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JEE PYQ: Units & Measurements - Question ID edaf72d53307 (JEE Main 2009)

ID: edaf72d53307JEE Main 2009Single Correct MCQ
In an experiment the angles are required to be measured using an instrument, 29 divisions of the main scale exactly coincide with the 30 divisions of the vernier scale. If the smallest division of the main scale is half-a degree(=0.5∘0.5^\circ), then the least count of the instrument is:

Select Option

Step-by-step Explanation

Core Formula & Concept:

In any vernier-type instrument (such as a vernier caliper or a vernier protractor), the least count (LC) is the smallest angle (or length) that can be measured with the instrument. It is defined by the difference between one division of the main scale and one division of the vernier scale.

Let

  • NmsN_{\text{ms}} = number of main-scale divisions that coincide with NvsN_{\text{vs}} vernier-scale divisions,
  • ss = value of one main-scale division (in degrees or minutes).
Then the least count is given by LC  =  sNvs−Nmsor equivalentlyLC  =  ∣  s−v∣,\text{LC} \;=\;\frac{s}{N_{\text{vs}} - N_{\text{ms}}} \quad\text{or equivalently}\quad \text{LC} \;=\;\bigl|\;s - v\bigr|, where vv is the value of one vernier-scale division.

Step-by-Step Derivation:

1. Identify the given data:

  • 29 divisions of the main scale coincide with 30 divisions of the vernier scale.
  • Smallest main-scale division s=0.5∘s = 0.5^\circ.
2. Compute the value of one vernier-scale division vv: 29 s=30 v⟹v=2930 s=2930×0.5∘=2960∘.29\,s = 30\,v \quad\Longrightarrow\quad v = \frac{29}{30}\,s = \frac{29}{30}\times 0.5^\circ = \frac{29}{60}^\circ. 3. The least count is the difference between one main-scale division and one vernier-scale division: LC=∣ s−v∣=∣ 0.5∘−2960∘∣=160∘.\text{LC} = \bigl|\,s - v\bigr| = \Bigl|\,0.5^\circ - \frac{29}{60}^\circ\Bigr| = \frac{1}{60}^\circ. 4. Convert 160∘\tfrac{1}{60}^\circ into minutes: 160∘×60  minutes1∘=1  minute.\frac{1}{60}^\circ \times 60\;\frac{\text{minutes}}{1^\circ} = 1\;\text{minute}. 5. Match with the given options: the least count is one minute, which corresponds to option A.

Common Traps & Exam Tip:

  • Misidentifying which scale is finer. Students sometimes confuse which scale has more divisions and end up computing vv incorrectly.
  • Forgetting to convert degrees to minutes. The question asks for the answer in minutes, so a final conversion is essential.
  • Sign errors in the difference. Always take the absolute value ∣s−v∣\bigl|s - v\bigr| to ensure a positive least count.
Remember: the least count is always the smallest measurable difference, so it must be positive and as small as possible.

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