N divisions on the main scale of a vernier callipers coincide with (N + 1) divisions on the vernier scale. If each division on the main scale is 'a' units, then the least count of the instrument is

Options

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

Answer

To find the least count of the vernier calipers given that N divisions on the main scale coincide with (N + 1) divisions on the vernier scale, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Main Scale and Vernier Scale Divisions**:
- Let the number of divisions on the main scale be \( N \).
- Let the number of divisions on the vernier scale be \( N + 1 \).

2. **Define the Size of Each Division**:
- The size of each division on the main scale is given as \( a \) units.
- Let the size of each division on the vernier scale be \( b \).

3. **Relate the Sizes of the Divisions**:
- Since \( N \) divisions on the main scale coincide with \( N + 1 \) divisions on the vernier scale, we can express this relationship mathematically:
\[ N \cdot a = (N + 1) \cdot b \]

4. **Solve for the Size of the Vernier Scale Division**:
- Rearranging the equation gives:
\[ b = \frac{N \cdot a}{N + 1} \]

5. **Calculate the Least Count**:
- The least count (LC) of the instrument is defined as the difference between the size of one main scale division and the size of one vernier scale division:
\[ LC = a - b \]
- Substituting the value of \( b \) from the previous step:
\[ LC = a - \frac{N \cdot a}{N + 1} \]

6. **Simplify the Expression**:
- To simplify, we can write:
\[ LC = a \left(1 - \frac{N}{N + 1}\right) \]
- This can be simplified further:
\[ LC = a \left(\frac{(N + 1) - N}{N + 1}\right) = a \left(\frac{1}{N + 1}\right) \]

7. **Final Result**:
- Therefore, the least count of the vernier calipers is:
\[ LC = \frac{a}{N + 1} \]

### Conclusion:
The least count of the instrument is \( \frac{a}{N + 1} \).

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