Problem 90 - Master Resource Book in JEE Main Physics

Question

One side of a cubical block is measured with the help of a vernier callipers of vernier constant \(0.01 \mathrm{~cm}\). This side comes out to be \(1.23 \mathrm{~cm}\). What is the percentage error in the measurement of area? (a) \(\frac{1.23}{0.01} \times 100\) (b) \(\frac{0.01}{1.23} \times 100\) (c) \(2 \times \frac{0.01}{1.23} \times 100\) (d) \(3 \times \frac{0.01}{1.23} \times 100\)

Step-by-Step Solution

Answer

Option (c): \(2 \times \frac{0.01}{1.23} \times 100\)

Step 1: Identify Variables

We need to determine the percentage error in the measurement of the area of a square face of the cube. The actual measured side length of the cube is given as \(1.23 \text{ cm}\) with a vernier constant (least count) of \(0.01 \text{ cm}\).

Step 2: Understand Percentage Error in Measurement

Percentage error is calculated as the ratio of the absolute error to the measured value, multiplied by 100%. For an area measurement \((x^2)\), the percentage error in measurement is twice that of the length measurement because the error propagates through both dimensions of the area.

Step 3: Calculate Percentage Error in Area

The percentage error in the measurement of one dimension of the cube is \(\frac{\text{vernier constant}}{\text{measured value}} = \frac{0.01}{1.23}\). Since area is calculated as \(x^2\) for one side of the cube, the percentage error in the area is \(2 \times \frac{0.01}{1.23} \times 100\).

Step 4: Choose the Correct Option

From the calculations, the correct expression for the percentage error in the measurement of area is \(2 \times \frac{0.01}{1.23} \times 100\), which matches option (c).

Key Concepts

Understanding Percentage Error

When you measure something using a tool like a vernier caliper, there will always be a slight uncertainty or error associated with the measurement. This is where percentage error comes into play. It gives us an idea of how significant the error is relative to the measured value.
To calculate the percentage error, you divide the absolute error (in this case, the smallest measurement increment your calipers can measure, known as the vernier constant) by the measured value and then multiply by 100%. This calculation is a way to express the discrepancy between the measured value and the actual value as a percentage of the measured value.

• Percentage Error = \(\frac{\text{Vernier Constant}}{\text{Measured Value}} \times 100\%\)
• In our example, for the side length measurement, it was \(\frac{0.01}{1.23} \times 100\%\)

By understanding percentage error, you can better grasp the limitations of your measurement tools and how they affect the final results.

Measurement Error Propagation

Error propagation is a critical aspect of measurement to understand because it tells us how uncertainties affect calculations derived from measurements. When measuring something more complicated than a single measurement, like calculating an area from a length, the error can increase as it's passed through each step of calculation.
In the case of the cubical block, the area of one face is calculated as the square of the side length: \(x^2\). The percentage error in squaring a measurement isn't additive but instead is multiplied by the number of times the measurement is factored. So, squaring the side measurement involves multiplying the side’s percentage error by 2, because each side measurement contributes to the overall error of the area formula:

• Area Error = \(2 \times \frac{\text{Vernier Constant}}{\text{Measured Value}} \times 100\%\)

This approach helps in acknowledging that complex calculations can magnify errors, making accurate initial measurements crucial.

Getting to Know the Vernier Constant

The vernier constant, at times also referred to as the least count, is a crucial value when working with vernier calipers. It represents the smallest value that can be accurately measured using the calipers.
A vernier caliper has two sets of scales: the main scale and the vernier scale. The vernier constant is determined through the difference between the smallest divisions on these two scales.

• If the main scale division is 1 mm and the vernier scale division is 0.9 mm, the vernier constant is the difference, which is 0.1 mm.

Understanding the vernier constant helps in estimating how precisely you can measure with your instruments.
This constant is directly used in calculating percentage error, which means that its accuracy directly influences the accuracy of the measurement itself. Knowing your vernier constant is key to achieving reliable and repeatable measurements.