JEE PYQ: Units & Measurements - Question ID c5d2daf7274d (JEE Main 2025)

In a vernier callipers, the key idea is to measure lengths more precisely than the smallest division on the main scale. This is achieved by using a secondary (vernier) scale that slides along the main scale.

• Least Count (Vernier Constant): The smallest length that can be measured with the instrument is called the least count or vernier constant. It is defined as: $\text{Least Count} = \text{Value of one main scale division} - \text{Value of one vernier scale division}$ Equivalently, if there are $n$ vernier divisions matching $(n-1)$ main scale divisions, then: $\text{Least Count} = \frac{\text{Value of one main scale division}}{n}$
• Vernier Scale Division Size: Since the vernier scale is designed to be slightly shorter than the main scale, each vernier division is always smaller than one main scale division. This ensures that the difference between the two scales (the least count) is positive and meaningful.

Analyzing Statement I:

Statement I claims: "In a vernier callipers, one vernier scale division is always smaller than one main scale division."

• By design, the vernier scale is constructed such that $n$ vernier divisions coincide with $(n-1)$ main scale divisions. Let:
  • $MSD$ = length of one main scale division
  • $VSD$ = length of one vernier scale division
  Then: $n \cdot VSD = (n-1) \cdot MSD$ Rearranging: $VSD = \frac{(n-1)}{n} \cdot MSD$ Since $\frac{n-1}{n} < 1$, it follows that $VSD < MSD$.
• Thus, Statement I is true.

Analyzing Statement II:

Statement II claims: "The vernier constant is given by one main scale division multiplied by the number of vernier scale divisions."

• From the core concept, the vernier constant (least count) is: $\text{Least Count} = MSD - VSD = MSD - \frac{(n-1)}{n} MSD = \frac{MSD}{n}$ Alternatively, it can also be expressed as: $\text{Least Count} = \frac{MSD}{n}$ where $n$ is the number of vernier divisions.
• The statement incorrectly claims that the vernier constant is $MSD \times n$, which would be $n \cdot MSD$. This is not the correct formula.
• Thus, Statement II is false.

Conclusion:

Since Statement I is true and Statement II is false, the correct option is:

B: Statement I is true but Statement II is false

• Misidentifying the Vernier Constant Formula: Many students confuse the vernier constant with the product of the main scale division and the number of vernier divisions. Remember: the vernier constant is the difference between one main scale division and one vernier scale division, or equivalently, $MSD/n$.
• Assuming Vernier Divisions Can Be Larger: Some students mistakenly think that vernier divisions could be larger than main scale divisions. However, the vernier scale is always designed to be slightly shorter to enable precise measurements.
• Exam Tip: Always recall that the vernier scale is a "subdivision" of the main scale. The key formula to remember is: $\text{Least Count} = \frac{\text{Value of one main scale division}}{\text{Number of vernier divisions}}$ This will help you avoid common pitfalls in such questions.