Question 80: Units , Dimensions and Error Analysis

Question

A vernier callipers has 20 divisions on the vernier scale, which coincides with $$19^{	ext {th }}$$ division on the main scale. The least count of the instrument is $$0.1 \mathrm{~mm}$$. One main scale division is equal to ________ mm.

Accepted Answer

To determine the value of one main scale division (MSD) for the given vernier calipers, we'll analyze the relationship between the main scale divisions, vernier scale divisions, and the least count of the instrument.
Given:
Number of vernier scale divisions (n):
20
Vernier scale coincides with the 19th division on the main scale.
Least count (LC):
$ 0.1 \, 	ext{mm} $
Understanding the Vernier Calipers Configuration:
Relation between Vernier and Main Scale Divisions:
The total length of the vernier scale (comprising 20 divisions) coincides with the length of 19 main scale divisions.
Mathematically, this is expressed as:
$      n 	imes 	ext{VSD} = (n - 1) 	imes 	ext{MSD}      $
where:
$ 	ext{VSD} $ is the value of one vernier scale division.
$ 	ext{MSD} $ is the value of one main scale division.
Calculating the Value of One Vernier Scale Division ($ 	ext{VSD} $):
$    	ext{VSD} = \frac{(n - 1) 	imes 	ext{MSD}}{n}    $
Least Count (LC):
The least count is the smallest measurement that can be accurately read using the instrument.
It is calculated as:
$      	ext{LC} = 	ext{MSD} - 	ext{VSD}      $
Substituting $ 	ext{VSD} $ from step 2:
$      	ext{LC} = 	ext{MSD} - \left( \frac{(n - 1) 	imes 	ext{MSD}}{n} ight)      $
$      	ext{LC} = 	ext{MSD} \left( 1 - \frac{n - 1}{n} ight)      $
$      	ext{LC} = 	ext{MSD} \left( \frac{n - (n - 1)}{n} ight)      $
$      	ext{LC} = 	ext{MSD} \left( \frac{1}{n} ight)      $
Solving for One Main Scale Division ($ 	ext{MSD} $):
Rearranging the equation:
$      	ext{LC} = \frac{	ext{MSD}}{n}      $
$      	ext{MSD} = 	ext{LC} 	imes n      $
Substituting the given values ($ 	ext{LC} = 0.1 \, 	ext{mm}, n = 20 $):
$      	ext{MSD} = 0.1 \, 	ext{mm} 	imes 20      $
$      	ext{MSD} = 2 \, 	ext{mm}      $
Conclusion:
The value of one main scale division is
2 mm
, which corresponds to
Option B
.

Answer

5

Answer

2

Answer

1

Answer

0.5

Units , Dimensions and Error Analysis

A vernier callipers has 20 divisions on the vernier scale, which coincides with $19^{\text {th }} division on the main scale. The least count of the instrument is 0.1 \mathrm{~mm}$. One main scale division is equal to ________ mm.

Choose the correct answer: