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Interactive Audio Lesson
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Introduction to Trigonometrical Levelling
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Today, we will be exploring trigonometrical levelling, an essential method used in surveying. Can anyone tell me why we might need to measure the heights of buildings or towers?
To understand the elevation for designing roads or other constructions!
Exactly! We need accurate height information for construction planning. Trigonometrical levelling helps find these heights using angles and distances. It's quicker than traditional methods, especially in complex terrains.
How do we actually measure that?
Great question! We measure the vertical angle from a distance along with the horizontal distance to calculate the height using trigonometry. Remember, 'H = h + h_s', where h is the height above the instrument axis and h_s is from the BM.
What if the structure is too far away?
If it's inaccessible, we can place instruments at two positions and measure angles from both to calculate the height. It involves some trigonometric calculations, which we will dive into shortly!
Sounds fascinating!
Method for Accessible Objects
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Let’s talk about measuring accessible objects. We set up the instrument at a known distance from the tower or building. How do we calculate the height?
We use the angle measured to the top?
That's correct! By measuring the angle and horizontal distance, we can find the height. The formula we use is h = D tan(α). Who can explain what D and α represent?
D is the horizontal distance, and α is the vertical angle to the top of the object!
Exactly! Once we find h using that formula, we add it to the height of the instrument axis above the BM to get the total height H.
What correction do we make for large distances?
Good catch! We need to factor in curvature and refraction corrections if D is large, which can be approximated by 0.00673D²/1000. It's crucial for accuracy!
Methods for Inaccessible Objects
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Now, let’s explore what we do when our target structure is inaccessible directly. Can anyone suggest a method?
We can take readings from two different stations and use the angles!
Exactly! We set up at two points and measure the vertical angles from each. We also need to measure the horizontal distance between the two points. This allows us to derive the height through trigonometric calculations.
What formulas do we use to solve this?
Using triangles and the relationships for angles, we calculate heights at each station and employ them to find the overall height of the structure. We weave through some trigonometric identities here.
Can you show us an example?
Definitely! Let's work through one together, applying both angle measurements to derive the correct height step-by-step!
Introduction & Overview
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Quick Overview
Standard
This section covers trigonometric levelling, a method of calculating the height of objects by measuring angles and distances. It is efficient for topographical surveys, particularly when determining the elevation of structures from a distance and provides methods for both accessible and inaccessible objects.
Detailed
Trigonometrical Levelling
Trigonometrical levelling is an indirect levelling technique used to determine the elevation of points by analyzing the observed vertical angles and the measured horizontal distances from the instrument to the object. This method is particularly useful in topographical surveys for determining the height of buildings, chimneys, or other elevated structures from a distance, and it requires knowledge of the benchmark (BM) elevation in the area.
1.20.1 Finding the Height of an Object Which is Accessible
To calculate the height of a tower or other accessible structures, the instrument is set up at a point where the top of the structure can be viewed. The vertical angle to the top of the structure is measured, alongside the distance to the structure and the staff reading at the BM. The relationship between the angle and distance is expressed as:
- Height of the object (H) = height above BM + height above instrument axis.
Using the equation: h = D tan(α), where D is the horizontal distance to the object and α is the measured vertical angle.
1.20.2 Finding the Height of an Object Which is Inaccessible
When a structure is inaccessible directly, two observation points can be utilized to measure vertical angles from both positions. The process involves recording horizontal distances and angles of elevation before applying trigonometric relations to calculate height.
Both of these methods are essential for surveying and construction and illustrate the practical application of trigonometry in civil engineering.
Audio Book
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Introduction to Trigonometrical Levelling
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It is an indirect method of levelling in which the elevation of the point is determined from the observed vertical angles and the measured distances. It is commonly used in topographical work to find out the elevations of the top of buildings, chimneys, churches etc., from a distance. This is a faster method to get the elevations of top of structures and objects. Elevation of a BM in the area must be known.
Detailed Explanation
Trigonometrical levelling is a technique used in surveying that relies on indirect measurements. Instead of measuring the height of an object directly, we determine its elevation based on vertical angles observed from a distance and the horizontal distance to the object. This method is particularly effective for assessing tall structures, such as buildings or churches, as it saves time and effort compared to traditional measuring techniques. However, to use trigonometrical levelling, we must first know the elevation of a benchmark (BM), which serves as a reference point.
Examples & Analogies
Imagine trying to find the height of a tall building without a ladder. Instead of climbing up, you stand at a distance and use a theodolite to measure the angle to the top of the building. By knowing how far away you are and using some math, you can calculate the height of the building without ever leaving the ground.
Finding Height of an Accessible Object
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Let PP’ is a tower whose elevation of the top is to be determined. Set up the theodolite at a convenient ground point A so that the top of tower and a staff kept on the BM can be bisected. Measure vertical angle α of the top of tower as well as take the staff reading at the BM. Measure D, the horizontal distance between theodolite station and tower.
Detailed Explanation
To find the height of a structure, we first set up our theodolite at a point where we can see both the top of the tower and a staff at the benchmark. We then measure the vertical angle from our position up to the top of the tower (denoted as α) and note the reading from the staff at the benchmark, which gives us the lower reference. Lastly, we measure the horizontal distance (D) from our position to the base of the tower. These measurements allow us to calculate the height of the tower using trigonometric functions.
Examples & Analogies
Think about using a smartphone app to measure the height of a tall tree. You stand a certain distance away from the tree and point your phone at the top of it. The app calculates the angle to the top and, knowing your distance from the tree, it automatically tells you how tall the tree is based on those numbers.
Height Calculation Formula
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To find the height of the object above a BM:
Let H = height of the object above the BM
h = height of the tower above the instrument axis
h_s = height of the instrument axis above the BM
s
α = vertical angle of the top of tower at the instrument station
D = horizontal distance from the instrument station to the base of the tower.
Then, h = D tan α
H = h + h_s = D tan α + h_s.
Detailed Explanation
In the calculations, we consider the height of the object (H) to be the sum of the height above the instrument axis (h) and the height of the instrument above the benchmark (h_s). The height above the instrument axis can be calculated using the horizontal distance (D) and the tangent of the vertical angle (α). Thus, the formula connects the angle measured, distance away, and known height of the instrument to ultimately derive the object's height.
Examples & Analogies
Imagine adjusting the height of your camera to capture the perfect photo of a mountain while standing a specific distance away. Using the angle you look up at and your distance from the mountain, you can calculate just how high it towers above you, much like measuring a tall building's height using specific angles and distances.
Finding Height of an Inaccessible Object
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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. 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
When the object cannot be accessed directly, we employ two separate observation stations. From each station (A and B), we measure both the vertical angles (α from A to the object and β from B to the object). The distance between the two stations is also measured. These angles and distances create a geometric relationship allowing us to determine the height of the object indirectly using trigonometric principles.
Examples & Analogies
Suppose you're trying to find the height of a tree growing on the edge of a steep cliff. You can't get close without risking a fall, so you position yourself at two safe spots (A and B) further back. By measuring the angles to the top of the tree from both points and knowing how far apart you are from each other, you can still work out how tall that tree really is!
Adjusting for Errors in Measurement
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If the distance D is large, correction for curvature and refraction, i.e., 0.00673(D^2 / 1000), is to be applied.
Detailed Explanation
When using trigonometrical levelling over large distances, it is important to account for errors caused by the Earth’s curvature and atmospheric refraction. These factors can slightly alter the angle measurements and distances. The formula given (0.00673 times D squared divided by 1000) provides a correction factor that ensures our height calculations remain accurate even over long distances.
Examples & Analogies
Think of trying to measure the distance on a large sports field. If you were at one end but couldn’t see the other due to the curvature of the ground, you would want to adjust your measurements to account for that curve, much like how the formula helps tweak readings for tall buildings or towers over long distances.
Definitions & Key Concepts
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Key Concepts
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Trigonometrical Levelling: An indirect method used to calculate elevations through measurement of angles and distances.
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Vertical Angle: The angle between the horizontal plane and the line of sight to the target object.
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Horizontal Distance: The distance from the surveying instrument to the object being measured.
Examples & Real-Life Applications
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Examples
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To find the height of a chimney from a distance, a surveyor measures the vertical angle to the top and the horizontal distance from the instrument to the base of the chimney.
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When measuring an inaccessible tower, two stations can be set up at a distance, measuring angles from each to derive the height using the distance between the two observation points.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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To measure the height of a tower tall, angle plus distance will give you all.
📖 Fascinating Stories
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Once in a land of surveyor's fight, they used angles to measure with might. Heights of towers they’d savvy, calculating away, using trig methods all day.
🧠 Other Memory Gems
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HAD: Height = angle (A) plus distance (D).
🎯 Super Acronyms
HAD - Height, Angle, Distance
- key elements of trigonometrical levelling.
Flash Cards
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Glossary of Terms
Review the Definitions for terms.
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Term: Trigonometrical Levelling
Definition:
An indirect method of determining point elevation using vertical angles and measured distances.
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Term: Vertical Angle
Definition:
The angle measured from the horizontal plane to the line of sight to the object.
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Term: Horizontal Distance (D)
Definition:
The straight-line distance between the observer and the object being measured, projected horizontally.