Engineering Scales: Plain, Diagonal & Vernier Scales

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1. Fundamental Principles of Engineering Scales

A scale is defined as the ratio of the linear dimension of an element of an object shown in the drawing to the real linear dimension of the same element of the object itself.

A. Representative Fraction (R.F.)

• Full Size Scale: $\text{R.F.} = 1 : 1$ (e.g., small machine components).
• Reducing Scale: $\text{R.F.} < 1$ (e.g., $1 : 2, 1 : 5, 1 : 50, 1 : 1000$; used for buildings, large machines, geographical maps).
• Enlarging Scale: $\text{R.F.} > 1$ (e.g., $2 : 1, 5 : 1, 10 : 1$; used for microchips, watch mechanisms, precision instruments).

B. Unit Conversion Hierarchy

C. Length of Scale (L.O.S.)

(Standard practice: L.O.S. should always be converted to centimeters and usually lies between $15\text{ cm}$ and $20\text{ cm}$ for convenient sheet layout).

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2. Plain Scale

A plain scale is a flat line graduated to measure two units (a main unit and its immediate subdivision, e.g., meters and decimeters, or kilometers and hectometers) or one unit with a single decimal place.

Subdivisions (dm)               Main Divisions (meters)
10 8 6 4 2 0         1              2              3              4
 | | | | | |---------|--------------|--------------|--------------|
[ ] [ ] [ ][         ][             ][             ][             ]

Standard Construction Rules:

1. Calculate $\text{R.F.}$ and $\text{L.O.S.}$.
2. Draw a rectangle of length equal to $\text{L.O.S.}$ and width between $5\text{ mm}$ and $10\text{ mm}$.
3. Divide the total length into $N$ equal parts corresponding to the maximum measurement capacity ($M.L.$).
4. Zero Placement: Number the first dividing line as $0$. Divisions to the right of $0$ represent main units ($1, 2, 3, \dots$).
5. Divide the first division (to the left of $0$) into smaller sub-units ($10$ parts for metric units) and number it from right to left ($0, 1, 2, \dots, 10$).
6. Write the R.F. clearly below or above the scale.

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3. Diagonal Scale

A diagonal scale is used to measure three successive units (e.g., meters, decimeters, and centimeters) or a dimension up to two decimal places (e.g., $4.76\text{ m}$).

Principle of Diagonal Division:

The construction relies on the geometric property of similar right triangles:

• In right triangle $ABC$, let base $BC = 1\text{ unit}$.
• Divide height $AB$ into $n = 10$ equal parts.
• Horizontal lines drawn through points $1, 2, \dots, 9$ have lengths proportional to their height:

   C
   |\
 9 |-|- 0.9
 8 |-|- 0.8
   | |
 2 |-|- 0.2
 1 |-|- 0.1
   +---+
   A   B

Step-by-Step Construction Procedure:

1. Draw length of scale $\text{L.O.S.}$, with height $H \approx 40\text{–}50\text{ mm}$.
2. Divide length into main units. Place $0$ at the first main mark.
3. Subdivide the leftmost block horizontally into $10$ intermediate units (e.g., decimeters).
4. Subdivide the vertical edge into $10$ smallest units (e.g., centimeters).
5. Draw Diagonals: Join point $0$ on the bottom line to point $1$ on the top line; point $1$ bottom to point $2$ top, and so on.
6. Reading a Dimension (e.g., $4.67\text{ m}$):
   • Move from $0$ rightward to $4\text{ m}$.
   • Move from $0$ leftward along the baseline to $6\text{ dm}$.
   • Trace up the diagonal line of $6\text{ dm}$ to the $7^{\text{th}}$ horizontal centimeter line.

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4. Vernier Scale

A Vernier scale consists of two parts: a Main Scale (MS) and a sliding auxiliary Vernier Scale (VS), enabling readings finer than the smallest main-scale division.

A. Direct (Forward) Vernier:

• $n$ Vernier divisions equal $(n - 1)$ Main Scale divisions:

• Least Count (L.C.):

• The Vernier graduations read in the same direction as the main scale.

B. Retrograde (Backward) Vernier:

• $n$ Vernier divisions equal $(n + 1)$ Main Scale divisions:

• Least Count (L.C.):

• The Vernier graduations read in the opposite direction to the main scale.

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5. Solved Standard Exam Problem

Problem: On a map, an area of $144\text{ cm}^2$ represents an actual area of $36\text{ km}^2$. Construct a diagonal scale to measure up to $10\text{ km}$, showing kilometers, hectometers, and decameters. Show a distance of $7.46\text{ km}$ on the scale.

Solution:

1. Find R.F.:

2. Calculate Length of Scale (L.O.S.):

3. Construction:
   • Draw a line of $20\text{ cm}$ and divide it into $10$ equal parts (each $= 1\text{ km}$).
   • Subdivide first $1\text{ km}$ block horizontally into $10$ parts (each $= 1\text{ hm} = 100\text{ m}$).
   • Divide vertical height ($50\text{ mm}$) into $10$ parts (each $= 1\text{ dam} = 10\text{ m}$).
   • Draw diagonals across the first block.
   • Mark $7.46\text{ km}$: $7\text{ km}$ to the right of $0$ + $4\text{ hm}$ to the left of $0$ + trace up diagonal to $6^{\text{th}}$ vertical line ($6\text{ dam}$).

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6. Summary of Key Exam Points

1. R.F. Calculation: Must always be reduced to identical units in numerator and denominator ($\text{R.F.}$ is dimensionless).
2. Area & Volume R.F.:
   • $\text{R.F.} = \sqrt{\text{Area on map} / \text{Actual area}}$
   • $\text{R.F.} = \sqrt[3]{\text{Volume on drawing} / \text{Actual volume}}$
3. Number of Units:
   • Plain scale = $2$ units.
   • Diagonal scale = $3$ units (or $2$ decimals).
   • Vernier scale = $3$ units (or precision fraction of smallest division).
4. Zero Position: Zero is strictly placed at the boundary between the first subdivision block and the main scale divisions.