JEE PYQ: Units & Measurements - Question ID c79e89f8b550 (JEE Main 2012)

In a vernier spectrometer, the angle is measured by combining a main scale and a vernier scale. The key concept is the least count (LC) of the vernier, which determines the smallest angle that can be measured beyond the main scale reading.

The least count is calculated as: $\text{LC} = \frac{\text{Value of 1 main scale division}}{\text{Total vernier divisions}}$ Here, 1 main scale division = $0.5^\circ$, and the vernier has 30 divisions matching 29 main scale divisions. Thus: $\text{LC} = \frac{0.5^\circ}{30} = \frac{1}{60}^\circ = 0.0167^\circ \approx 0.0167^\circ$ The total measured angle is: $\text{Angle} = \text{Main scale reading} + (\text{Vernier scale reading} \times \text{LC})$

Step 1: Identify the given data
- Main scale reading = $58.5^\circ$ - Vernier scale reading = 9 divisions - 1 main scale division = $0.5^\circ$ - Vernier scale: 30 divisions = 29 main scale divisions

Step 2: Calculate the least count (LC)
Since 30 vernier divisions match 29 main scale divisions, the difference per vernier division is: $\text{LC} = \frac{0.5^\circ}{30} = 0.0167^\circ$

Step 3: Compute the vernier correction
Vernier scale reading = 9 divisions
Correction = $9 \times 0.0167^\circ = 0.15^\circ$

Step 4: Add the main scale and vernier readings
Total angle = $58.5^\circ + 0.15^\circ = 58.65^\circ$

Step 5: Match with the given options
The calculated angle is $58.65^\circ$, which corresponds to Option C.

Trap 1: Incorrect least count calculation
Students often mistakenly take the LC as $0.5^\circ / 29$ instead of $0.5^\circ / 30$. Remember, the LC is the value of one vernier division, not the difference between main and vernier scales.

Trap 2: Misinterpreting the vernier reading
Some students multiply the vernier reading by $0.5^\circ$ instead of the LC. Always use the LC for vernier corrections.

Exam Tip:
Double-check the least count formula and ensure the vernier reading is multiplied by the correct LC. A quick sanity check: The vernier correction should always be smaller than the main scale division value.