A vernier calipers 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. `(a)n =10, m =0.5 cm`.
B. `(b) n = 9, m = 0.4 cm`.
C. ` (c) n = 10,m = 0.8 cm`.
D. ` (d)n = 10, m = 0.2 cm`.

Explanation

To solve the problem, we need to find the relationship between the main scale divisions, the least count of the Vernier calipers, and the Vernier scale divisions. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understand the given data**:
- One main scale division (MSD) = 0.1 cm
- Least count (LC) = 0.02 cm

2. **Use the formula for least count**:
The least count of a Vernier caliper is given by the formula:
\[ \text{Least Count} = \text{One Main Scale Division} - \text{One Vernier Scale Division} \]
Let the length of one Vernier scale division be denoted as \( V \).

3. **Set up the equation**:
From the formula, we can write:
\[ LC = MSD - V \]
Substituting the known values:
\[ 0.02 = 0.1 - V \]

4. **Solve for \( V \)**:
Rearranging the equation gives:
\[ V = 0.1 - 0.02 = 0.08 \text{ cm} \]
Thus, one Vernier scale division is 0.08 cm.

5. **Relate \( n \) and \( m \)**:
If \( n \) is the number of divisions on the Vernier scale and \( m \) is the total length of the Vernier scale, then:
\[ V = \frac{m}{n} \]
Substituting \( V = 0.08 \) cm into the equation:
\[ 0.08 = \frac{m}{n} \]
Rearranging gives:
\[ m = 0.08n \]

6. **Conclusion**:
The relationship between the number of divisions \( n \) and the length of the Vernier scale \( m \) is:
\[ m = 0.08n \]