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Interactive Audio Lesson
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Setting Up the Instrument
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Today, we will learn about calculating reduced levels using angle of depression. We start by understanding how to set the instrument at the observation point P.
What do we need to set up the instrument at point P?
You need a stable base, a leveling staff, and a clear view of the point Q. Once everything is set up, we can measure the angle of depression.
How do we measure that angle?
Using a theodolite, you'll align the line of sight to the staff at Q and note the angle. In this example, it's measured at 5° 36′.
What do we do with that angle?
We use it to calculate the height difference using the tangent function. Remember the acronym SOHCAHTOA for sine, cosine, and tangent!
So tangent helps us find height changes?
Exactly! Let's move forward with our calculations.
Calculating the Height Difference
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Now that we have the angle of depression, let's calculate the difference in elevation.
What is the formula we are using here?
The formula is D * tan(α), where D is the distance from P to Q, and α is the angle of depression.
In this case, D is 3000 m?
Correct! And our angle α is 5° 36′. Let's calculate that now.
Why do we need to consider curvature and refraction?
These factors can affect our distance measurements, making our results less accurate. We apply a correction to account for them.
And what is the correction for curvature and refraction?
For the distance of 3000 m, the combined correction is about 0.606 m. We subtract this from our height calculation.
Finalizing the R.L.
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Now that we have the height adjusted from the angle of depression, we can find the R.L. of Q.
What is the formula for that?
We have the R.L. of the instrument axis, which is the B.M. elevation plus the staff reading minus the height difference.
So it becomes 436.050 + 2.865 - 293.547?
Exactly! This gives us the R.L. for the staff station Q.
And we get 143.368 m for the R.L. of Q. Is that correct?
Yes! Excellent work! Understanding these calculations is crucial for accuracy in surveying.
Introduction & Overview
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Quick Overview
Standard
The example illustrates how to compute the R.L. of a staff station using known distances and angles of depression from an observation point. The methodology involves calculating the difference in elevation due to distance and angles, as well as adjustments for curvature and refraction.
Detailed
In this example, an instrument is set up at point P with a known elevation of 436.050 meters. By measuring the angle of depression to a vane at a distance of 3000 meters, several calculations are made to find the Reduced Level (R.L.) of the staff station Q, which is positioned 2 meters above the base of the staff. The calculation involves using trigonometric functions (specifically tangent) to determine the vertical height difference due to the angle of depression. Additionally, corrections for curvature and refraction are applied to ensure accuracy. This section emphasizes the importance of understanding how different factors affect the measurement of distances and elevations in surveying practice.
Audio Book
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Instrument Setup and Measurements
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An instrument was set up at P and the angle of depression to a vane 2 m above the foot of the staff held at Q was 5° 36′. The horizontal distance between P and Q was known to be 3000 metres.
Detailed Explanation
In this first part of the problem, we are given a scenario where a survey instrument is placed at point P, and it is used to measure an angle of depression to a point (vane) located at Q. The vane is positioned 2 meters above the foot of the staff that is held vertically. The angle of depression is the angle between the horizontal line from the instrument and the line of sight to the vane. Knowing the horizontal distance between the two points (P and Q) allows us to relate this angle and distance to the height difference between the instrument and the vane, which is critical for calculating the reduced level (R.L.) of the staff station at Q.
Examples & Analogies
Imagine you are standing on a hill (point P) and looking down at a friend (the vane) standing at the bottom of the hill (point Q) holding a stick. The angle at which you are looking down is similar to the angle of depression. The distance you are from your friend helps you understand how high you are standing relative to them.
Elevation Difference Calculation
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Determine the R.L. of the staff station Q given that staff reading on a B.M. of elevation 436.050 was 2.865 metres. The difference in elevation between the vane and the instrument axis = D tan α = 3000 tan 5° 36′ = 294.153 m.
Detailed Explanation
To find the difference in elevation between the vane and the instrument, we apply the formula: difference = horizontal distance x tan(angle of depression). In this case, we calculate that the elevation difference amounts to 294.153 meters. This information is crucial as it informs us how much lower the staff is positioned compared to the height of the instrument itself. This calculation is foundational as it exemplifies how trigonometric functions can help resolve vertical height issues in surveying.
Examples & Analogies
Consider a flat piece of ground where you have a tall tree (the instrument). If you are 3000 meters away from the tree and observe that you need to look down at an angle of 5° 36’ to see the top of an object below, this angle helps you calculate how much lower the object is than your eye level. Just like with measuring the tree's height, the angle and distance help you determine how high off the ground the top of the object must be.
Correction for Curvature and Refraction
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Combined correction due to curvature and refraction C = 0.06735 D² metres when D is in km = 0.606 m. Since the observed angle is negative, the combined correction due to curvature and refraction is subtractive.
Detailed Explanation
The correction for curvature and refraction accounts for the fact that the Earth is not flat, and light bends when passing through different atmospheric layers. Here, we calculate C using the formula where D is the distance in kilometers. Once we find C to be approximately 0.606 m, we apply this correction to ensure our final elevation account reflects the true height of the point being measured. It's essential to subtract this value, as it adjusts our previous positive difference in elevation.
Examples & Analogies
Imagine looking at a distant mountain through a thick layer of air filled with heat from the ground; just as the air can distort your view, the Earth’s curve can affect how we observe heights over long distances. The correction is akin to taking off sunglasses that distort your view, so you can see the actual height of the mountain more clearly.
Final Calculation of Reduced Level
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RL of instrument axis = 436.050 + 2.865 = 438.915, RL of the vane = RL of instrument axis – h = 438.915 – 293.547 = 145.368, RL of Q = 145.368 – 2 = 143.368 m.
Detailed Explanation
In this final part, we calculate the Reduced Level (R.L.) of the instrument axis by adding the B.M. elevation to the staff reading. The R.L. of the vane is then calculated by subtracting the height difference (h) found earlier from the instrument’s R.L. Finally, we determine the R.L. for the staff station Q by subtracting the height of the vane from the R.L. of the vane. This step-by-step approach reveals the central process of deriving R.L.s in surveying.
Examples & Analogies
Think of it like figuring out the total height of a multi-story building. First, you find how high you are above the ground (the instrument's R.L.), then you measure how far you need to go down to reach a balcony (height of the vane) before finally determining how high the ground level is compared to your starting point. This method helps you 'connect the dots' to see the full elevation picture.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Angle of Depression: The measure of the angle formed between a horizontal line and the line of sight to a point below.
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Tangent Function: A fundamental trigonometric ratio used in calculating heights and distances.
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Curvature Correction: An adjustment made to account for Earth curvature in measurement.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Example of calculating height difference using the tangent of the angle of depression.
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Example of applying curvature and refraction corrections in surveying.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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Angle down, look around, tangent helps you measure ground.
📖 Fascinating Stories
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Imagine you're on a tower, looking down at a ship with a friend measuring the angle.
🧠 Other Memory Gems
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Remember: T for Tangent, H for Height, D for Distance. THD helps in calculation.
🎯 Super Acronyms
RLT - Reduced Level Tangent - to remember the components needed for calculations.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Reduced Level (R.L.)
Definition:
The height of a point relative to a reference level, typically mean sea level.
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Term: Angle of Depression
Definition:
The angle formed between the horizontal line from an observer and the line of sight down to an object below.
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Term: Tangent
Definition:
In trigonometry, the ratio of the opposite side to the adjacent side of a right triangle.
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Term: Curvature Correction
Definition:
Adjustments made to measurements to account for the curvature of the Earth.
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Term: Refraction Correction
Definition:
Adjustments made to measurements to account for the bending of light rays in the atmosphere.