Question

One main scale division of a vernier caliper is equal to \( m \) units. If the \( m \) divisions of the main scale coincide with the \( (n + 1)^{\text{th}} \) division of the 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. \( \frac{m}{n(n+1)} \)

Hint:

Least count is the smallest measurement a device can accurately measure; it’s crucial for precision in scientific experiments.

Answer 1

The Correct Option is B

Step 1: {Understanding Vernier Caliper} 
The vernier scale least count is given by: \[ LC = {Main Scale Division} - {Vernier Scale Division} \] Step 2: {Applying the Given Condition} 
Since \( n \) main scale divisions equal \( (n+1) \) vernier scale divisions, \[ n \times MSD = (n+1) \times VSD \] Step 3: {Solving for Least Count} 
\[ VSD = \frac{n}{n+1} \times MSD \] \[ LC = MSD - VSD = m - \frac{n}{n+1} m \] \[ LC = m \left(1 - \frac{n}{n+1} \right) \] \[ LC = \frac{m}{n+1} \] Thus, the correct answer is (B)
 

Answer 2

Step 1: Understand the Given Information
We are given that the n^th division of the main scale (MS) coincides with the (n + 1)^th division of the vernier scale (VS). Also, 1 main scale division (1 MSD) is equal to \( a \) units.

Step 2: Write the Relationship
From the problem, we write: \[ n \times \text{MSD} = (n + 1) \times \text{VSD} \] This means the length of \( n \) main scale divisions is equal to the length of \( n + 1 \) vernier scale divisions.

Step 3: Express 1 Vernier Scale Division (VSD)
Rewriting the above equation: \[ \text{VSD} = \frac{n \times \text{MSD}}{n + 1} \] Since \( \text{MSD} = a \), we get: \[ \text{VSD} = \frac{n \cdot a}{n + 1} \]

Step 4: Use the Formula for Least Count
The least count (LC) of the vernier scale is given by: \[ \text{LC} = \text{MSD} - \text{VSD} \]

Step 5: Substitute the Values
\[ \text{LC} = a - \frac{n \cdot a}{n + 1} \]

Step 6: Simplify the Expression
\[ \text{LC} = a \left( 1 - \frac{n}{n + 1} \right) = a \left( \frac{n + 1 - n}{n + 1} \right) = a \left( \frac{1}{n + 1} \right) \]

Step 7: Final Result
Therefore, the least count of the vernier scale is: \[ \boxed{\text{LC} = \frac{a}{n + 1}} \text{ units} \]