In a vernier calliper, N divisions of vernier scale coincide with (N-1) divisions of main scale (in which division represent 1mm). The least count of the instrument in cm. should be

Options

A. N
B. `N-1`
C. `(1)/(10N)`
D. `(1)/((N-1))`

Explanation

To find the least count of a Vernier caliper where N divisions of the Vernier scale coincide with (N-1) divisions of the main scale, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- Let \( N \) be the number of divisions on the Vernier scale (VSD).
- Let \( (N-1) \) be the number of divisions on the main scale (MSD).
- Each division on the main scale represents \( 1 \, \text{mm} \).

2. **Determine the Length of the Main Scale and Vernier Scale**:
- The total length of \( (N-1) \) divisions of the main scale is \( (N-1) \, \text{mm} \).
- The total length of \( N \) divisions of the Vernier scale is equal to the length of the \( (N-1) \) divisions of the main scale, which is \( (N-1) \, \text{mm} \).

3. **Calculate the Value of One Vernier Scale Division**:
- The length of one Vernier scale division (VSD) can be calculated as:
\[ \text{Length of one VSD} = \frac{(N-1) \, \text{mm}}{N} \]

4. **Find the Least Count**:
- The least count (LC) of the Vernier caliper is defined as:
\[ \text{LC} = \text{MSD} - \text{VSD} \]
- Since 1 MSD = 1 mm, we can substitute the values:
\[ \text{LC} = 1 \, \text{mm} - \frac{(N-1) \, \text{mm}}{N} \]

5. **Simplify the Expression**:
- Now, simplifying the expression for LC:
\[ \text{LC} = 1 - \frac{(N-1)}{N} = \frac{N - (N-1)}{N} = \frac{1}{N} \, \text{mm} \]

6. **Convert to Centimeters**:
- To convert the least count from mm to cm, we divide by 10:
\[ \text{LC in cm} = \frac{1}{N} \, \text{mm} \times \frac{1 \, \text{cm}}{10 \, \text{mm}} = \frac{1}{10N} \, \text{cm} \]

7. **Final Result**:
- Thus, the least count of the Vernier caliper in centimeters is:
\[ \text{LC} = \frac{1}{10N} \, \text{cm} \]