Question

A Vernier caliper has 20 divisions on the Vernier scale, which coincide with 19 on the main scale. The least count of the instrument is $0.1\,{\text{mm}}$. The main scale divisions are of:
A. $0.5\,{\text{mm}}$
B. $1\,{\text{mm}}$
C. $2\,{\text{mm}}$
D. $\dfrac{1}{4}\,{\text{mm}}$

Answer 1

Use the formula for least count of the Vernier calipers. This formula gives the relation between the least count of the Vernier calipers, mains scale main scale division and Vernier scale division. Determine the value of the Vernier scale division and substitute in this formula to determine the value of one main scale division.

Complete step by step answer:
The least count of the Vernier calipers is 0.1mm.
$LC = 0.1\,{\text{mm}}$
We have given that there are 20 divisions on the Vernier scale and one of the line on the Vernier scale coincide with the 19th line on the main scale of the Vernier calipers.
Hence, the value of the one Vernier scale division becomes
$1{\text{VSD}} = \dfrac{{19}}{{20}}{\text{MSD}}$

We have the formula for least count as
$LC = 1{\text{MSD}} - 1{\text{VSD}}$
Here, ${\text{MSD}}$ is the value of one main scale division on Vernier calipers and ${\text{VSD}}$ is the value of the one Vernier scale division on Vernier caliper.
Substitute $\dfrac{{19}}{{20}}{\text{MSD}}$ for $1{\text{VSD}}$ in the above equation.
$LC = 1{\text{MSD}} - \dfrac{{19}}{{20}}{\text{MSD}}$
$\Rightarrow LC = \dfrac{1}{{20}}{\text{MSD}}$
Substitute $0.1\,{\text{mm}}$ for $LC$ in the above equation.
$\Rightarrow 0.1\,{\text{mm}} = \dfrac{1}{{20}}{\text{MSD}}$
Rearrange the above equation for ${\text{MSD}}$.
$\Rightarrow {\text{MSD}} = 0.1\,{\text{mm}} \times 20$
$\therefore {\text{MSD}} = 2\,{\text{mm}}$
Therefore, the main scale divisions are of $2\,{\text{mm}}$.

Hence, the correct option is C.

Note: The students should be careful while determining the values of the Vernier scale divisions on the Vernier calipers. If this value is not taken correctly, the final answer will be incorrect. The students should also keep in mind that we have given the least count with units. So, the final value of the main scale division value is also in the same given unit.