Problem 2 - Master Resource Book in JEE Main Physics

Question

On measuring the diameter of a spherical body using vernier callipers, main scale reading \(=1.3 \mathrm{~cm}, 5\) th vernier scale division is coinciding with any main scale division and zero error is \(-0.03 \mathrm{~cm}\), what will be corrected reading? (a) \(1.38 \mathrm{~cm}\) (b) \(1.32 \mathrm{~cm}\) (c) \(1.35 \mathrm{~cm}\) (d) \(-1.38 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The corrected reading is 1.38 cm.

Step 1: Identify the Main Scale Reading

The main scale reading given is 1.3 cm. This is the linear measurement on the main scale of the vernier calipers.

Step 2: Calculate Vernier Scale Reading

Since the 5th vernier scale division coincides with a main scale division, the vernier scale reading is 0.05 cm (each scale division is assumed 0.01 cm).

Step 3: Calculate Actual Measurement

The actual measurement is found by adding the main scale reading and the vernier scale reading: \(1.3 + 0.05 = 1.35\) cm.

Step 4: Account for Zero Error

Since the zero error is \(-0.03\) cm, add the absolute value of the zero error to the actual measurement to correct for it: \(1.35 + 0.03 = 1.38\) cm.

Key Concepts

Understanding the Vernier Scale

Vernier calipers are an essential tool in precision measurement, which use two scales known as the main scale and the vernier scale. The vernier scale is a smaller, auxiliary scale that slides along the main scale. It's crucial because it allows us to measure lengths with much finer precision, beyond the limitations of the main scale alone.

Here's how it works:

• Each division on the vernier scale is a fraction smaller than a division on the main scale.
• The idea is to find a point where a vernier division aligns perfectly with a division on the main scale.
• In this problem, the 5th division of the vernier scale aligns with a division on the main scale, which translates to 0.05 cm when you multiply 5 by 0.01 cm (assuming each vernier division is 0.01 cm).

This method enhances measurement precision because it reduces the chance of human error in estimating small divisions on the main scale.

The Importance of Zero Error Correction

When using vernier calipers, zero error is a common occurrence. Zero error happens when the zero marks on the vernier scale and main scale do not coincide when the jaws are fully closed. If unnoticed, it could lead to inaccurate readings.

Here's what you should know:

• Zero error can be positive or negative. If the zero of the vernier scale is to the right of the main scale zero, the error is positive. If it’s to the left, it’s negative.
• In this exercise, the given zero error is \(-0.03\) cm, meaning you need to add this value to the measurement as a mathematical correction.
• To correct for this in our measurement, we take the actual measurement found and add this zero error correction:\((1.35 + 0.03 = 1.38\) cm).

Correcting for zero error ensures that the measurement starts from the true zero point of the measuring device.

Main Scale Reading Explained

The main scale of the vernier calipers provides the primary measurement in centimeters or millimeters, as applicable. It is the linear, clearly marked scale on the body of the calipers and typically determines tenths or whole units of a measurement.

While taking a measurement:

• Read the main scale measurement before noting any vernier scale alignment.
• In this exercise, the main scale reading is clearly marked as 1.3 cm.
• This basic reading serves as the foundation to which any additional precision from the vernier scale is added.

Understanding the main scale reading is the first crucial step in achieving a precise measurement with vernier calipers, as it provides the significant figures before factoring in the vernier's enhanced precision.