# One main scale division of a vernier callipers is 'a' cm and nth division of the vernier scale coincide with (n – 1)th division of the main scale.

Question:

One main scale division of a vernier callipers is ' $\mathrm{a}$ ' $\mathrm{cm}$ and $\mathrm{n}^{\text {th }}$ division of the vernier scale coincide with $(\mathrm{n}-1)^{\text {th }}$ division of the main scale. The least count of the callipers in $\mathrm{mm}$ is :

1. $\frac{10 n a}{(n-1)}$

2. $\frac{10 a}{(n-1)}$

3. $\left(\frac{\mathrm{n}-1}{10 \mathrm{n}}\right) \mathrm{a}$

4. $\frac{10 a}{n}$

Correct Option: , 4

Solution:

$(\mathrm{n}-1) \mathrm{a}=\mathrm{n}\left(\mathrm{a}^{\prime}\right)$

$a^{\prime}=\frac{(n-1) a}{n}$

$\therefore$ L.C. $=1 \mathrm{MSD}-1 \mathrm{VSD}$

$=\left(a-a^{\prime}\right) c m$

$=a-\frac{(n-1) a}{n}$

$=\frac{n a-n a+a}{n}=\frac{a}{n} \mathrm{~cm}$

$=\left(\frac{10 \mathrm{a}}{\mathrm{n}}\right) \mathrm{mm}$