In a vernier callipers, N divisions of vernier scale coincide with (N-1) divisions of main scale (in which 1 division represents 1 mm). The least count of the instrument in cm should be

Options

A. `N`
B. `N-1`
C. `(1)/(10N)`
D. `(1)/(N-1)`

Explanation

To find the least count of the vernier calipers in centimeters, we can follow these steps:

### Step 1: Understand the relationship between the divisions of the scales
In the given problem, we know that N divisions of the vernier scale coincide with (N - 1) divisions of the main scale. This means that:

- N divisions of the vernier scale = (N - 1) divisions of the main scale

### Step 2: Calculate the value of one vernier scale division
The length of one main scale division (MSD) is given as 1 mm. Therefore, the total length of (N - 1) main scale divisions is:

\[ \text{Total length of (N - 1) MSD} = (N - 1) \times 1 \text{ mm} = (N - 1) \text{ mm} \]

Since N divisions of the vernier scale coincide with this length, the length of one vernier scale division (VSD) can be calculated as:

\[ \text{Length of one VSD} = \frac{(N - 1) \text{ mm}}{N} = \frac{(N - 1)}{N} \text{ mm} \]

### Step 3: Use the formula for least count
The least count (LC) of the instrument is given by the formula:

\[ \text{Least Count (LC)} = \text{1 MSD} - \text{1 VSD} \]

Substituting the values we have:

\[ \text{LC} = 1 \text{ mm} - \frac{(N - 1)}{N} \text{ mm} \]

### Step 4: Simplify the expression
Now, we simplify the expression for LC:

\[ \text{LC} = 1 \text{ mm} - \frac{(N - 1)}{N} \text{ mm} = \frac{N}{N} \text{ mm} - \frac{(N - 1)}{N} \text{ mm} \]
\[ \text{LC} = \frac{N - (N - 1)}{N} \text{ mm} = \frac{1}{N} \text{ mm} \]

### Step 5: Convert the least count to centimeters
Since we need the least count in centimeters, we convert mm to cm:

\[ \text{LC in cm} = \frac{1}{N} \text{ mm} \times \frac{1 \text{ cm}}{10 \text{ mm}} = \frac{1}{10N} \text{ cm} \]

### Final Answer
Thus, the least count of the vernier calipers in centimeters is:

\[ \text{Least Count} = \frac{1}{10N} \text{ cm} \]

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