Problem 47 - Master Resource Book in JEE Main Physics

Question

In an experiment, the angles are required to be measured using an instrument. 29 divisions of the main scale coincide with 30 divisions of the vernier scale. If the smallest division of the main scale is half a degree \(\left(=0.5^{\circ}\right)\), then the least count of the instrument is (a) half minute (b) one degree (c) half degree (d) one minute

Step-by-Step Solution

Answer

The least count of the instrument is one minute (option d).

Step 1: Understand the Relationship Between the Scales

In vernier calipers, the relationship between the main scale and vernier scale is important. Here, 29 divisions of the main scale align with 30 divisions of the vernier scale. This discrepancy indicates the alignment needed to measure smaller increments.

Step 2: Calculate the Value of Each Division

The smallest division of the main scale is given as \(0.5^{\circ}\). Therefore, 1 division of the main scale equals 0.5 degrees.

Step 3: Formulate the Vernier Scale Division Value

Given that 29 main scale divisions (MSD) coincide with 30 vernier scale divisions (VSD), the size of 1 VSD is:\[1 \text{ VSD} = \frac{29 \times 0.5^{\circ}}{30} = \frac{14.5}{30}^{\circ}\]

Step 4: Compute the Least Count

The least count is the difference between one main scale division and one vernier scale division. Let's calculate this:\[\text{Least Count} = 1 \text{ MSD} - 1 \text{ VSD} = 0.5^{\circ} - \frac{14.5}{30}^{\circ}\]Solving this gives:\[\text{Least Count} = 0.5^{\circ} - 0.4833^{\circ} = 0.0167^{\circ}\]Convert degrees to minutes where \(1^{\circ} = 60\) minutes:\[0.0167^{\circ} \times 60 = 1 \text{ minute}\]

Step 5: Select the Correct Answer

Based on the calculations, the least count of the instrument is 1 minute. Therefore, the correct option is (d) one minute.

Key Concepts

Least Count Calculation

Understanding the least count of an instrument is crucial because it defines the smallest value that can be measured accurately. For vernier calipers, the least count is calculated as the difference between one main scale division (MSD) and one vernier scale division (VSD). This is indicative of the precision level the caliper can achieve. In this exercise, we're given that the smallest division of the main scale is 0.5 degrees. Thus, one MSD equals 0.5 degrees. To find the size of one VSD, use the given relationship: 29 MSD equals 30 VSD. This tells us how two scales relate for finer measurements. First, calculate the vernier scale division value:\[ 1 \text{ VSD} = \frac{29 \times 0.5^{\circ}}{30} = \frac{14.5^{\circ}}{30} \approx 0.4833^{\circ} \]Then, calculate the least count itself:\[ \text{Least Count} = 1 \text{ MSD} - 1 \text{ VSD} \]\[ = 0.5^{\circ} - 0.4833^{\circ} = 0.0167^{\circ} \]This difference in degrees translates to minutes, giving us a least count of 1 minute, thus enhancing measurement precision.

Main Scale and Vernier Scale Relationship

The interplay between the main scale and the vernier scale is foundational to understanding how vernier calipers improve measurement accuracy. In many instruments, the main scale provides a coarse measurement, while the vernier scale offers a finer reading by subdividing the smallest main scale division. In our problem, 29 divisions of the main scale match precisely with 30 divisions of the vernier scale. This mismatch enables the vernier calipers to distinguish very fine differences, demonstrating the high precision of the instrument. The more vernier scale divisions compared to main scale divisions, the finer and more precise the instrument measures. This alignment pattern allows for minute changes in measurement to be detected, which would otherwise remain unnoticed with only a main scale. This relationship, wherein a slightly varied count between scales helps in extrapolating very small values, forms the crux of the vernier caliper's ability to provide high precision readings.

Angle Measurement Precision

Precision in angle measurement is vital for many scientific and engineering applications. Instruments like vernier calipers are designed to enhance accuracy so that even the smallest angle adjustments can be detected. Achieving high precision is particularly important when dealing with angles as even a minor discrepancy can cause significant errors in alignment or construction. In our example, the main scale reads in degrees, but the refinement offered by the vernier scale narrows it down to a minute, showcasing why precision reading tools are indispensable in accurate angle measurement. Vernier calipers achieve this by allowing tiny adjustments to be measured through their least count, translating into more precise angle readings. This improved precision is realized through the systematic difference between the scales, facilitating enhanced measurement capabilities. Thus, vernier calipers are quintessential when tasks demand high precision and minimal tolerance in angular measurements.