In a vernier callipers, $(N+1)$ divisions of the vernier scale coincide with $N$ divisions of the main scale.

Question

In a vernier callipers, $(N+1)$ divisions of the vernier scale coincide with $N$ divisions of the main scale. If $1 	ext{ MSD}$ represents $0.1 	ext{ mm}$, the vernier constant (in cm) is:

Accepted Answer

The Vernier constant (VC), or least count, is defined as the difference between the value of one main scale division (MSD) and one vernier scale division (VSD) .

1. **Establish the relationship between VSD and MSD:** The problem states that $(N+1)$ VSDs coincide with $N$ MSDs. Therefore, $1 	ext{ VSD} = \frac{N}{N+1} 	ext{ MSD}$.

2. **Apply the Vernier Constant formula:** 
   $VC = 1 	ext{ MSD} - 1 	ext{ VSD}$ 
   $VC = 1 	ext{ MSD} - \left(\frac{N}{N+1}ight) 	ext{ MSD}$ 
   $VC = 	ext{MSD} \left(1 - \frac{N}{N+1}ight) = \frac{	ext{MSD}}{N+1}$.

3. **Substitute given values and convert units:** We are given $1 	ext{ MSD} = 0.1 	ext{ mm}$. To find the constant in cm, we use the conversion $1 	ext{ mm} = 0.1 	ext{ cm}$, thus $0.1 	ext{ mm} = 0.01 	ext{ cm}$ .
   $VC = \frac{0.01}{N+1} 	ext{ cm} = \frac{1}{100(N+1)} 	ext{ cm}$.

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