Problem 52 - Master Resource Book in JEE Main Physics

Question

Two full turns of the circular scale of a screw gauge cover a distance of \(1 \mathrm{~mm}\) on its main scale. The total number of divisions on the circular scale is 50 . Further, it is found that the screw gauge has a zero error of \(-0.03 \mathrm{~mm}\). While measuring the diameter of a thin wire, a student notes the main scale reading of \(3 \mathrm{~mm}\) and the number of circular scale divisions in line with the main scale as 35. The diameter of the wire is (a) \(3.32 \mathrm{~mm}\) (b) \(3.37 \mathrm{~mm}\) (c) \(3.67 \mathrm{~mm}\) (d) \(3.38 \mathrm{~mm}\)

Step-by-Step Solution

Answer

The diameter of the wire is (d) 3.38 mm.

Step 1: Calculate the Least Count

The least count of the screw gauge is the smallest measurement that can be accurately obtained with it. The distance covered by 1 full turn of the circular scale = \(\frac{1}{2}\mathrm{~mm}\) (since 2 turns cover 1 mm)The least count (LC) is given by:\[\text{LC} = \frac{\text{Distance per full turn}}{\text{Total divisions on circular scale}} = \frac{0.5 \mathrm{~mm}}{50} = 0.01 \mathrm{~mm}\]

Step 2: Determine Circular Scale Reading

To find the contribution of the circular scale to the measurement, we multiply the number of divisions aligned with the main scale by the least count (LC).Circular scale reading = \(35 \times 0.01 \mathrm{~mm} = 0.35 \mathrm{~mm}\)

Step 3: Calculate Total Reading Without Error

Add the main scale reading and the circular scale reading to find the total measurement without considering zero error.Total reading = Main scale reading + Circular scale reading \[3 \mathrm{~mm} + 0.35 \mathrm{~mm} = 3.35 \mathrm{~mm}\]

Step 4: Correct for Zero Error

The gauge has a zero error of \(-0.03 \mathrm{~mm}\). Therefore, we need to add this error (since it is negative) to the total reading to correct it.Corrected reading = Total reading + Zero error \[3.35 \mathrm{~mm} + 0.03 \mathrm{~mm} = 3.38 \mathrm{~mm}\]

Key Concepts

Least Count

The least count of a screw gauge is a concept that defines the smallest measurement increment that the instrument can precisely indicate. In simpler terms, it’s the tiniest value between two scale marks on the gauge. Understanding least count is essential because:

• It determines the precision of your measurements. The smaller the least count, the more precise the screw gauge.
• It helps in gauging measurement uncertainties.

To calculate the least count for a screw gauge, you need two critical pieces of information: the distance the screw covers in one complete turn and the total number of divisions on its circular scale.
In the given problem, two complete turns of the circular scale cover 1 mm on the main scale. This means one full turn covers 0.5 mm. With 50 divisions on the circular scale, the least count becomes \[ \text{Least Count (LC)} = \frac{\text{Distance per full turn}}{\text{Total divisions on circular scale}} = \frac{0.5 \mathrm{~mm}}{50} = 0.01 \mathrm{~mm} \]This means the smallest measurement you can reliably obtain with this screw gauge is 0.01 mm.

Zero Error Correction

Zero error in a screw gauge occurs when the zero mark on the circular scale does not align perfectly with the zero mark on the main scale. This misalignment causes inaccuracies if left uncorrected. Correcting for zero error is crucial to ensure that your measurements are accurate and trustworthy. Here's why it's essential:

• It ensures that any systematic biases in your measurements are accounted for and corrected.
• It prevents erroneous readings that can lead to significant errors in subsequent calculations or assessments.

In cases where there is a negative zero error, such as \(-0.03 \mathrm{~mm}\) in our exercise, you need to add this value when calculating the accurate reading since it underestimates the measurement.
For the measurement conducted in the exercise, correcting the zero error means:
\[ \text{Corrected Reading} = \text{Total Reading} + \text{Zero Error} = 3.35 \mathrm{~mm} + 0.03 \mathrm{~mm} = 3.38 \mathrm{~mm} \]This correction leads to the final diameter measurement being more accurate and reliable.

Circular Scale Reading

The circular scale reading is a crucial component in using a screw gauge. It allows for fine precision in measuring dimensions by providing additional granular readings with the circular scale, which complements the main scale readings. Here’s a straightforward breakdown:

• It fills the gap between the larger units measured on the main scale, making the overall instrument more accurate.
• It relies on the least count to correlate the divisions on the circular scale to a measurable value in millimeters or another unit.

In our exercise, the circular scale reading directly affects the final measurement of the diameter of the wire. To compute this, multiply the number of circular scale divisions that align with the main scale by the least count:
\[ \text{Circular scale reading} = 35 \times 0.01 \mathrm{~mm} = 0.35 \mathrm{~mm} \]
This measurement is then added to the main scale reading to get the total length measurement from the screw gauge before any error correction.