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JEE PYQ: Units & Measurements - Question ID 49f361bfba10 (JEE Main 2024)

ID: 49f361bfba10JEE Main 2024Single Correct MCQ

Least count of a vernier caliper is 120 N cm\frac{1}{20 \mathrm{~N}} \mathrm{~cm}. The value of one division on the main scale is 1 mm1 \mathrm{~mm}. Then the number of divisions of main scale that coincide with N\mathrm{N} divisions of vernier scale is :

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Step-by-step Explanation

Core Formula & Concept:

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

Let:

  • MSDMSD = value of one main-scale division (given as 1 mm1\,\text{mm})
  • VSDVSD = value of one vernier-scale division
  • NN = number of divisions on the vernier scale that coincide with a certain number of main-scale divisions
The least count is: LC=MSD−VSDLC = MSD - VSD Additionally, when NN divisions of the vernier scale coincide with MM divisions of the main scale, we have: N⋅VSD=M⋅MSDN \cdot VSD = M \cdot MSD Rearranging gives the relationship between MM, NN, and the least count.

Step-by-Step Derivation:

1. Given data:

  • Least count LC=120N cm=120N×10 mm=12N mmLC = \frac{1}{20N}\,\text{cm} = \frac{1}{20N} \times 10\,\text{mm} = \frac{1}{2N}\,\text{mm}
  • Main scale division MSD=1 mmMSD = 1\,\text{mm}
2. From the definition of least count: LC=MSD−VSD⇒VSD=MSD−LC=1−12NLC = MSD - VSD \quad \Rightarrow \quad VSD = MSD - LC = 1 - \frac{1}{2N} 3. When NN divisions of the vernier scale coincide with MM divisions of the main scale: N⋅VSD=M⋅MSDN \cdot VSD = M \cdot MSD Substituting VSDVSD and MSDMSD: N(1−12N)=M⋅1N \left(1 - \frac{1}{2N}\right) = M \cdot 1 Simplify: N−12=MN - \frac{1}{2} = M Hence: M=2N−12M = \frac{2N - 1}{2} 4. The question asks for the number of main-scale divisions (MM) that coincide with NN vernier-scale divisions. From the above, we have: M=2N−12M = \frac{2N - 1}{2} This matches option D.

Common Traps & Exam Tip:

  • Unit confusion: Students often forget to convert the least count from cm to mm, leading to incorrect values for VSDVSD. Always ensure consistent units.
  • Misinterpretation of coincidence: Some confuse whether MM or NN is the number of divisions coinciding. Remember: NN vernier divisions coincide with MM main divisions.
  • Algebraic errors: Incorrectly rearranging N⋅VSD=M⋅MSDN \cdot VSD = M \cdot MSD can lead to wrong expressions. Double-check each algebraic step.

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