Question

In an usual vernier, 10 vernier scale divisions, coin side with 8 main scale divisions, then what is the least count of the vernier? (A) \(0.1 \mathrm{~mm}\) (B) \(0.2 \mathrm{~mm}\) (C) \(0.8 \mathrm{~mm}\) (D) \((1 / 8) \mathrm{mm}\)

Short answer

The least count of the vernier is (B) \(0.2 \mathrm{~mm}\).

Step-by-step solution

Determine the value of 1 vernier scale division

As we know, 8 main scale divisions are equal to 10 vernier scale divisions. We can write this as a simple equation: \(8 \times MS = 10 \times VS\) where MS is the value of 1 main scale division, and VS is the value of 1 vernier scale division. Now, we can solve for the value of 1 vernier scale division (VS) in terms of main scale divisions: \( VS = \frac{8 \times MS}{10}\)

Plug the value of 1 vernier scale division into the least count formula

We have determined the value of 1 vernier scale division in terms of main scale divisions. Now, we can use the least count formula to find the least count of the vernier: Least Count = (Value of 1 main scale division) - (Value of 1 vernier scale division) Least Count = \(MS - \frac{8 \times MS}{10}\)

Simplify and solve for the least count

Now, we can simplify the equation and solve for the least count: Least Count = \(MS(1 - \frac{8}{10})\) Least Count = \(MS(\frac{10 - 8}{10})\) Hence, the least count of the vernier is: Least Count = \(\frac{2}{10} \times MS\) Since we are not given the value of one main scale division (MS) in any units, we can assume it to be in millimeters (mm) and write the least count as: Least Count = \(\frac{2}{10} \mathrm{~mm}\) This matches the second answer choice, so the correct answer is: (B) \(0.2 \mathrm{~mm}\)

Key concepts

Least Count

The least count of a measuring instrument is the smallest value that it can measure accurately. For a vernier caliper, this is determined by using the main scale divisions and the vernier scale divisions. More specifically, the least count is the difference between the value of one main scale division and one vernier scale division. To determine the least count:

• Calculate one vernier scale division in terms of main scale divisions.
• Subtract this from the value of one main scale division.

This difference is the least count. It tells us the smallest increment that can be measured with precision using the caliper. In the given exercise, it is found to be 0.2 mm.

Vernier Scale

The vernier scale is a crucial part of a vernier caliper. It consists of a series of divisions that slide along the main scale. It allows precise measurements by helping to pinpoint an exact reading between the smallest marks on the main scale. Each division on the vernier scale typically spans less space than those on the main scale, creating a finer resolution when measuring.

To understand this better, consider that in the exercise, 10 vernier scale divisions line up with 8 main scale divisions. This setup is used to determine how much each vernier scale division represents in terms of main scale space. By doing so, one can find how close the vernier can get to pinpointing measurements—leading us to the least count.

Main Scale Divisions

Main scale divisions are the fixed marks on the main body of the vernier caliper. They provide the primary measurement workspaces, typically calibrated in millimeters. Each increment represents a standard quantity, such as 1 mm or 0.1 cm, that aids the user in assessing larger measurements efficiently.

In our context, the exercise specifies that 8 of these divisions equate to 10 vernier scale divisions. Using this relationship, you can determine a single vernier scale division in terms of these main scale units, which helps in calculating the least count. Therefore, understanding main scale divisions is essential for utilizing a vernier caliper to its full potential, as they form the base for all measurements.