
Then we would measure the angles from both stations to the top of the tower.
Correct! With these angles, how can we relate the triangles formed to find the height?
We can use the sine and cosine relationships to figure out the height from each angle!
Exactly! For each angle measured, we will have to calculate two heights, and then adjust to find the final height. Let's summarize: measure the angles from both instruments and utilize trigonometric relationships to calculate height.
Introduction & Overview
Read a summary of the section's main ideas. Choose from Basic, Medium, or Detailed.
Quick Overview
Standard
In this section, the process of finding the height of an inaccessible object, such as a tower, is discussed through two scenarios: when the instrument can be set up in the same vertical plane as the object and when it cannot. Key formulas for calculating height based on vertical angles and horizontal distances are provided.
Detailed
Finding Height of an Object Which Is Inaccessible
In this section, we explore how to determine the height of a tower or other object that is not directly accessible using trigonometric levelling techniques. Two primary scenarios are presented:
1. When the Base is Inaccessible but Instrument Stations Are in the Same Vertical Plane
- Select two stations, A and B, on level ground such that they align vertically with the tower. Measure the horizontal distance between the two stations.
- Set up the theodolite at station A and take a staff reading at the BM, as well as the vertical angle α to the top of the tower. Repeat the process at station B to obtain the vertical angle β.
- From these measurements, we can establish relationships between the angles and the distances involved to derive the height of the tower relative to the BM:
- Distance D to the tower from point A can be calculated from the angles,
- Compute the height using the formulae:
- If the instrument at A: h = D tan α
- Account for differences in instrument heights between A and B.
2. When the Instrument Stations Are Not in the Same Vertical Plane as the Object
- Measure the horizontal distance b between the two stations on the level ground. At each station, obtain the angles of elevation (α from A to P, and β from B to P) as well as the horizontal angles (θ1, θ2).
- Using triangulation, relate the distances AC and BC to derive the height of the tower above the BM using:
- h = AC tan α
- h = BC tan β
- Finally, calculate the height of the tower with respect to the benchmark using the instrument's elevation data.
This chapter emphasizes the importance of accurate measurements and understanding trigonometric relationships to overcome challenges in surveying inaccessible heights.
Audio Book
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Selection of Stations
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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).
Detailed Explanation
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.
Examples & Analogies
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.
Taking Readings from Stations
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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.
Detailed Explanation
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.
Examples & Analogies
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.
Setting Equations for Height Calculation
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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
Detailed Explanation
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.
Examples & Analogies
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.
Using Trigonometric Relationships
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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
Detailed Explanation
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.
Examples & Analogies
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.
Final Calculation of Tower Height
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Height of the tower above the BM,
H = h + h d
Detailed Explanation
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.
Examples & Analogies
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.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Trigonometric Relationship: The use of angles and distances to find heights.
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Instrument Setup: Properly configuring instruments in either straight vertical alignment or different planes.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example 1: Measuring the height of a church tower using theodolites from two ground stations while taking vertical angles.
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Example 2: Finding the height of a lighthouse located along a coast using angles from two different points on the shore.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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To find a tower's height, just take a sight! Measure the angle and distance tight.
📖 Fascinating Stories
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Once a curious surveyor needed to measure a tall tower. With a theodolite and some clever thinking, they measured angles from two spots until they reached the height they sought.
🧠 Other Memory Gems
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Remember AHT: Angles, Heights, and Trig! Always think angles for heights in trig.
🎯 Super Acronyms
DASH
- Distance And Slope Heights - everything you need to remember when calculating height!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Theodolite
Definition:
An instrument for measuring angles of horizontal and vertical planes.
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Term: Vertical Angle
Definition:
The angle measured upward or downward from the horizontal level.
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Term: Staff Reading
Definition:
The measurement of the rod used in conjunction with a theodolite to determine height.