JEE PYQ: Units & Measurements - Question ID 4aa0ae59d16c (JEE Main 2020)

A screw gauge is a precision instrument used to measure small lengths. Its working principle relies on the conversion of rotational motion into linear displacement. The key concepts involved are:

• Pitch of the screw: The linear distance moved by the screw per complete rotation of the circular scale. Mathematically, $\text{Pitch} = \frac{\text{Linear displacement per rotation}}{\text{Number of rotations}}$ Here, since one complete rotation causes a displacement of 0.5 mm, the pitch is 0.5 mm.
• Least Count (LC): The smallest measurement that can be made with the instrument. It is given by the ratio of the pitch to the number of divisions on the circular scale: $\text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}}$ For this screw gauge, the circular scale has 50 divisions.
• Zero Error: The error present when the instrument reads a non-zero value at the zero position. If the circular scale is ahead of the pitch scale marking (i.e., the zero of the circular scale is below the reference line of the pitch scale), the zero error is positive. Conversely, if it is behind, the zero error is negative.

Step 1: Determine the Pitch

Given that one complete rotation of the circular scale causes a displacement of 0.5 mm on the pitch scale, the pitch of the screw gauge is: $\text{Pitch} = 0.5 \text{ mm}$

Step 2: Calculate the Least Count

The circular scale has 50 divisions. The least count is: $\text{Least Count} = \frac{\text{Pitch}}{\text{Number of divisions}} = \frac{0.5 \text{ mm}}{50} = 0.01 \text{ mm}$ Convert 0.01 mm to micrometers (since 1 mm = 1000 μm): $0.01 \text{ mm} = 10 \text{ μm}$

Step 3: Determine the Nature of Zero Error

The problem states that the circular scale is 4 units ahead of the pitch scale marking prior to use. This means the zero of the circular scale is below the reference line of the pitch scale, indicating a positive zero error.

Step 4: Match with Given Options

From the above: - Nature of zero error: Positive - Least count: 10 μm The correct option is C: Positive, 10 μm.

Students often make the following mistakes in this question:

1. Misinterpreting the zero error: Confusing "ahead" with a negative error. Remember: if the circular scale is ahead (i.e., the zero mark is below the reference line), the zero error is positive.
2. Unit confusion: Forgetting to convert the least count from millimeters to micrometers. Always check the units in the options and convert accordingly.
3. Incorrect pitch calculation: Some students assume the pitch is equal to the number of divisions or misapply the formula. The pitch is the linear displacement per rotation, not the number of divisions.
4. Least count formula error: Using the number of divisions on the pitch scale instead of the circular scale. The least count depends on the circular scale divisions.

Exam Tip: Always draw a mental or physical sketch of the screw gauge to visualize the zero error. For least count, memorize the formula: $\text{Least Count} = \frac{\text{Pitch}}{\text{Circular scale divisions}}$ and double-check units before selecting the answer.