Problem 22 - Master Resource Book in JEE Main Physics

Question

The dimensions of a rectangular block measured with callipers having least count of \(0.01 \mathrm{~cm}\) are \(5 \mathrm{~mm} \times 10 \mathrm{~mm} \times 5 \mathrm{~mm} .\) The maximum percentage error in the measurement of the volume of the block is (a) \(5 \%\) (b) \(10 \%\) (c) \(15 \%\) (d) 2096

Step-by-Step Solution

Answer

The maximum percentage error in the volume measurement is 5%, so the answer is (a) 5%.

Step 1: Understanding the Problem

We have a rectangular block with dimensions given in millimeters: \(5 \, \mathrm{mm} \times 10 \, \mathrm{mm} \times 5 \, \mathrm{mm}\). The least count of the calipers is \(0.01 \, \mathrm{cm}\). We are asked to find the maximum percentage error in the measurement of the volume of the block.

Step 2: Convert Units

First, we need to convert the dimensions from millimeters to centimeters, as the least count is given in centimeters. Recall that \(1 \, \mathrm{cm} = 10 \, \mathrm{mm}\). Thus: - Length: \(5 \, \mathrm{mm} = 0.5 \, \mathrm{cm}\) - Breadth: \(10 \, \mathrm{mm} = 1.0 \, \mathrm{cm}\) - Height: \(5 \, \mathrm{mm} = 0.5 \, \mathrm{cm}\).

Step 3: Calculate Absolute Error

The absolute error for each dimension is equal to the least count of the measurement device, \(\pm 0.01 \, \mathrm{cm}\).

Step 4: Calculate Percentage Error for Each Dimension

Calculate the percentage error for each dimension: - For length: \[ \left( \frac{0.01}{0.5} \right) \times 100\% = 2\% \] - For breadth: \[ \left( \frac{0.01}{1.0} \right) \times 100\% = 1\% \] - For height: \[ \left( \frac{0.01}{0.5} \right) \times 100\% = 2\% \]

Step 5: Calculate Maximum Percentage Error in Volume

The maximum percentage error in the measurement of the volume of the block is the sum of the percentage errors of each dimension: \(2\% + 1\% + 2\% = 5\%\).

Step 6: Identify the Correct Option

The calculated maximum percentage error in the volume measurement is \(5\%\). Thus, the correct answer is option (a) \(5\%\).

Key Concepts

Percentage Error Calculation

Percentage error is a valuable concept in measurements as it gives us insight into how much a calculated measurement might deviate from the true value. It's a vital parameter for understanding uncertainty in experimental data. For calculating this error, you employ the formula:

• Percentage Error = \( \left( \frac{\text{Absolute Error}}{\text{Measured Value}} \right) \times 100\% \)

In the exercise provided, the measurements are taken with calipers. Each dimension has a known least count, which serves as the absolute error. By applying the percentage error formula to each dimension—length, breadth, and height—we can compute their individual errors:
These are then summed to obtain the maximum potential error in calculating the volume.
This method helps pinpoint weaknesses in precision and allows for better predictions of measurement inaccuracies.

Measurement Conversion

Conversion of units is a critical step when working with measurement data, especially when different parts of an experiment or calculation require specific units. In the exercise, measurements given in millimeters needed conversion to centimeters to match the precision which the calipers can effectively resolve.
When converting units:

• Remember the conversion factors, such as \(1 \text{ cm} = 10 \text{ mm}\).
• Apply these consistently across your measurements to maintain uniformity in your calculation.

Converting the dimensions from mm to cm helped in aligning the unit of measurement with the least count given in cm. This adjustment is crucial for accurate error calculation and to prevent discrepancies in outcomes. Measurement conversion ensures clarity and consistency in experimental procedures.

Calipers in Physics

Calipers are precise measurement tools used extensively in physics and engineering to gauge accurate dimensions of objects. The type of calipers referenced here uses a least count of 0.01 cm, indicating its precision level.

• Least Count: The smallest measurement that can be reliably read. In this case, it is 0.01 cm.
• Usage: They are handy for measuring internal and external dimensions and depths.

Understanding how to use calipers correctly is key to minimizing errors in measurements. Their precision plays an important role in reducing the uncertainty of experimental data. By knowing the limitations presented by the least count, one can optimize the reliability of measurements.
Calipers help ensure that the measured values are as close as possible to the true dimensions of an object, which is especially important in experimental physics where precise data leads to better understanding and analysis.