JEE PYQ: Units & Measurements - Question ID 20d41cc1ed71 (JEE Main 2021)

In a screw gauge (or micrometer), two scales are used to measure small lengths with high precision:

• Main scale: A linear scale marked along the axis of the screw.
• Circular scale: A rotating scale attached to the screw, which moves along the main scale as the screw is turned.

The key definitions are:

• Pitch: The distance moved by the screw (and hence the circular scale) along the main scale in one complete rotation of the circular scale. It is the smallest division on the main scale that the screw advances per full turn.
• Least Count (LC): The smallest length that can be measured with the instrument. It is given by: $\text{Least Count} = \frac{\text{Pitch}}{\text{Total divisions on circular scale}}$ This formula arises because the circular scale subdivides the pitch into finer parts, allowing measurement of fractions of the pitch.

Step 1: Determine the Pitch

The question states that in 5 complete rotations of the circular scale, the distance travelled on the main scale is 5 mm.

Since pitch is the distance moved per one complete rotation, we calculate: $\text{Pitch} = \frac{\text{Distance moved on main scale}}{\text{Number of rotations}} = \frac{5\,\text{mm}}{5} = 1\,\text{mm}$ So, the pitch is 1 mm.

Step 2: Use the Least Count Formula

The circular scale has 50 total divisions. Using the formula: $\text{Least Count} = \frac{\text{Pitch}}{\text{Total divisions on circular scale}} = \frac{1\,\text{mm}}{50} = 0.02\,\text{mm}$

Step 3: Convert Units for Comparison

The assertion claims the least count is 0.001 cm. Let's convert our result to cm: $0.02\,\text{mm} = 0.02 \times 0.1\,\text{cm} = 0.002\,\text{cm}$ This is not equal to 0.001 cm. Hence, Assertion A is incorrect.

Step 4: Evaluate Reason R

Reason R states: $\text{Least Count} = \frac{\text{Pitch}}{\text{Total divisions on circular scale}}$ This is the correct formula for least count in a screw gauge. So, Reason R is correct.

Step 5: Match with Given Options

• A: A is not correct but R is correct. ✅
• B: Both A and R are correct and R explains A. ❌
• C: A is correct but R is not correct. ❌
• D: Both A and R are correct but R does not explain A. ❌

Since A is incorrect and R is correct, the correct option is A.

Trap 1: Misinterpreting Pitch
Students often confuse the total distance moved with the number of rotations. They might divide 5 mm by 50 divisions (instead of 5 rotations), leading to an incorrect pitch of 0.1 mm. Always remember: pitch is distance per one full rotation.

Trap 2: Unit Conversion Errors
The assertion gives the least count in cm, while the calculation is often done in mm. Forgetting to convert units (1 mm = 0.1 cm) can lead to incorrect comparison. Always double-check units.

Trap 3: Misapplying the Least Count Formula
Some students mistakenly use: $\text{Least Count} = \frac{\text{Total distance}}{\text{Total divisions}}$ This is wrong. The correct formula uses pitch, not total distance.

Exam Tip:
When solving screw gauge problems, always:

1. Find the pitch first (distance per rotation).
2. Use the correct least count formula.
3. Convert units consistently (preferably to mm or cm).
4. Verify the assertion and reason separately.