JEE PYQ: Units & Measurements - Question ID 4f5edb452a4b (JEE Main 2022)

Core Formula & Concept:

In precision measurement instruments like a travelling microscope, the least count (LC) is the smallest length that can be measured accurately. It is determined by the difference between one main scale division (MSD) and one vernier scale division (VSD).

Key formulas:

• $\text{Least Count (LC)} = \text{Value of 1 MSD} - \text{Value of 1 VSD}$
• If $n$ vernier divisions coincide with $m$ main scale divisions, then: $n \cdot \text{VSD} = m \cdot \text{MSD}$ $\Rightarrow \text{VSD} = \frac{m}{n} \cdot \text{MSD}$

The least count is then: $\text{LC} = \text{MSD} - \text{VSD} = \text{MSD} \left(1 - \frac{m}{n}\right)$

Step-by-Step Derivation:

Step 1: Determine the value of 1 main scale division (MSD)

The main scale has 20 divisions per cm. So, $\text{MSD} = \frac{1 \text{ cm}}{20} = 0.05 \text{ cm}$

Step 2: Relate vernier scale divisions to main scale divisions

Given: 25 vernier scale divisions (VSD) = 24 main scale divisions (MSD) So, $25 \cdot \text{VSD} = 24 \cdot \text{MSD}$ $\Rightarrow \text{VSD} = \frac{24}{25} \cdot \text{MSD}$

Step 3: Compute the least count (LC)

$\text{LC} = \text{MSD} - \text{VSD}$ Substitute VSD: $\text{LC} = \text{MSD} - \frac{24}{25} \cdot \text{MSD} = \text{MSD} \left(1 - \frac{24}{25}\right) = \text{MSD} \cdot \frac{1}{25}$ Substitute MSD = 0.05 cm: $\text{LC} = 0.05 \cdot \frac{1}{25} = 0.002 \text{ cm}$

Step 4: Match with given options

The least count is 0.002 cm, which corresponds to option C.

Common Traps & Exam Tip:

Students often confuse the number of divisions and their values. A frequent mistake is misinterpreting the relation between vernier and main scale divisions, leading to incorrect least count calculations. Always ensure:

• Correctly identify MSD and VSD values.
• Use the relation $n \cdot \text{VSD} = m \cdot \text{MSD}$ accurately.
• Double-check units (cm vs mm) to avoid mismatches with options.

In this question, option B (0.002 mm) is a trap for those who misconvert units. Always verify the final unit consistency.