One main scale division of a vernier callipers is

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 $(\mathrm{n}-1)^{\mathrm{th}}$ division of the main scale. The least count of the callipers in $\mathrm{mm}$ is :

  1. (1) $\frac{\text { 10na }}{(\mathrm{n}-1)}$

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

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

  4. (4) $\frac{10 \mathrm{a}}{\mathrm{n}}$


Correct Option: 4

Solution:

(4)

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

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

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

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

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

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

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

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