Given:
- 20 vernier scale divisions (VSD) coincide with 19 main scale divisions (MSD).
- Least count (L.C.) of the instrument is \( 0.1 \, \text{mm} \).
Step 1: Relation Between VSD and MSD
From the given data:
\[ 20 \, \text{VSD} = 19 \, \text{MSD}. \]
The value of one vernier scale division (1 VSD) is:
\[ 1 \, \text{VSD} = \frac{19}{20} \, \text{MSD}. \]
Step 2: Calculating the Least Count
The least count (L.C.) is given by:
\[ \text{L.C.} = 1 \, \text{MSD} - 1 \, \text{VSD}. \]
Substituting the value of 1 VSD:
\[ \text{L.C.} = 1 \, \text{MSD} - \frac{19}{20} \, \text{MSD} = \frac{1}{20} \, \text{MSD}. \]
Given that the least count is \( 0.1 \, \text{mm} \):
\[ 0.1 \, \text{mm} = \frac{1}{20} \, \text{MSD}. \]
Multiplying both sides by 20:
\[ 1 \, \text{MSD} = 2 \, \text{mm}. \]
Therefore, one main scale division is equal to \( 2 \, \text{mm} \).
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