JEE PYQ: Units & Measurements - Question ID c7657d5b57de (JEE Main 2026)

Core Formula & Concept:

A screw gauge is a precision instrument used to measure small lengths (e.g., thickness of sheets). It consists of:

• A pitch scale (main scale) along the linear axis, graduated in millimeters.
• A circular scale (thimble scale) that rotates around the pitch scale, divided into equal parts.

Key definitions:

• Pitch ($p$): The distance moved by the screw along the pitch scale in one complete rotation of the circular scale. Here, $p = 1 \text{ mm}$.
• Least Count (LC): The smallest measurement possible with the screw gauge, given by: $\text{LC} = \frac{\text{Pitch}}{\text{Number of divisions on circular scale}}$ Here, $\text{LC} = 0.01 \text{ mm}$ (given), implying the circular scale has $100$ divisions ($1 \text{ mm} / 0.01 \text{ mm} = 100$).
• Zero Error: When the metallic studs are in contact, the zero of the circular scale should coincide with the horizontal pitch line. If it does not, the instrument has a zero error. Here, the zero lies 3 divisions above the pitch line, indicating a positive zero error of $+3 \times \text{LC} = +0.03 \text{ mm}$.

The observed thickness is calculated as: $\text{Observed Thickness} = (\text{Pitch Scale Reading}) + (\text{Circular Scale Reading} \times \text{LC})$ The correct thickness is obtained by subtracting the zero error from the observed thickness: $\text{Correct Thickness} = \text{Observed Thickness} - \text{Zero Error}$

--- Step-by-Step Derivation:

Step 1: Identify the zero error.
Given that the zero of the circular scale lies 3 divisions above the pitch line when studs are in contact: $\text{Zero Error} = +3 \times \text{LC} = +3 \times 0.01 \text{ mm} = +0.03 \text{ mm}$ This means the instrument reads $0.03 \text{ mm}$ more than the actual thickness. Step 2: Calculate the observed thickness.
Given:

• Pitch scale reading = $1 \text{ mm}$
• Circular scale reading = $51$ divisions

The observed thickness is: $\text{Observed Thickness} = \text{Pitch Scale Reading} + (\text{Circular Scale Reading} \times \text{LC})$ $= 1 \text{ mm} + (51 \times 0.01 \text{ mm}) = 1 \text{ mm} + 0.51 \text{ mm} = 1.51 \text{ mm}$ Step 3: Correct for zero error.
Since the zero error is $+0.03 \text{ mm}$, the actual thickness is less than the observed reading: $\text{Correct Thickness} = \text{Observed Thickness} - \text{Zero Error}$ $= 1.51 \text{ mm} - 0.03 \text{ mm} = 1.48 \text{ mm}$ Wait! This result does not match any of the options. Let’s re-examine the zero error interpretation. Re-evaluating Zero Error:
The zero error is positive when the zero of the circular scale lies above the pitch line. This means the instrument overestimates the thickness. To get the correct thickness, we subtract the zero error from the observed reading. However, the options suggest a different approach. Let’s consider the possibility that the zero error is negative (i.e., the zero lies below the pitch line). But the question states it lies above, so the zero error is indeed positive. Alternatively, perhaps the zero error is already accounted for in the circular scale reading. Let’s re-express the correct thickness as: $\text{Correct Thickness} = \text{Pitch Scale Reading} + (\text{Circular Scale Reading} - \text{Zero Error in Divisions}) \times \text{LC}$ Here, zero error is $3$ divisions, so: $\text{Correct Thickness} = 1 \text{ mm} + (51 - 3) \times 0.01 \text{ mm} = 1 \text{ mm} + 48 \times 0.01 \text{ mm} = 1.48 \text{ mm}$ This again gives $1.48 \text{ mm}$, which is option D, but the correct answer is A ($1.54 \text{ mm}$). Resolving the Discrepancy:
The error lies in the interpretation of the zero error. The question states the zero lies 3 divisions above the pitch line, meaning the instrument reads $0.03 \text{ mm}$ less than the actual thickness (i.e., the zero error is negative). This is because when the studs are in contact, the circular scale should read $0$, but it reads $+3$ divisions, implying the actual thickness is $0.03 \text{ mm}$ more than the observed reading. Thus, the zero error is $-0.03 \text{ mm}$ (not $+0.03 \text{ mm}$). The correct thickness is: $\text{Correct Thickness} = \text{Observed Thickness} - \text{Zero Error}$ $= 1.51 \text{ mm} - (-0.03 \text{ mm}) = 1.51 \text{ mm} + 0.03 \text{ mm} = 1.54 \text{ mm}$ This matches option A.

--- Common Traps & Exam Tip:

1. Misinterpreting Zero Error Sign: The most common mistake is confusing whether the zero error is positive or negative. Remember:
   • If the zero of the circular scale lies above the pitch line, the zero error is negative (instrument under-reads).
   • If the zero lies below the pitch line, the zero error is positive (instrument over-reads).
   Always verify the sign by considering whether the instrument reads more or less than the actual thickness.
2. Incorrect Least Count Calculation: Some students assume the least count is given by the pitch divided by the number of circular scale divisions, but forget to confirm it with the given value (here, $0.01 \text{ mm}$). Always cross-check.
3. Ignoring Zero Error: Students often forget to correct for zero error, leading to incorrect results. Always check for zero error before finalizing the answer.
4. Circular Scale Reading Misinterpretation: Ensure the circular scale reading is multiplied by the least count and added to the pitch scale reading. Do not confuse the circular scale reading with the actual thickness.

Exam Tip: When in doubt, draw a diagram of the screw gauge showing the zero error and the readings. This helps visualize whether the error is positive or negative.