Question

In a vernier caliper, 20 the division on the vernier scale coincides with the seventeenth division on the main scale If one cm on the main scale is divided into 20 equal parts then what is the least count on the calipers.
a) 0.085mm
b) 0.0075mm
c) 0.0085mm
d) 0.075mm

Answer 1

The least count is basically given as the difference between a single division on main scale and the single division on the vernier scale. Using this definition we will understand how much division on the given vernier scale makes up to how much on the main scale. Consider there is zero error in the given vernier calliper.

Complete answer: To begin with let us first understand what is a vernier scale. A scale basically consists of its minimum division being 1mm. Let us say we want to measure the diameter of a circular plate. The reading lies between say 8.9 – 9 cm and we cannot make out where it lies. Hence if we divide that 1mm into say n parts than the reading would be more precise and accurate. Hence a vernier scale is introduced on the main scale to increase its least count.

Least count of any scale is the smallest measurement the scale can make.
As mentioned in the hint the least count in a vernier caliper is given as,
$\text{Least Count}=M.S.D-V.S.D......(1)$ where M.S.D is the main scale division and V.S.D is the vernier scale division.
It is mentioned that 1cm is divided into 20 parts. Hence 1 division on the main scale is $\dfrac{1cm}{20}=0.05cm$.
It is given that 20 the division on the vernier scale coincides with the seventeenth division on the main scale. Hence we can write 20 V.S.D = 17 M.S.D , Further we can write $V.S.D=\dfrac{17}{20}M.S.D....(2)$
Substituting 2 in equation 1 we get,

$\text{Least Count}=M.S.D-V.S.D$
$\text{Least Count}=M.S.D-\dfrac{17}{20}M.S.D$
$\text{Least Count}=M.S.D\left( \dfrac{20-17}{20} \right)$
Since 1 M.S.D= 0.05 cm
$\text{Least Count}=0.05\left( \dfrac{20-17}{20} \right)cm$
$\text{Least Count}=0.0075cm=0.075mm$
So, the correct answer is “Option d”.

Note:
In the above vernier caliper the zero division of the main scale coincides with the zero division on the vernier scale. Hence there was no error in the instrument. But if there was an error we would have to subtract, if the zero division of the vernier scale lies ahead the zero division of main scale or add if the zero division of the vernier scale lies behind the zero division of main scale on the vernier scale reading.