Problem 20 - Master Resource Book in JEE Main Physics

Question

\(N\) divisions on the main scale of a vernier callipers coincide with \(N+1\) divisions of the vernier scale. If each division of main scale is \(a\) units, then least count of the instrument is (a) \(\frac{a}{N+1}\) (a) \(a\) (c) \(\frac{N}{N+1} \times a\) (d) \(\frac{a}{N}\)

Step-by-Step Solution

Answer

The least count is \(\frac{a}{N+1}\), which corresponds to option (a).

Step 1: Understand the Problem

We have a vernier caliper scale where \(N\) divisions on the main scale are equal to \(N+1\) divisions on the vernier scale. Each division on the main scale corresponds to \(a\) units. We need to find the least count of the instrument.

Step 2: Determine the Length of One Division on Each Scale

The total length of \(N\) divisions on the main scale is \(N \times a\). Since \(N\) main scale divisions are equal to \(N+1\) vernier divisions, each vernier division equals \(\frac{N \times a}{N+1}\).

Step 3: Calculate the Least Count

The least count is the difference between one main scale division and one vernier scale division. So, we have:\[\text{Least Count} = a - \frac{N \times a}{N+1}\]

Step 4: Simplify the Least Count

Simplify the expression to find the least count:\[\text{Least Count} = a - \frac{N \times a}{N+1} = a\left(1 - \frac{N}{N+1}\right) = a \cdot \frac{1}{N+1}\]

Step 5: Identify the Correct Option

The least count of the vernier caliper is \(\frac{a}{N+1}\). This matches option (a).

Key Concepts

Least Count

The least count of a vernier caliper refers to the smallest measurement it can accurately read. This precision is fundamental when you're dealing with instruments that require fine measurements. Essentially, it's the difference between one main scale division and one vernier scale division. In our exercise, we calculated it as \( a - \frac{N \times a}{N+1} \), which simplifies further to \( \frac{a}{N+1} \). This shows how the precision of measurements increases with a smaller least count. A vernier caliper with a lesser least count can measure smaller increments, allowing for more detailed and accurate readings. Always ensure you know your instrument's least count before measuring, so you are aware of its precision capability.

Main Scale

The main scale of a vernier caliper is the fixed scale that provides the baseline measurement. It generally resembles a regular ruler with evenly spaced lines, each representing a certain unit of measurement. In our problem, each division of this main scale corresponds to \( a \) units. Thus, to find the distance covered by multiple divisions, you multiply the number of divisions by \( a \). For instance, if there were \( N \) divisions, the total length would be \( N \times a \). The main scale provides a foundation upon which more exact measurements are determined, and it is combined with the vernier scale to achieve even finer precision.

Vernier Scale

The vernier scale is a secondary scale that encompasses the enhancements needed for exact measurement. It slides alongside the main scale, aligning its own divisions next to those on the main scale to improve precision. In the exercise, initially, \( N+1 \) divisions on the vernier scale equate to \( N \) divisions on the main scale. The key role of the vernier scale is to facilitate identification of the fractional reading between two main scale divisions. Using the alignment between these two scales, you can determine the accurate value down to a fraction of the unit represented by the main scale.

Units of Measurement

A crucial component when working with any measuring instrument is the unit of measurement it utilizes. With vernier calipers, this typically could be millimeters, centimeters, or inches, depending on the specific tool and what you are measuring. The exercise references units as \( a \), meaning each main scale division is equated to \( a \) units. It's vital to grasp these units because they influence the interpretation of any measurement taken. Understand these units typical applications can vary, as scientific measurements might prefer the metric system, while some industrial applications may opt for imperial units. Always ensure the unit of measurement aligns with the purpose of your task for accuracy.