JEE PYQ: Units & Measurements - Question ID 601694c52201 (JEE Main 2021)

In measurements using a vernier scale, the following key concepts are involved:

• Least Count (LC): The smallest measurement that can be accurately read using the vernier scale. It is calculated as: $\text{LC} = \frac{\text{Value of 1 main scale division}}{\text{Number of vernier divisions}}$ In this problem, the least count is given as $0.01 \text{ cm}$.
• Vernier Coincidence (VC): The number on the vernier scale that aligns exactly with a mark on the main scale. This value is multiplied by the least count to get the fractional part of the measurement.
• Zero Error: A systematic error where the zero of the vernier scale does not coincide with the zero of the main scale when the jaws are closed. A positive zero error means the vernier zero is ahead of the main scale zero, so the measured value is larger than the actual value. To correct this, the zero error must be subtracted from the observed reading.
• Measurement Formula: The correct measurement $M$ is given by: $M = \text{Main Scale Reading} + (\text{Vernier Coincidence} \times \text{Least Count}) - \text{Zero Error}$

Step 1: Identify the main scale reading (MSR)
The '0' on the vernier scale lies between $8.5 \text{ cm}$ and $8.6 \text{ cm}$. This means the main scale reading is $8.5 \text{ cm}$, as the vernier zero has not yet reached $8.6 \text{ cm}$.

Step 2: Calculate the vernier scale contribution
The vernier coincidence is given as $6$. Since the least count is $0.01 \text{ cm}$, the fractional part of the measurement is: $6 \times 0.01 \text{ cm} = 0.06 \text{ cm}$

Step 3: Compute the observed reading (before zero error correction)
Add the main scale reading and the vernier contribution: $8.5 \text{ cm} + 0.06 \text{ cm} = 8.56 \text{ cm}$

Step 4: Apply zero error correction
The vernier scale has a positive zero error of $0.2 \text{ mm}$. Convert this to cm: $0.2 \text{ mm} = 0.02 \text{ cm}$ Since the zero error is positive, the observed reading is larger than the actual value. Thus, subtract the zero error from the observed reading: $8.56 \text{ cm} - 0.02 \text{ cm} = 8.54 \text{ cm}$

Step 5: Match with the given options
The correct value of the measurement is $8.54 \text{ cm}$, which corresponds to option B.

Students often make the following mistakes in such problems:

• Misidentifying the main scale reading: Some students mistakenly take $8.6 \text{ cm}$ as the main scale reading because the vernier zero lies "between" $8.5 \text{ cm}$ and $8.6 \text{ cm}$. However, the main scale reading is always the lower value before the vernier zero.
• Incorrect zero error correction: A positive zero error means the instrument reads higher than the actual value, so the error must be subtracted. Some students add it instead, leading to an incorrect result.
• Unit inconsistency: The zero error is given in $\text{mm}$, while the main scale is in $\text{cm}$. Failing to convert units properly can lead to errors. Always ensure all values are in the same unit before performing calculations.
• Ignoring the least count: Some students forget to multiply the vernier coincidence by the least count, leading to an incorrect fractional part.

Exam Tip: Always follow a structured approach:

1. Note the main scale reading (lower value).
2. Multiply the vernier coincidence by the least count.
3. Add the two to get the observed reading.
4. Apply zero error correction (subtract for positive error, add for negative error).
5. Double-check units and arithmetic.