One main scale division of a vernier caliper is equal to m units. If nth division of main scale coincides with (n + 1)th division of vernier scale,
the least count of the vernier caliper is:
the least count of the vernier caliper is:
- \(\frac{n}{n+1}\)
- \(\frac{m}{n+1}\)
- \(\frac{1}{n+1}\)
- \(\frac{m}{n(n+1)}\)
The Correct Option is B
Approach Solution - 1
To solve this question, we need to understand the concept of a vernier caliper and how the least count (LC) is determined. The least count is the smallest measurement a device can accurately measure. For a vernier caliper, the least count is determined by the difference between one main scale division (MSD) and one vernier scale division (VSD).
Given:
- One main scale division = \( m \) units
- The \( n^{th} \) division of the main scale coincides with the \( (n + 1)^{th} \) division of the vernier scale.
This means, in mathematical terms:
\(n \cdot \text{MSD} = (n + 1) \cdot \text{VSD}\)
From the given information:
\(n \cdot m = (n + 1) \cdot \text{VSD}\)
\(\text{VSD} = \frac{n \cdot m}{n + 1}\)
The least count (LC) is then given by the difference between one main scale division and one vernier scale division:
\(\text{LC} = \text{MSD} - \text{VSD}\)
\(\text{LC} = m - \frac{n \cdot m}{n + 1}\)
Simplifying the expression:
\(\text{LC} = \frac{(n + 1) \cdot m - n \cdot m}{n + 1}\)
\(\text{LC} = \frac{m}{n + 1}\)
Therefore, the least count of the vernier caliper is \(\frac{m}{n+1}\).
Conclusion: The correct answer is \(\frac{m}{n+1}\), which matches with the given correct answer option.
This is calculated based on how vernier scales are designed to introduce finer measurements by using a secondary scale (vernier scale) that helps measure fractions of the smallest unit on the main scale.
Approach Solution -2
Step 1: Relationship between main scale and vernier scale Given that:
\[ n \, \text{MSD} = (n + 1) \, \text{VSD}. \]
From this:
\[ 1 \, \text{VSD} = \frac{n}{n + 1} \, \text{MSD}. \]
Step 2: Least count formula The least count (L.C.) of a vernier caliper is given by:
\[ \text{L.C.} = 1 \, \text{MSD} - 1 \, \text{VSD}. \]
Substitute \( 1 \, \text{VSD} \) from Step 1:
\[ \text{L.C.} = m - m \left( \frac{n}{n + 1} \right). \]
Simplify:
\[ \text{L.C.} = m \left[ 1 - \frac{n}{n + 1} \right]. \]
\[ \text{L.C.} = m \left( \frac{n + 1 - n}{n + 1} \right). \]
\[ \text{L.C.} = \frac{m}{n + 1}. \]
Final Answer: \( \frac{m}{n + 1} \).
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