A vernier callipers having `1` main scale division `= 0.1 cm` to have a least count of `0.02 cm`.If `n` be the number of divisions on vernier scale and `m` be the length of vernier scale, then.

Options

A. `n =10, m =0.5 cm`.
B. ` n = 9, m = 0.4 cm`.
C. ` n = 10,m = 0.8 cm`.
D. ` n = 10, m = 0.2 cm`.

Explanation

To solve the problem, we need to determine the relationship between the number of divisions on the vernier scale (n), the length of the vernier scale (m), and the least count of the vernier callipers.

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The least count (LC) of a measuring instrument is defined as the smallest value that can be measured with it. For the vernier callipers, we are given that the least count is 0.02 cm.
- The main scale division (MSD) is given as 0.1 cm.

2. **Formula for Least Count**:
- The least count can be calculated using the formula:
\[ \text{Least Count (LC)} = \text{1 MSD} - \text{1 VSD} \]
where VSD is the value of one division on the vernier scale.

3. **Substituting Known Values**:
- We know:
\[ \text{LC} = 0.02 \text{ cm} \]
\[ \text{1 MSD} = 0.1 \text{ cm} \]
- Substituting these values into the formula:
\[ 0.02 = 0.1 - \text{1 VSD} \]

4. **Solving for 1 VSD**:
- Rearranging the equation to find 1 VSD:
\[ \text{1 VSD} = 0.1 - 0.02 = 0.08 \text{ cm} \]

5. **Finding the Length of the Vernier Scale (m)**:
- If there are n divisions on the vernier scale, then the total length of the vernier scale (m) can be expressed as:
\[ m = n \times \text{1 VSD} \]
- Substituting the value of 1 VSD:
\[ m = n \times 0.08 \text{ cm} \]

6. **Conclusion**:
- The relationship between the number of divisions (n) and the length of the vernier scale (m) is given by:
\[ m = 0.08n \text{ cm} \]