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# Tacheometer

Tacheometer

Contents

Define Tachometer 1

Describe different Systems of Tacheometric Measurement 2

Recall the Principle of the Stadia Method 3

Describe the Distance and Elevation Formulae for Staff Vertical-Inclined Sight with respect to the Ground surface 4

Describe the Distance and Elevation Formulae for Staff Normal to Inclined Sight 6

Define Anallactic Lens 7

Describe the following terms: i. Additive Constant ii. Multiplying Constant 7

Recall the Principle of Subtense Method 8

Define Subtense Diaphragm 9

Recall the Horizontal Subtense Measurement 11

Recall the Effect of Angular Error on the Horizontal Distance 11

Describe the General Arrangement of Field Work 12

Recall the Tacheometric Observations 13

Derive the expressions for Horizontal and Vertical Distance when: i. Both angles are at an angle of Elevation ii. Both angles are at an angle of Depression iii. One angle is at elevation and the other is at depression 14

Recall the Effect of Angular Error on Tangential Measurement 16

Classify the Errors in Tacheometry 17

Describe the Effect of Error in Stadia Tacheometry due to Manipulation and Sighting 18

# Define Tachometer

A tachometer is an instrument that measures the speed of an object, typically in rotations per minute (RPM). The word “tachometer” comes from the Greek words “tachos,” meaning speed, and “metron,” meaning measure.

A tachometer is commonly used to measure the speed of rotating components, such as engines, motors, fans, and turbines. It is used in a wide range of applications, including automotive engines, industrial machinery, and aircraft.

There are several types of tachometers, including mechanical, electrical, and digital. Mechanical tachometers use a mechanical linkage to measure the speed of rotation, while electrical tachometers use electromagnetic induction to measure the speed. Digital tachometers use electronic sensors to measure the speed and display the results on a digital display.

Tachometers play an important role in ensuring the safe and efficient operation of machinery and engines. By accurately measuring the speed of rotation, they allow operators to monitor the performance of machinery and identify potential issues, such as overspeed conditions that can cause damage or unsafe operating conditions.

In conclusion, a tachometer is a device used to measure the speed of rotating components, typically in rotations per minute (RPM). It is used in a wide range of applications and plays an important role in ensuring the safe and efficient operation of machinery and engines.

# Describe different Systems of Tacheometric Measurement

Tacheometry is a surveying method that involves the use of instruments to measure horizontal and vertical distances, angles and slopes. There are various systems of tacheometry, each with its own unique characteristics and applications. Some of the most commonly used systems are:

1. Stadia Tacheometry: This system uses a sighting device (stadia hair) which is mounted on a vertical scale to measure the horizontal and vertical distances between points. The stadia hair provides an accurate measurement of the distance between two points by using the difference in height of the two points and the angle of elevation.
2. Subtense method : Subtense method involves measuring the angle subtended by two points on a graduated instrument, such as a theodolite or a transit. The distance between the two points can then be calculated using trigonometric functions. The Subtense method is commonly used for measuring distances between points that are inaccessible or difficult to access, such as points on opposite sides of a river or a valley.
3. Stadia: Stadia is a technique that uses a stadia rod, which is marked with graduations that represent the distance between the marks as viewed through a surveying instrument. The marks are viewed through a stadia telescope, which has two crosshairs. The distance between the crosshairs corresponds to a known distance, which is typically one hundredth or one thousandth of the distance between the marks on the stadia rod. The distance to the target can then be calculated using the known distance and the observed stadia interval. Stadia is commonly used for rapid surveying of distances over relatively flat terrain, such as in construction projects.

# Recall the Principle of the Stadia Method

The stadia method is a surveying technique that uses a stadia rod or tape to measure horizontal and vertical distances between points. The principle of the stadia method is based on the relationship between the difference in height between two points, the angle of elevation, and the horizontal distance between the points.

The stadia method uses a sighting device, known as a stadia hair, which is mounted on a vertical scale. The stadia hair is used to measure the difference in height between two points, and the angle of elevation between the two points is measured using a theodolite or transit. The horizontal distance between the two points can then be calculated using trigonometry, based on the measured angle of elevation and the difference in height.

The stadia method provides a quick and accurate method for measuring distances, as well as for determining the elevation of points in the field. It is particularly useful in areas where it is difficult to directly measure distances, such as in rough terrain or in areas with obstructions.

In conclusion, the principle of the stadia method is based on the relationship between the difference in height between two points, the angle of elevation, and the horizontal distance between the points. The stadia method provides a quick and accurate method for measuring distances and elevations in the field, and is particularly useful in areas where direct distance measurement is difficult.

# Describe the Distance and Elevation Formulae for Staff Vertical-Inclined Sight with respect to the Ground surface

In stadia tacheometry, a staff is used to measure horizontal and vertical distances between two points, with one point being the instrument and the other point being a target on the ground. When the staff is in a vertical position, the difference in height between the two points can be determined. However, when the staff is in an inclined position, the difference in height between the two points is not immediately apparent. In these cases, the distance and elevation between the two points must be calculated using mathematical formulae.

The distance formula for a staff vertical-inclined sight with respect to the ground surface is given by:

d = H / tan (θ + φ)

where:

d = the horizontal distance between the instrument and the target

H = the difference in height between the instrument and the target

θ = the angle of elevation between the instrument and the target

φ = the angle between the staff and the ground surface

The elevation formula for a staff vertical-inclined sight with respect to the ground surface is given by:

h = H / cos (θ + φ)

where:

h = the vertical distance between the instrument and the target

H = the difference in height between the instrument and the target

θ = the angle of elevation between the instrument and the target

φ = the angle between the staff and the ground surface

In conclusion, when a stadia tacheometry staff is in an inclined position, the distance and elevation between two points must be calculated using the distance and elevation formulae. These formulas take into account the difference in height between the two points, the angle of elevation, and the angle between the staff and the ground surface, to determine the horizontal and vertical distances between the points.

# Describe the Distance and Elevation Formulae for Staff Normal to Inclined Sight

In stadia tacheometry, the staff can be positioned in various ways to measure horizontal and vertical distances between two points. When the staff is in a vertical position, the difference in height between the two points can be directly determined. However, when the staff is in an inclined position, the distance and elevation between the two points must be calculated using mathematical formulae.

When the staff is positioned in a normal to inclined sight, the distance formula is given by:

d = H / tan (θ)

where:

d = the horizontal distance between the instrument and the target

H = the difference in height between the instrument and the target

θ = the angle of elevation between the instrument and the target

The elevation formula for a staff normal to an inclined sight is given by:

h = H / cos (θ)

where:

h = the vertical distance between the instrument and the target

H = the difference in height between the instrument and the target

θ = the angle of elevation between the instrument and the target

In conclusion, when a stadia tacheometry staff is positioned in a normal to inclined sight, the distance and elevation between two points must be calculated using the distance and elevation formulae. These formulas take into account the difference in height between the two points and the angle of elevation, to determine the horizontal and vertical distances between the points.

# Define Anallactic Lens

An anallactic lens, also known as a compensating lens, is a type of lens used in surveying and other optical instrumentation. It is designed to correct for a phenomenon known as chromatic aberration, which is a type of distortion that occurs when light passes through a lens and is separated into its component colours. Chromatic aberration can result in the image appearing blurred or fringed with colour.

An anallactic lens works by bending the light passing through it in such a way that it compensates for chromatic aberration, resulting in a clear, sharp image. This is achieved by using a special type of glass that has different refractive indices for different wavelengths of light, which allows the lens to bend the light in a way that corrects for chromatic aberration.

Anallactic lenses are commonly used in tacheometry, theodolites, and other surveying instruments, where precise and accurate measurements are essential. They are also used in some types of camera lenses to improve image quality.

In conclusion, an anallactic lens is a type of lens that is designed to correct for chromatic aberration, resulting in a clearer, sharper image. It is used in a variety of surveying and optical instruments where accurate and precise measurements are essential.

# Describe the following terms: i. Additive Constant ii. Multiplying Constant

In tacheometry, the additive constant and multiplying constant are two important terms that are used to correct for the effects of the instrument and its lenses on the measurements taken.

i. Additive Constant: The additive constant is a correction factor that is used to account for any errors in the instrument that result in incorrect readings. It is usually applied to the tacheometer’s reading to correct for any systematic error, such as the effect of parallax, a misalignment of the reticule, or a discrepancy in the zero point of the instrument.

ii. Multiplying Constant: The multiplying constant is a correction factor that is used to compensate for any errors in the instrument that result in incorrect readings. It is usually applied to the tacheometer’s reading to correct for any systematic error, such as an incorrect focus, a deviation in the lens system, or an incorrect setting of the instrument’s scale.

Both the additive and multiplying constants are used to ensure that the measurements taken with the tacheometer are accurate and reliable. The correction factors are determined through a calibration process and are specific to each individual instrument.

In conclusion, the additive constant and multiplying constant are two important terms in tacheometry that are used to correct for systematic errors in the instrument. These correction factors are determined through a calibration process and ensure that the measurements taken with the tacheometer are accurate and reliable.

# Recall the Principle of Subtense Method

The subtense method is a surveying technique used to measure distances and elevations. It is based on the principle of triangulation, which is the process of determining the position of a point by measuring angles to it from two known points.

In the subtense method, a tacheometer is used to measure the angle between two points, and the length of a perpendicular line (subtense) drawn from one of the points to the line connecting the two points. By using trigonometry, the distance between the two points can then be calculated.

The subtense method can be used to measure both horizontal and vertical distances, as well as elevations. It is particularly useful for measuring distances to objects that are not easily accessible or in areas where other methods, such as direct measurement or EDM, are not practical.

The subtense method is relatively simple to use and can be performed with a single person and a tacheometer. However, it does require a clear line of sight between the two points being measured and accurate measurements of the angles and subtense.

In conclusion, the subtense method is a surveying technique that is based on the principle of triangulation. It uses a tacheometer to measure the angle between two points and the length of a perpendicular line, and then uses trigonometry to calculate the distance between the two points. The subtense method can be used to measure both horizontal and vertical distances and elevations, and is particularly useful in situations where other methods are not practical.

# Define Subtense Diaphragm

A subtense diaphragm is a device used in tacheometry and surveying to measure the distance between two points by means of the subtense method. The subtense diaphragm is a disc with a central hole and several equally spaced radial lines that are used to measure angles.

The subtense diaphragm is mounted on a tacheometer and is used to measure the angle between two points. The tacheometer is then used to measure the length of the perpendicular line (subtense) drawn from one of the points to the line connecting the two points. By using trigonometry, the distance between the two points can then be calculated.

The subtense diaphragm is a simple and effective tool for measuring distances using the subtense method. It is particularly useful in situations where a clear line of sight between the two points being measured is not possible, as the subtense diaphragm allows the measurement to be taken from a distance. In conclusion, a subtense diaphragm is a device used in tacheometry and surveying to measure distances using the subtense method. It is a disc with a central hole and radial lines used to measure angles, and is mounted on a tacheometer to measure the length of the subtense. The subtense diaphragm is a simple and effective tool for measuring distances in situations where a clear line of sight is not possible.

# Recall the Horizontal Subtense Measurement

The horizontal subtense measurement is a surveying technique used to measure the distance between two points that are not in line with the observer. This method is based on the principle of triangulation and uses a tacheometer and subtense diaphragm to measure the angles and length of the subtense.

In the horizontal subtense measurement, the tacheometer is set up at one of the points and aimed at the other point. The subtense diaphragm is used to measure the angle between the two points, and the tacheometer is used to measure the length of the subtense. By using trigonometry, the distance between the two points can then be calculated.

The horizontal subtense measurement is a useful tool for measuring distances in areas where a direct line of sight is not possible, such as in dense forests or in urban areas with tall buildings. It is also useful for measuring distances to objects that are at different elevations, as the subtense method takes into account the difference in elevation between the two points.

In conclusion, the horizontal subtense measurement is a surveying technique used to measure the distance between two points that are not in line with the observer. It uses a tacheometer, subtense diaphragm, and trigonometry to calculate the distance between the two points. The horizontal subtense measurement is useful for measuring distances in areas where a direct line of sight is not possible and for measuring distances to objects that are at different elevations.

# Recall the Effect of Angular Error on the Horizontal Distance

Angular error is a common problem in tacheometry and surveying that can have a significant impact on the accuracy of horizontal distance measurements. Angular error occurs when the measurement of the angle between two points is not accurate, which can lead to errors in the calculated distance.

In the horizontal subtense measurement, the angle between the two points is a critical component of the calculation of the horizontal distance. If the angle measurement is not accurate, this will result in an error in the calculated distance. For example, if the angle measurement is off by 1 degree, the error in the calculated distance could be several centimeters or even meters, depending on the distance being measured.

Angular error can occur due to a variety of factors, including improper calibration of the tacheometer, human error in taking the measurements, or the presence of obstructions that affect the line of sight. To minimize the impact of angular error on the accuracy of horizontal distance measurements, it is important to ensure that the tacheometer is properly calibrated and that the measurements are taken with care and attention to detail.

In conclusion, angular error is a problem in tacheometry and surveying that can have a significant impact on the accuracy of horizontal distance measurements. The angle between the two points is a critical component of the calculation of the horizontal distance, and if the angle measurement is not accurate, this will result in an error in the calculated distance. To minimize the impact of angular error, it is important to ensure proper calibration of the tacheometer and careful measurement practices.

# Describe the General Arrangement of Field Work

The general arrangement of field work in tacheometry and surveying refers to the overall layout and organization of the survey equipment, personnel, and procedures in the field. This includes the placement and setup of the tacheometer and other survey equipment, the assignment of tasks to personnel, and the overall plan for carrying out the survey.

In the field, the tacheometer is typically set up on a tripod and levelled using a spirit level. The surveyor then takes measurements using the tacheometer, either by sighting directly at the object being measured or by using the subtense method. The surveyor may also use additional equipment, such as a rangefinder or total station, to assist in taking measurements.

The general arrangement of field work should be planned in advance to ensure that the survey is conducted efficiently and effectively. This may include establishing a grid system for taking measurements, determining the order in which measurements will be taken, and determining how data will be recorded and processed.

In conclusion, the general arrangement of field work in tacheometry and surveying refers to the overall layout and organization of the survey equipment, personnel, and procedures in the field. This includes the placement and setup of the tacheometer and other survey equipment, the assignment of tasks to personnel, and the overall plan for carrying out the survey. Proper planning of the general arrangement of field work is important to ensure that the survey is conducted efficiently and effectively.

# Recall the Tacheometric Observations

Tacheometric observations refer to the measurements taken using a tacheometer, a surveying instrument used to measure distances and elevations. Tacheometric observations are a critical component of surveying and are used to gather data about the environment and objects being measured.

Tacheometric observations can be taken in a variety of ways, including direct sighting, stadia methods, subtense methods, and others. The tacheometer uses a combination of optical and mechanical components to accurately measure angles and distances, and the observations are used to calculate the actual distance or elevation between two points.

In order to ensure the accuracy of tacheometric observations, it is important to follow proper measurement procedures and to use a well-calibrated tacheometer. The observations should also be recorded in a systematic and consistent manner, using standardised forms and procedures.

In conclusion, tacheometric observations refer to the measurements taken using a tachometer in surveying. Tacheometric observations are used to gather data about the environment and objects being measured, and they are a critical component of surveying. Proper measurement procedures and a well-calibrated tacheometer are important to ensure the accuracy of tacheometric observations, and the observations should be recorded in a systematic and consistent manner.

# Derive the expressions for Horizontal and Vertical Distance when: i. Both angles are at an angle of Elevation ii. Both angles are at an angle of Depression iii. One angle is at elevation and the other is at depression

In tacheometry and surveying, horizontal and vertical distances are calculated based on the angles measured by the tacheometer. These distances are used to determine the actual location and elevation of objects in the environment.

i. Both angles are at an angle of Elevation:

When both angles are at an angle of elevation, the horizontal and vertical distances can be calculated using the following expressions:

Horizontal Distance (HD):

HD = (h1 + h2) / tan (θ1 + θ2)

Where h1 and h2 are the heights of the instrument and the staff, respectively, and θ1 and θ2 are the angles of elevation.

Vertical Distance (VD):

VD = h1 + h2 – HD * tan (θ1 + θ2)

ii. Both angles are at an angle of Depression:

When both angles are at an angle of depression, the horizontal and vertical distances can be calculated using the following expressions:

Horizontal Distance (HD):

HD = (h1 + h2) / tan (θ1 + θ2)

Where h1 and h2 are the heights of the instrument and the staff, respectively, and θ1 and θ2 are the angles of depression.

Vertical Distance (VD):

VD = -(h1 + h2) + HD * tan (θ1 + θ2)

iii. One angle is at elevation and the other is at depression:

When one angle is at elevation and the other is at depression, the horizontal and vertical distances can be calculated using the following expressions:

Horizontal Distance (HD):

HD = (h1 – h2) / tan (θ1 – θ2)

Where h1 and h2 are the heights of the instrument and the staff, respectively, and θ1 and θ2 are the angles of elevation and depression, respectively.

Vertical Distance (VD):

VD = (h1 + h2) * tan θ1 – HD * tan θ2

In conclusion, the expressions for the horizontal and vertical distances in tacheometry and surveying can be derived based on the angles measured by the tacheometer. The expressions for the horizontal and vertical distances change based on whether both angles are at an angle of elevation, both angles are at an angle of depression, or one angle is at elevation and the other is at depression.

# Recall the Effect of Angular Error on Tangential Measurement

The “Effect of Angular Error on Tangential Measurement” refers to the impact that errors in angular measurement have on the accuracy of tangential measurements in tacheometry.

In tacheometry, measurements of horizontal and vertical distances are made using angular measurements obtained through instruments such as theodolites, transit levels, and tachometers. When taking these measurements, there is always the possibility of error, either in the instruments or in the operator’s ability to take precise measurements.

Angular errors in tacheometry can result from various factors such as misalignment of the instrument, incorrect sighting, or incorrect reading of the scale. These errors affect the accuracy of the tangential measurements, as the horizontal and vertical distances are calculated using trigonometric relationships between the angles and the distances.

For example, if an angular measurement error occurs, it can lead to a proportional error in the calculated horizontal distance. Similarly, an error in the vertical angle measurement results in a proportional error in the calculated vertical distance. The magnitude of the error in the tangential measurement increases with the size of the measurement.

Therefore, it is important to be aware of the potential impact of angular error on tangential measurements and to take steps to minimize it, such as through regular calibration of the instruments, precise sighting, and proper handling of the instruments.

# Classify the Errors in Tacheometry

Errors in tacheometry can be classified into two main categories: systematic and random errors.

1. Systematic Errors: These are errors that are consistent in magnitude and direction and affect the accuracy of the measurements in a systematic manner. Examples of systematic errors include instrument errors, such as misalignment or incorrect zero setting, and observer errors, such as parallax errors.
2. Random Errors: These are errors that are unpredictable and vary in magnitude and direction. They are usually the result of fluctuations in the environment, such as temperature changes, wind, and atmospheric refraction. Random errors can also be due to human error, such as incorrect reading of the scales or misalignment of the instrument.

Both systematic and random errors can have a significant impact on the accuracy of the measurements in tacheometry. It is important to be aware of these errors and to take steps to minimize them in order to obtain accurate and reliable results in tacheometric measurements. This can be achieved through proper calibration of the instruments, precise sighting, and regular maintenance of the instruments.

# Describe the Effect of Error in Stadia Tacheometry due to Manipulation and Sighting

Errors in stadia tacheometry can occur due to incorrect manipulation and sighting during the measurement process. These errors can have a significant impact on the accuracy of the results and should be taken into consideration when performing tacheometric measurements.

1. Manipulation Errors: These are errors that occur due to incorrect handling of the tacheometer or the staff. For example, if the staff is not held level, or if the tacheometer is not set at the correct height, this can result in errors in the measurement.
2. Sighting Errors: These are errors that occur due to incorrect sighting of the staff or the target. This can be caused by incorrect alignment of the instrument, incorrect reading of the scales, or misalignment of the reticle. Parallax error, where the sight line is not coincident with the axis of the instrument, is another common sighting error in tacheometry.

To minimize these errors, it is important to follow proper procedures when manipulating the tacheometer and the staff, and to ensure that the instrument is properly aligned and that the reticle is accurately aligned with the target. Additionally, regular calibration of the instruments, precise sighting, and regular maintenance of the instruments can also help minimize these errors in tacheometric measurements.