Question

N divisions on the main scale of a vernier calipers coin sides with \((\mathrm{N}+1)\) divisions on the vernier scale. If each division on the main scale is of a units, the least of count of instrument is................ (A) \(\\{\mathrm{a} /(\mathrm{N}+1)\\}\) (B) \(\\{\mathrm{a} /(\mathrm{N}-1)\\}\) (C) \(\\{(N+1) / a\\}\) (D) \(\\{(N-1) / a\\}\)

Short answer

The least count of the instrument is (A) \(\\{\mathrm{a} /(\mathrm{N}+1)\\}\).

Step-by-step solution

Finding the value of one division on the vernier scale

The least count can be calculated as the difference between one main scale reading and one vernier scale reading. From the given information, we know that N divisions on the main scale coincide with (N+1) divisions on the vernier scale. Let's denote the value of one division on the vernier scale as "b" units. Therefore: \(Nb = (N+1)b\)

Solve for the value of b

Now, in order to find the value of one vernier scale reading, we will solve for b in the above equation. Divide both sides of the equation by N(N+1): \(\frac{Nb}{N(N+1)} = \frac{(N+1)b}{N(N+1)}\) This simplifies to: \(\frac{b}{N+1} = a\) Now, solve for b: \(b = a(N+1)\)

Calculate the least count

The least count of the instrument is the difference between one main scale reading (a) and one vernier scale reading (b). Least count = \(|a - b|\) Substitute the value of b from step 2: Least count = \(|a - a(N+1)|\) Least count = \(|a(1 - (N+1))|\) Least count = \(|a(N+1 - N - 1)|\) Least count = \(a / (\mathrm{N} + 1)\) Comparing our result with the given options, we find that the correct option is: (A) \(\\{\mathrm{a} /(\mathrm{N}+1)\\}\)

Key concepts

main scale divisions

Main scale divisions are the evenly spaced markings on the main scale or ruler of the vernier calipers. These divisions represent the primary measurement units in which the instrument measures. Each division typically equals a certain unit of measurement, which could be millimeters or centimeters, depending on the calipers. Understanding these divisions is crucial for interpreting readings accurately.
For instance, if each main scale division represents 1 mm, this means that every interval or step on the scale measures exactly 1 mm. When using the calipers, the measurement taken from the main scale involves counting these known divisions and adding any additional, smaller subdivisions measured by the vernier scale.

vernier scale divisions

The vernier scale divisions exist to provide more precise readings than the main scale alone. This smaller scale slides alongside the main scale and has more divisions compared to a similar span on the main scale. It is designed to match with the main scale such that one complete rotation or movement of the vernier scale shows a slight difference against the main scale, enabling measurements of fractions of the smallest main scale division.
The number of vernier scale divisions is typically one more than those on a corresponding span on the main scale. For example, the vernier scale may have \(N+1\) divisions over a distance equal to N divisions on the main scale. This setup allows the vernier calipers to read smaller increments, increasing precision by creating very close approximations.

least count calculation

The least count of a vernier calipers is the smallest measurement it can accurately take. It represents the difference between the smallest main scale division and the vernier scale.This is calculated using the formula:\[\text{Least Count} = \text{Value of one main scale division} - \text{Value of one vernier scale division}\]Given that N divisions on the main scale correspond to \(N+1\) divisions on the vernier scale, and if one main scale division is 'a' units, then the least count is calculated as follows:\[\text{Least Count} = a - \left(\frac{a}{N+1}\right)\]Simplifying this results in the least count:\[\text{Least Count} = \frac{a}{N+1}\]This formula shows how using more vernier divisions allows more precise measurements by making the smallest measurable difference smaller.

vernier calipers measurement

Vernier calipers use both the main scale and vernier scale to measure an object's dimension with high precision. To measure, align the object between the jaws of the calipers, then check the alignment on both the main and vernier scales.

• Firstly, look at the main scale reading just before the 'zero' mark of the vernier scale as the main scale reading (MSR).
• Secondly, locate the vernier scale division that aligns or matches closely with one of the lines on the main scale, providing the vernier scale reading (VSR).

Adding the MSR and the difference indicated by the VSR gives the exact measurement. For enhanced accuracy, always ensure the calipers are clean and free of obstructions, and hold the object securely during the measurement.

N divisions

N divisions refer to the specific number of divisions on the main scale that are used to calibrate the vernier scale. In the context of vernier calipers, these divisions provide the basis for determining how the vernier scale is designed.When we say there are 'N divisions' on the main scale corresponding to \(N+1\) divisions on the vernier scale, it signifies the method of creating these precise measurements. The main scale's divisions essentially set a baseline for the vernier scale's divisions, allowing the tool to fine-tune measurements by slightly offsetting these divisions to increase accuracy and provide precise measurements.