Length of a main scale division of vernier calliper is `a` cm & (n-1) division of main scale are equal to n-division of vernier scale. Find least count

Answer

To find the least count of a vernier caliper given that the length of one main scale division is `a` cm and (n-1) divisions of the main scale are equal to n divisions of the vernier scale, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We know that the length of one main scale division (MSD) is `a` cm.
- We are given that (n-1) divisions of the main scale are equal to n divisions of the vernier scale.

2. **Expressing the Length of the Main Scale**:
- The total length of (n-1) divisions of the main scale can be expressed as:
\[ \text{Length of } (n-1) \text{ MSD} = (n-1) \times a \text{ cm} \]

3. **Expressing the Length of the Vernier Scale**:
- This length is equal to the length of n divisions of the vernier scale (VSD):
\[ \text{Length of } n \text{ VSD} = n \times \text{Length of one VSD} \]

4. **Setting Up the Equation**:
- From the information given, we can equate the two lengths:
\[ (n-1) \times a = n \times \text{Length of one VSD} \]

5. **Finding the Length of One Vernier Scale Division**:
- Rearranging the equation gives us:
\[ \text{Length of one VSD} = \frac{(n-1) \times a}{n} \]

6. **Calculating the Least Count**:
- The least count (LC) of the vernier caliper is defined as:
\[ \text{LC} = \text{Length of one MSD} - \text{Length of one VSD} \]
- Substituting the values we have:
\[ \text{LC} = a - \frac{(n-1) \times a}{n} \]
- Simplifying this expression:
\[ \text{LC} = a - \left(a - \frac{a}{n}\right) = a - a + \frac{a}{n} = \frac{a}{n} \]

7. **Final Result**:
- Therefore, the least count of the vernier caliper is:
\[ \text{Least Count} = \frac{a}{n} \text{ cm} \]