JEE PYQ: Units & Measurements - Question ID 0e5de45d91ff (JEE Main 2024)

A vernier caliper measures lengths with higher precision than a simple main scale. Its least count (LC) is the smallest length it can measure accurately. The key idea is:

• The main scale has divisions of size $m$ units.
• The vernier scale has divisions slightly smaller than the main scale.
• When the $n^{\text{th}}$ division of the main scale coincides with the $(n+1)^{\text{th}}$ division of the vernier scale, the difference between one main scale division and one vernier scale division gives the least count.

Mathematically, if $v$ is the size of one vernier division, then: $n \cdot m = (n+1) \cdot v$ The least count is the difference between one main scale division and one vernier scale division: $\text{LC} = m - v$

1. Let $m$ be the size of one main scale division (given). 2. Let $v$ be the size of one vernier scale division (unknown). 3. According to the problem, the $n^{\text{th}}$ main scale division coincides with the $(n+1)^{\text{th}}$ vernier scale division. This means: $n \cdot m = (n+1) \cdot v$ 4. Solve for $v$: $v = \frac{n \cdot m}{n+1}$ 5. The least count (LC) of the vernier caliper is the difference between one main scale division and one vernier scale division: $\text{LC} = m - v$ 6. Substitute $v$ from step 4: $\text{LC} = m - \frac{n \cdot m}{n+1}$ 7. Simplify the expression: $\text{LC} = m \left(1 - \frac{n}{n+1}\right) = m \left(\frac{n+1 - n}{n+1}\right) = m \left(\frac{1}{n+1}\right)$ 8. Thus, the least count is: $\text{LC} = \frac{m}{n+1}$

This matches option B.

1. Misinterpreting the coincidence condition: Students often confuse which divisions coincide. The problem states the $n^{\text{th}}$ main scale division aligns with the $(n+1)^{\text{th}}$ vernier division, not the other way around. 2. Incorrect formula for least count: Some students mistakenly use $\text{LC} = \frac{m}{n}$ instead of $\frac{m}{n+1}$. Always derive it from first principles to avoid this error. 3. Ignoring units: While the question is unit-agnostic, ensure that $m$ and the least count have consistent units in practical problems. 4. Algebraic simplification errors: When simplifying $m - \frac{n \cdot m}{n+1}$, students may forget to factor $m$ or make sign errors. Double-check each step.

Exam Tip: For vernier caliper problems, always write down the coincidence condition as an equation first. This ensures clarity and reduces mistakes.