## Scale, Types of Scale and Accuracy & Errors in Surveying

## SCALE

- It is the fixed ratio that every distance on the plan/map bears with the corresponding distance on the ground.
- Scale can be represented in the following manners:

### a) Engineer's scale:

Here one 'cm' on the plan represents some whole number of meters on the ground.

Example: 1 cm = 100 m

### b) Representative fraction (RF):

Here one unit of length on the plan represents some number of the same units of length on the ground.

Example: 1 : 10000 or 1/10000**Note: Here, the distance is independent of units of measurement.**

### c) Graphical scale:

This is the most suitable method used to represent the scale of a map. It is a line drawn on the map so that its distance on the map corresponds to a convenient unit of length on the ground.

### Important question

## What is the advantage of graphical scale over numerical scales?

With the passage of time, map or plan paper undergoes to dimensional changes (shrink/expand) due to which lesser accurate the result obtained by numerical scales than the graphical scales.

Graphical scales shrink or expand proportionally, and the distance can thus be found more accurately.

## Shrunk scale:

- Shrunk factor/shrunk ratio is defined as the ratio of shrunk length to the original length

SHRUNK FACTOR (SF) = SHRUCK LENGTH / ORIGINAL LENGTH = SHRUNK FACTOR / ORIGINAL LENGTH

Corrected Distance on the map in terms of original scale = Shrunk Length / Shrunk Factor

Corrected Area on the map in terms of original scale = Shrunk Area / (Shrunk Factor)^{2}

## Error due to wrong measuring scale:

- If a wrong measuring scale is used to measure the length of an already drawn line, the measured length will be incorrect similarly if the wrong measuring scale is used to measure the area of an already drawn area on the plan, the measured area will be incorrect.

Corrected length = RF of Wrong Scale / RF of Correct Scale * Measured Length

Corrected Area = (RF of Wrong scale / RF of Correct Scale)^{2} * Measured Area

### important question

## The distance between the two points marked on the plane drawn to a scale of 1cm = 1 m is measured by surveyor and found to be 60 m. Later it was found that the wrong scale of 1 cm = 50 cm was used for the measurement. Then what will be the correct length and correct area if the measured area is 70m?

Corrected length = RF of Wrong Scale / RF of Correct Scale * Measured Length

Corrected length = (1/50) / (1/100) * 60 = 120 m

Corrected Area = (RF of Wrong scale / RF of Correct Scale)^{2}* Measured Area

Corrected length = {(1/50) / (1/100)}^{2}* 70 = 280 m^{2}

## Choice of the scale of a map:

The choice of scale of a map to be selected is dependent upon:**A)** The use of map**B)** The extent of territory to be represented

While deciding the scale of a map, the following points are taken into consideration:

- i) Scale should be chosen larger enough such that in scaling or plotting distance from the finished map, it will not be necessary to read the scale closer than 0.25 mm.
- il) Choose as small a scale as is consistent with a clear representation of the smallest detail desired.

Purpose of Survey | Scale | R.F |

A) Geographical Survey | 1 cm = 5 km to 160 km | 1/500000 to 1/16000000 |

B) Cross Section (Both Horizontal and Vertical Scale are Equal) | 1 cm = 1 m to 2 m | 1/100 to 1/200 |

C) Longitudinal Section | ||

i) Horizontal Scale | 1 cm = 10 m to 200 m | 1/1000 to 1/20000 |

ii) Vertical Scale | 1 cm = 1 m to 2 m | 1/100 to 1/200 |

D) Topographical Survey | ||

i) Building Sites | 1 cm = 10 m or less | 1/1000 or less |

ii) Town Planning Schemes, Reservoirs etc | 1 cm = 50 m to 100 m | 1/5000 to 1/10000 |

iii) Horizontal Scale | 1 cm = 50 to 200 m | 1/5000 to 1/20000 |

iv) Horizontal Scale | 1 cm = 0.25 km to 2.5 km | 1/25000 to 1/250000 |

## Types of scale:

### A) Plain scale:

- The scale on which only two dimensions can be measured (i.e units and tenths) is known as the plane scale.
- Example -
**cm**and**mm**,**m**and**dm**

### b) diagonal scale:

- The scale on which three dimensions (i.e. units, tenths, hundredth) can be measured is known as a diagonal scale.
- Example-
**m, dm, cm**

Draw line AB, at B draw perpendicular BC to AB of any convenient length. Divide it into ten equal parts. Join a diagonal AC. From each of the divisions 1, 2, 3' etc, draw lines parallel to AB to cut the diagonal in corresponding points 1,2, 3, etc, thus dividing the diagonal into ten equal parts.

∴ **△** C 1'1 and **△** CAB (from the law of similarity)

C1/CB = 11'/AB

1/10 = 11'/AB

11' = (1/10)AB

So, 1-1' represent (1/10)AB

Similarity, 2-2' = (2/10)AB

9-9' = (9/10)AB

### c) vernier scale:

- The device is used for measuring accurately the fractional part of the smallest division on a graduated scale or main scale is known as the vernier scale. In the vernier scale, the readings are taken closer than the smallest reading on the graduated scale.
- The vernier consists of a small scale called the "Vernier scale", which moves along the graduated scale / main scale.
- The divisions of the vernier scale are made either slightly longer or shorter than that o the main scale, and it has an index mark (arrow) which represents zero of the vernier scale.

## important question

## What is the least count of vernier scale ?

The difference in the length of one division of main scale and one division of vernier scale is defined as the vernier scale.

### types of vernier scale:

### direct vernier

### extended vernier

### retrograde vernier

### A) direct vernier :

- In direct vernier, the reading on both the vernier & main scale increases in the same direction and the divisions on the vernier scale is slightly shorter than the division of the main scale.
- Considering the n divisions on the vernier scale are equal to the length of (n-1) division on the main scale.

nV = (n-1) S

V = {(n-1)S}/n

V = Length of one division on a vernier scale.

S = Length of one division on the main scale.

Least Count (LC) = S-V

= S- {(n-1)S}/n = {nS-(n-1)S}/n = LC = S/n

### double direct vernier:

- It is the type of vernier in which the vernier extends on both sides of the index mark & the main scale is also figured in both the directions.

LC = S/n

### b) retrograde vernier :

- In retrograde vernier, the divisions on the vernier scale are slightly longer than those on the main scale.
- Here reading in case of retrograde vernier increases in the direction opposite to that of the main scale.
- Considering the n divisions of vernier scale are equal to (n+1) division of the main scale.

nV = (n+1) S

V = {(n+1)n}/S

LC = V-S (The smallest division of the vernier scale is longer than the smallest division on the main scale)

nV = (n+1) S-S

LC = S/n

### c) extended vernier :

- It may happen that the divisions on the main scale are very close, and it would then be difficult, if the vernier were of normal length, to judge the exact graduations where coincidence occurs. In this case, extended vernier may be used.
- Considering the (2n-1) divisions on the main scale are equal to n divisions of vernier.

(2n-1) S = nV

V= (2n-1) S / n

LS = 2S - V = 2S - {( 2n-1)S}/n

LC = 2nS - {(2n-1)S}/n

LC = S/n

## accuracy and error:

**i) Precision:**

The degree of perfection or degree of fineness used in instruments, methods of observation and observation is termed as precision.

**iI) Accuracy:**

The degree of perfection obtained in using instruments, methods of observation and observation is termed as accuracy.

**Note:**

- a) The difference between two measured values of the same quantity is known as the discrepancy.
- b) It is not the same as an Error.
- c)A small discrepancy may have a great errors and a large discrepancy may have a small errors.

**iII) Error:**

It is the difference between a measured value and the true value of quantity.

Since the true value of the quantity is not known, the true error of measurement cannot be found, hence measurement is done within a certain limit of error prescribed.

Error = Measured value - True value

Correction = True value - Measured value

Error = (-) correction

**The sources of arising error are as follows:**

**a) Instrument:**This error may arise due to faulty or imperfect instrument used for the measurement. Example - A Chain/tape being used that is too short or too long than actual.**b) Personal:**This error arises due to imperfection in human response like weak eyesight, imperfection of touch etc.**c) Natural:**The error caused due to natural phenomena such as humidity, refraction, wind, magnetic declination, temperature, gravity etc.

**Errors are classified as follows:**

**a) Mistake:**

- These are errors which arise from inattention, inexperience, carelessness, poor judgement, and confusion in the mind of the observer.
- It can be identified by taking the same measurement from an another observer.
- It can also be overcome by comparing several measurements of the same quantity and discarding the odd value.

Example - A reading of 70 m is booked as 17 m.

Miscounting of the number of tape lengths.

**b) Systematic (Cumulative error):**

- It arises from the source that acts in a similar manner on observation.
- The method of measurement, the instrument used and the physical condition at the time of measurement causes the systematic error.

Example - Expansion of steel tapes

Using a compass in a particular magnetic field. - These errors are cumulative in nature.
- These can be eliminated by applying the correction.
- The proper way to check systematic error is to measure the same quantity by an entirely different method.

**c) Compensation / Accidental / Random error:**

- The errors which remain after the mistakes and systematic error are removed is known as compensation error. These are random in nature and are caused mainly due to the limitations of the observer and instruments.
- These errors follow the law of probability:
- Small error occurs more frequently than large ones.
- Positive and negative errors are equally likely.
- Very large errors are seldom. (rare in occurrence)
- It can be represented by a normal distribution curve.
- Random error is proportional to 1/√N, Where N is the number of observation.

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