Question

The main scale of a vernier callipers has $n\text{divisions/cm}$. $n$ divisions of the vernier scale coincide with $(n-1)$ divisions of main scale. The least count of the vernier callipers is

• A. $\frac{1}{n(n+1)}\text{cm}$
• B. $\frac{1}{n}\text{cm}$
• C. $\frac{1}{(n+1)(n-1)}\text{cm}$
• D. $\frac{1}{n^2}\text{cm}$

Answer 1

The correct option is D $\frac{1}{n^2}\text{cm}$

If $n$ divisions of vernier scale coincides with $(n-1)$ divisions of main scale.
Therefore,
$n\text{VSD}=(n-1)\text{MSD}$
$\Rightarrow 1\text{VSD}=\frac{(n-1)}{n}\text{MSD}$
$\therefore$ Least count $=1\text{MSD}-1\text{VSD}=1\text{MSD}-\frac{(n-1)}{n}\text{MSD}$
$\Rightarrow 1\text{MSD}-1\text{MSD}+\frac{1}{n}\text{MSD}=\frac{1}{n}\text{MSD}$
$\Rightarrow \frac{1}{n}\times \frac{1}{n}=\frac{1}{n^2}\text{cm}[\because 1\text{MSD}=\frac{1}{n}\text{cm}]$