Vernier Caliper Least Count Calculation

This solution explains how to calculate the least count of a Vernier caliper, a precision measuring instrument used to measure linear dimensions accurately. We will use the information provided in the question to determine the minimum measurement the instrument can detect.

Understanding the Vernier Caliper Parameters

The question provides the following key details about the Vernier caliper:

• Number of divisions on the Vernier scale: $N = 20$ divisions.
• These $20$ Vernier scale divisions coincide exactly with $19$ divisions on the main scale.
• The value of one main scale division (MSD): $1 \text{ MSD} = 0.5 \text{ mm}$.

What is Least Count?

The least count (LC) of a measuring instrument is the smallest measurement that can be measured accurately with it. For a Vernier caliper, it represents the difference between the value of one main scale division and the value of one Vernier scale division.

Formula for Least Count

There are a couple of common ways to calculate the least count of a Vernier caliper. The most direct formula, when the total number of divisions on the Vernier scale ($N$) is known, is:

$$ \text{Least Count (LC)} = \frac{\text{Value of one Main Scale Division (MSD)}}{\text{Total number of divisions on the Vernier scale (N)}} $$

Alternatively, it can be calculated as:

$$ \text{LC} = \text{MSD} - \text{VSD} $$

where VSD is the value of one Vernier scale division. To use this second formula, we first need to find the value of VSD. Since $N$ divisions on the Vernier scale coincide with $M$ divisions on the main scale, we have:

$$ N \times \text{VSD} = M \times \text{MSD} $$

In this specific problem, $N=20$ and $M=19$. So:

$$ 20 \times \text{VSD} = 19 \times \text{MSD} $$

$$ \text{VSD} = \frac{19}{20} \times \text{MSD} $$

Calculating the Least Count

Let's use the direct formula first, as it's simpler with the given information.

Using the Direct Formula (LC = MSD / N)

We are given:

• Value of one Main Scale Division (MSD) = $0.5 \text{ mm}$
• Total number of divisions on the Vernier scale (N) = $20$

Plugging these values into the formula:

$$ \text{LC} = \frac{0.5 \text{ mm}}{20} $$

$$ \text{LC} = 0.025 \text{ mm} $$

Using the Alternative Formula (LC = MSD - VSD)

First, calculate the value of one Vernier scale division (VSD):

$$ \text{VSD} = \frac{19}{20} \times \text{MSD} $$

$$ \text{VSD} = \frac{19}{20} \times 0.5 \text{ mm} $$

$$ \text{VSD} = \frac{9.5}{20} \text{ mm} $$

$$ \text{VSD} = 0.475 \text{ mm} $$

Now, calculate the least count:

$$ \text{LC} = \text{MSD} - \text{VSD} $$

$$ \text{LC} = 0.5 \text{ mm} - 0.475 \text{ mm} $$

$$ \text{LC} = 0.025 \text{ mm} $$

Both methods yield the same result.

Conclusion

The least count of the Vernier caliper is $0.025$ mm. This means the instrument can measure dimensions accurately up to two decimal places in millimeters, with the smallest readable value being $0.025$ mm.