Problem 61 - Master Resource Book in JEE Main Physics

Question

A screw gauge gives the following reading when used to measure the diameter of a wire. Main scale reading : \(0 \mathrm{~mm}\) Circular scale reading : 52 divisions Given that \(1 \mathrm{~mm}\) on main scale corresponds to 100 divisions of the circular scale. The diameter of wire from the above data is (a) \(0.052 \mathrm{~cm}\) (b) \(0.026 \mathrm{~cm}\) (c) \(0.005 \mathrm{~cm}\) (d) \(0.52 \mathrm{~cm}\)

Step-by-Step Solution

Answer

(a) \(0.052 \mathrm{~cm}\)

Step 1: Understanding the Screw Gauge

The screw gauge has a main scale and a circular scale. Each division on the main scale corresponds to 1 mm, and each division on the circular scale contributes to the total measurement. Our task is to find the actual diameter using both scale readings.

Step 2: Convert Circular Scale Reading

The circular scale reading is given as 52 divisions. Since 1 mm on the main scale corresponds to 100 divisions on the circular scale, the circular reading in mm is \( \frac{52}{100} \mathrm{~mm} = 0.52 \mathrm{~mm} \).

Step 3: Calculate Total Measurement in mm

Add the main scale reading and the converted circular scale reading to get the total measurement in mm. Main scale reading is \(0 \mathrm{~mm}\), so the total measurement is \(0 + 0.52 = 0.52 \mathrm{~mm}\).

Step 4: Convert mm to cm

To convert the measurement from mm to cm, divide by 10 (since there are 10 mm in a cm). Thus, \(0.52 \mathrm{~mm} = \frac{0.52}{10} \mathrm{~cm} = 0.052 \mathrm{~cm}\).

Step 5: Determine Correct Answer

Compare the calculated diameter \(0.052 \mathrm{~cm}\) with the given options. The option that matches is (a) \(0.052 \mathrm{~cm}\).

Key Concepts

Understanding Circular Scale Reading

Circular scale reading is a crucial part of using a screw gauge effectively. The screw gauge has two primary scales: the main scale and the circular scale. The main scale measures whole millimeters, while the circular scale assists in obtaining finer measurements. The circular scale is designed with a set number of divisions, often around 100, allowing very precise readings.

When measuring with a screw gauge, if the circular scale shows a reading of 52 divisions, it means the measurement is 52 divisions beyond the main scale reading. Typically, the relationship is such that 1 mm on the main scale equals 100 divisions on the circular scale. To translate the division reading into millimeters, divide the number of divisions by the total divisions per millimeter:

• Circular scale reading in mm = \( \frac{52}{100} \text{ mm} = 0.52 \text{ mm} \)

Understanding this conversion is key to getting accurate measurements with a screw gauge.

Main Scale Reading Explained

The main scale reading of a screw gauge is simpler than the circular scale. It represents whole millimeter increments directly and is used alongside the circular reading to provide a complete measurement. In many cases, the main scale reading starts at zero, meaning the user is only measuring additional values presented by the circular scale.

For our exercise, the main scale reading is 0 mm. No whole millimeters are added here. In tasks where more intricate measurements are required, the main scale would show values like 1 mm, 2 mm, etc. These values are then combined with the circular scale reading to determine the total measurement.

Remember these key points:

• Main scale reading provides the base measurement in mm.
• It helps adjust and refine the readings provided by the circular scale, ensuring all aspects of a measurement are captured.

This combination allows the screw gauge to be an incredibly precise instrument.

Unit Conversion Essentials

The final step in screw gauge measurement is converting units to the desired form, often from millimeters to centimeters. In many real-world applications, especially in academics and industry, it's important to express measurements in suitable units.

To convert from millimeters to centimeters, use this straightforward calculation since 1 cm equals 10 mm:

• Total measurement in cm = Total measurement in mm / 10

Using the example from our exercise where the total reading was 0.52 mm, we convert:\[0.52 \text{ mm} = \frac{0.52}{10} \text{ cm} = 0.052 \text{ cm}\]
Thus, properly converting these values not only completes your measurement but ensures consistency and accuracy in record-keeping and analysis.