JEE PYQ: Units & Measurements - Question ID db5a3e35111c (JEE Main 2023)

Core Formula & Concept:

In vernier callipers, the least count (LC) is the smallest length that can be measured accurately. It is calculated as: $\text{LC} = \frac{\text{Value of 1 main scale division}}{\text{Number of vernier divisions}}$ However, in this problem the least count is already given as $0.1 \text{ mm}$.

When the jaws are closed, the zero of the vernier scale does not coincide with the zero of the main scale. This introduces a zero error. If the vernier zero lies to the right of the main scale zero, the zero error is positive. The magnitude of the zero error is: $\text{Zero error} = (\text{Number of coinciding vernier division}) \times \text{LC}$

When measuring an object, the observed reading is the sum of the main scale reading (MSR) and the vernier scale reading (VSR): $\text{Observed reading} = \text{MSR} + (\text{Vernier coinciding division} \times \text{LC})$ The true reading is then obtained by correcting for the zero error: $\text{True reading} = \text{Observed reading} - \text{Zero error}$

Step-by-Step Derivation:

Step 1: Determine the zero error
When the jaws are closed, the zero of the vernier scale lies to the right of the main scale zero, and the 6th vernier division coincides with a main scale division. Given least count $\text{LC} = 0.1 \text{ mm} = 0.01 \text{ cm}$, the zero error is: $\text{Zero error} = 6 \times 0.01 \text{ cm} = 0.06 \text{ cm}$ Since the vernier zero is to the right, the zero error is positive.

Step 2: Record the observed reading while measuring the bob
The zero of the vernier scale lies between $3.2 \text{ cm}$ and $3.3 \text{ cm}$ on the main scale. Hence, the main scale reading (MSR) is $3.2 \text{ cm}$. The 4th vernier division coincides with a main scale division. Thus, the vernier scale reading (VSR) is: $\text{VSR} = 4 \times 0.01 \text{ cm} = 0.04 \text{ cm}$ The observed reading is: $\text{Observed reading} = \text{MSR} + \text{VSR} = 3.2 \text{ cm} + 0.04 \text{ cm} = 3.24 \text{ cm}$

Step 3: Correct for zero error to find the true diameter
Since the zero error is positive, we subtract it from the observed reading: $\text{True diameter} = \text{Observed reading} - \text{Zero error} = 3.24 \text{ cm} - 0.06 \text{ cm} = 3.18 \text{ cm}$

Common Traps & Exam Tip:

Trap 1: Students often confuse the sign of the zero error. If the vernier zero lies to the right of the main scale zero, the zero error is positive, and must be subtracted from the observed reading. Conversely, if it lies to the left, the zero error is negative and must be added.

Trap 2: Misidentifying the coinciding vernier division. Ensure you count divisions from the vernier zero, not from the main scale zero.

Exam Tip: Always convert all measurements to the same unit (preferably cm) before performing calculations to avoid unit mismatch errors.

Thus, the correct diameter of the bob is $3.18 \text{ cm}$, corresponding to option A.