JEE PYQ: Units & Measurements - Question ID 00333d0b3ae0 (JEE Main 2020)

In vernier callipers, the least count (LC) is the smallest length that can be measured accurately. It is determined by the difference between one main scale division (MSD) and one vernier scale division (VSD). The key formula is:

When the vernier scale is divided into $n$ divisions that coincide with $(n-1)$ divisions of the main scale, the least count becomes:

Additionally, the zero error must be accounted for. Zero error occurs when the zero of the vernier scale does not align with the zero of the main scale when the jaws are closed. It can be positive or negative depending on the direction of misalignment.

The actual measurement is calculated as:

Where the sign depends on the nature of the zero error.

Step 1: Determine the Least Count

Given: - Least count of main scale = 1 mm = 0.1 cm - Vernier scale has 10 divisions coinciding with 9 divisions of the main scale

Since 10 VSD = 9 MSD,

Least Count (LC) = 1 MSD – 1 VSD = 0.1 cm – 0.09 cm = 0.01 cm

So, the vernier callipers can measure up to 0.01 cm accuracy.

Step 2: Determine Zero Error

When jaws are touching (zero position): - The 7^th division of the vernier scale coincides with a main scale division - Zero of vernier scale lies to the right of the zero of the main scale

This indicates a positive zero error.

Zero error = (Coinciding VSD number) × LC

Since the zero of the vernier is to the right of the main scale zero, the instrument reads more than the actual length. Hence, the zero error is positive and must be subtracted from the observed reading.

Step 3: Measure the Length of the Cylinder

When measuring the cylinder: - Zero of vernier scale lies between 3.1 cm and 3.2 cm on the main scale - 4^th VSD coincides with a main scale division

Observed Reading = Main Scale Reading + (Coinciding VSD × LC)

Main scale reading just before zero of vernier = 3.1 cm

Step 4: Apply Zero Error Correction

Actual Length = Observed Reading – Zero Error

Conclusion:

The length of the cylinder is 3.07 cm, which corresponds to option C.

1. Misidentifying Zero Error Sign: Many students confuse whether to add or subtract the zero error. Remember: - If the zero of the vernier is to the right of the main scale zero, the zero error is positive, and must be subtracted from the observed reading. - If to the left, it's negative, and must be added.

2. Incorrect Least Count Calculation: Students often misapply the formula for least count. Always use:

But ensure the logic matches the physical setup (10 VSD = 9 MSD).

3. Ignoring Units: Mixing mm and cm can lead to decimal errors. Always convert all measurements to the same unit (preferably cm for this question).

Exam Tip: Always sketch the vernier scale alignment for both zero position and measurement. Visualizing the coincidence helps avoid sign errors.