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

Answer

To solve the problem, we need to find the relationship between the main scale division, the Vernier scale division, the number of divisions on the Vernier scale (n), and the length of the Vernier scale (m).

### Step-by-Step Solution:

1. **Understand the Least Count**:
The least count (LC) of a measuring instrument is defined as the smallest measurement that can be accurately read. For the Vernier caliper, it is given by the formula:
\[ \text{Least Count} = 1 \text{ Main Scale Division} - 1 \text{ Vernier Scale Division} \]
Given that the least count is \(0.02 \, \text{cm}\) and \(1 \text{ Main Scale Division} = 0.1 \, \text{cm}\), we can set up the equation:
\[ 0.02 = 0.1 - \text{Vernier Scale Division} \]

2. **Calculate the Vernier Scale Division**:
Rearranging the equation gives us:
\[ \text{Vernier Scale Division} = 0.1 - 0.02 = 0.08 \, \text{cm} \]

3. **Relate Vernier Scale Division to Length and Number of Divisions**:
The Vernier scale division can also be expressed in terms of the length of the Vernier scale (m) and the number of divisions on the Vernier scale (n):
\[ \text{Vernier Scale Division} = \frac{m}{n} \]
From the previous step, we know that:
\[ \frac{m}{n} = 0.08 \]

4. **Substituting Values**:
Now we can express \(m\) in terms of \(n\):
\[ m = 0.08n \]

5. **Analyzing the Options**:
We need to check the given options to find the correct values of \(n\) and \(m\):
- Option A: \(n = 10\), \(m = 0.5 \, \text{cm}\) → \(0.08 \times 10 = 0.8 \, \text{cm}\) (Incorrect)
- Option B: \(n = 9\), \(m = 0.4 \, \text{cm}\) → \(0.08 \times 9 = 0.72 \, \text{cm}\) (Incorrect)
- Option C: \(n = 10\), \(m = 0.8 \, \text{cm}\) → \(0.08 \times 10 = 0.8 \, \text{cm}\) (Correct)
- Option D: \(n = 10\), \(m = 0.2 \, \text{cm}\) → \(0.08 \times 10 = 0.8 \, \text{cm}\) (Incorrect)

6. **Conclusion**:
The correct option is C, where \(n = 10\) and \(m = 0.8 \, \text{cm}\).