JEE PYQ: Units & Measurements - Question ID 7ebbc9e74f07 (JEE Main 2021)

In vernier callipers, the least count (LC) is the smallest length that can be measured accurately. It is defined as the difference between one main scale division and one vernier scale division.

Let:

• $a$ = length of one main scale division (in cm)
• $n$ = the vernier scale division that coincides with the $(n-1)$^th main scale division

1. Express the main scale division:
One main scale division = $a$ cm. 2. Express the vernier scale division:
Since the $n$^th vernier division coincides with the $(n-1)$^th main scale division, the total length covered by $n$ vernier divisions equals the length covered by $(n-1)$ main scale divisions: $n \times \text{(one vernier division)} = (n-1) \times a$ Solving for one vernier division: $\text{One vernier division} = \frac{(n-1)}{n} \times a \text{ cm}$ 3. Calculate the least count (LC):
$LC = \text{One main scale division} - \text{One vernier division}$ Substituting the values: $LC = a - \frac{(n-1)}{n} a = \left(1 - \frac{n-1}{n}\right) a = \frac{a}{n} \text{ cm}$ 4. Convert cm to mm:
Since $1 \text{ cm} = 10 \text{ mm}$, $LC = \frac{a}{n} \times 10 = \frac{10a}{n} \text{ mm}$

• Misidentifying coinciding divisions: Students often confuse which division coincides with which, leading to incorrect expressions for the vernier division length.
• Unit inconsistency: Forgetting to convert the final answer from cm to mm (as the options are in mm) is a frequent error.
• Algebraic simplification: Errors in simplifying $\left(1 - \frac{n-1}{n}\right)$ can lead to incorrect options like $D$.