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Question: The main scale division of a vernier calipers is equal to \( p\,\,unit \) and the \( {(m + 1)^{th}} ...

The main scale division of a vernier calipers is equal to p  unitp\,\,unit and the (m+1)th{(m + 1)^{th}} division of a vernier scale coincides with the mth division of the main scale. The least count of the vernier is:
(A) mam+1\dfrac{{ma}}{{m + 1}}
(B) mm+1\dfrac{m}{{m + 1}}
(C) am+1\dfrac{a}{{m + 1}}
(D) pm+1\dfrac{p}{{m + 1}}

Explanation

Solution

To solve this question, we must first understand the whole concept of vernier calipers. Then we need to assess them in a correct manner to find the least count and then only we can conclude the correct answer.

Complete step by step solution:
Before we move forward with the solution of this given question, let us first understand some basic concepts about vernier calipers:
Vernier scale is an important tool to take an accurate measurement reading between two graduation markings on a linear scale by using mechanical interpolation; thereby increasing resolution and reducing measurement uncertainty by using vernier acuity to reduce human estimation error.
Let’s discuss that how to calculate the least count of Vernier Calipers:
Least Count or LC  =  1  MSD  −  1  VSDLC\,\, = \,\,1\,\,MSD\,\, - \,\,1\,\,VSD ; where MSDMSD stands for main scale division and VSDVSD stands for vernier scale division.
Now we will calculate the Least count of the given Vernier calipers:
Step 1: It is given that (m+1)th{(m + 1)^{th}} division of the vernier scale division coincides with the mth division of the main scale, and therefore:
1  VSD  =  mm+1  MSD1\,\,VSD\,\, = \,\,\dfrac{m}{{m + 1}}\,\,MSD
Least count of the main scale is pp unit i.e. 1  MSD  =  p  unit1\,\,MSD\,\, = \,\,p\,\,unit
So, least count of vernier caliper, LC  =  1MSD  −  mm+1  ×1  MSDLC\,\, = \,\,1MSD\,\, - \,\,\dfrac{m}{{m + 1}}\,\, \times 1\,\,MSD
⇒  LC  =  p−  mm+1×p  =  p(m+1)− mpm+1  =  \Rightarrow \,\,LC\,\, = \,\,p - \,\,\dfrac{m}{{m + 1}} \times p\,\, = \,\,\dfrac{{p(m + 1) - \,mp}}{{m + 1}}\,\, = \,\, pm+1  unit\dfrac{p}{{m + 1}}\,\,unit
Hence we got our required least count i.e. pm+1  unit\dfrac{p}{{m + 1}}\,\,unit
So, clearly we can conclude that the correct answer is Option D.

Note:
The Vernier Calliper consists of a main scale fitted with a jaw at one end. Another jaw, containing the vernier scale, moves over the main scale. When the two jaws are in contact, the zero of the main scale and the zero of the Vernier scale should coincide. If both the zeros do not coincide, there will be a positive or negative zero error.