Problem 6 - Master Resource Book in JEE Main Physics

Question

Two screw gauges \(A\) and \(B\) have equal number of divisions on circular scale. \(A\) has pitch \(1 \mathrm{~mm}\) and \(B\) has pitch \(0.5 \mathrm{~mm}\). Which one is more accurate? (a) \(A\) (b) \(\underline{B}\) (c) Both (d) Can't say

Step-by-Step Solution

Answer

Screw Gauge B is more accurate because it has a smaller least count.

Step 1: Understanding Pitch and Least Count

The pitch of a screw gauge is the distance it moves forward when the screw is rotated one full turn. The least count, which measures the smallest length that can be measured with the screw gauge, is calculated as the pitch divided by the number of divisions on the circular scale.

Step 2: Calculate Least Count for Screw Gauge A

For screw gauge A, we use the formula: Least Count (LC) = Pitch / Number of divisions. Given, Pitch for A = 1 mm and assuming the number of divisions is N, \( LC_A = \frac{1 \text{ mm}}{N} \).

Step 3: Calculate Least Count for Screw Gauge B

Similarly, for screw gauge B, the Least Count is given by \( LC_B = \frac{0.5 \text{ mm}}{N} \). Again, assuming both gauges have the same number of divisions N, B's least count depends entirely on the smaller pitch, meaning \( LC_B < LC_A \) as 0.5 mm is lesser than 1 mm.

Step 4: Compare Least Counts

With smaller least count, Screw Gauge B can measure smaller increments than A. This means screw gauge B is more precise in its measurements compared to gauge A.

Key Concepts

Understanding Least Count

The least count is a crucial concept when it comes to measuring instruments like screw gauges. It denotes the smallest measurement a device can accurately read. In the case of a screw gauge, the least count is determined by the ratio of the pitch to the number of divisions on its circular scale. To clarify, consider the formula:\[\text{Least Count (LC)} = \frac{\text{Pitch}}{\text{Number of Divisions}}\]This formula reveals that if the pitch remains constant, a greater number of divisions on the circular scale will result in a smaller least count, thereby increasing resolution. Therefore, the least count is inversely proportional to the number of divisions, and this is why it is instrumental in determining how finely an instrument can measure. Understanding this allows us to recognize that instruments with a lower least count are capable of providing more precise measurements.

Pitch and Its Role in Screw Gauges

Pitch, an integral part of the screw gauge, is defined as the distance a screw moves linearly along its axis with one complete rotation of the screw. This movement forms the basic mechanical translation involved in the gauge's operation. The pitch of a screw gauge dictates the incremental steps a screw takes per full turn, hence directly impacting the least count of the gauge. For instance, if a screw gauge has a pitch of 1 mm, the screw moves forward 1 mm per full turn. If another has a pitch of 0.5 mm, it moves just 0.5 mm per full turn. Understanding pitch and its interaction with the circular scale is vital for anyone looking to wield their screw gauge with precision. Smaller pitches usually imply a more delicate and detailed measurement capability, provided the divisions on the circular scale are ample enough to reflect this finer gradation.

Accuracy in Measuring Instruments

Accuracy in measuring instruments, like screw gauges, involves understanding both the precision and the reproducibility of the measurements they provide. A measuring instrument's accuracy is largely influenced by its least count. When two instruments have the same number of divisions, the one with a smaller least count is inherently more accurate. This is because it can measure smaller increments and thus reflect more minor changes or differences in the object being measured. For screw gauges specifically, accuracy also hinges on correctly aligning the instrument and eliminating parallax errors during measurement. Taking repeated measurements and averaging them can further enhance accuracy. Consistency across independent measurements is a hallmark of an accurate instrument. Ultimately, ensuring that both the pitch and the number of scale divisions are optimal will contribute to the high accuracy of measurements, allowing precise and reliable results each time.