Question

The smallest division on the main scale of a vernier calipers is $1mm$, and $10$ vernier division coincide with $9$ main scale division. while measuring the diameter of a sphere, the zero mark of the vernier. scale lies between $2.0$ and $2.1\,cm$ the fifth vernier scale coincides with a scale division. Then diameter of the sphere is:
A) $2.05\,cm$
B) $3.05\,cm$
C) $2.50\,cm$
D) $\text{None of these}$

Answer 1

Here we find the reading of the scale, the above statement considers that we know the magnitude of the scale, and then we begin to measure the diameter of the sphere, we proceed with the formula for measuring the diameter. By using the method of the vernier.

Formula used:
$LC = \dfrac{{Smaller\,reading\,on\,main\,scale}}{{No.of\,division\,on\,vernier\,scale}}$
$\Rightarrow$ $MSD$ is the main scale division
$\Rightarrow$ $VSD$ is the vernier scale division
$\Rightarrow$ $LC$ is the least count

Complete step by step solution:
Given by,
Main scale division,
$1MSD = 1mm$
Vernier scale division,
$1VSD = 9MSD$
We find the diameter of the sphere,
Therefore,
$1VSD = \dfrac{9}{{10}}MSD$
On simplifying,
$\Rightarrow$ $1VSD = .9MSD$
Now,
We find the least count,
We know that,
$Least\,Count\,of\,the\,vernier\,calipers = \dfrac{{Smaller\,reading\,on\,main\,scale}}{{No.of\,division\,on\,vernier\,scale}}$
Rearranging the above formula,
We get,
$Least\,Count = 1Main\,Scale\,Division - 1Vernier\,Scale\,Division$
$\Rightarrow$ $LC = 1 - .9 = .1mm$
Here,
$\Rightarrow$ $LC = .01\,cm$
Then we find the diameter of the sphere,
$\Rightarrow$ $Diameter = 2.0 + VSR$
$VSR$ vernier scale reading is $5$ and multiplying with least count of vernier calipers
We get,
$\Rightarrow$ $Diameter = 2.0 + 5 \times .01$
On simplifying,
$\Rightarrow$ $Diameter = 2.05\,cm$

Hence, the option A is the correct answer, the diameter of the sphere is $2.05\,cm$.

Note: Whenever this kind of problem It slides parallel to the main scale and allows for readings on the main scale to be rendered to a fraction of a division. With the assistance of a meter scale, the duration is measured. It is graduated in cm and mm such that one small division's value is one millimeter, so a meter scale can be used to accurately measure a length of up to one millimeter.