Question

One main scale division of a vernier caliper is equal to m units. If n^th division of main scale coincides with (n + 1)^th division of vernier scale,
the least count of the vernier caliper is:

• A. \(\frac{n}{n+1}\)
• B. \(\frac{m}{n+1}\)
• C. \(\frac{1}{n+1}\)
• D. \(\frac{m}{n(n+1)}\)

Answer 1

The Correct Option is B

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.

Answer 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: \( \frac{m}{n + 1} \).