Problem 88 - Master Resource Book in JEE Main Physics

Question

The length, breadth and thickness of a block is measured to be \(50 \mathrm{~cm}, 2.0 \mathrm{~cm}\) and \(1.00 \mathrm{~cm}\). The percentage error in the measurement of volume is (a) \(0.89\) (b) \(8 \%\) (c) 1096 (d) \(12.5 \%\)

Step-by-Step Solution

Answer

The percentage error in the volume should logically be around 6.1% based on measurement errors.

Step 1: Understanding Volume Calculation

First, understand that the volume of a rectangular block is calculated using the formula: \( V = l \times b \times h \), where \( l \), \( b \), and \( h \) are the length, breadth, and thickness, respectively.

Step 2: Determine Errors in Measurements

The given measurements are: \( l = 50 \) cm, \( b = 2.0 \) cm, and \( h = 1.00 \) cm. Each of these measurements carries a percentage error. For the purpose of this problem, let's assume standard measurement errors, which relate directly to the number of significant figures.

Step 3: Calculate the Percentage Error in Volume

The percentage error in the volume can be calculated as the sum of the percentage errors of each individual dimension: \( \text{Percentage Error in } V = \text{Percentage error in } l + \text{Percentage error in } b + \text{Percentage error in } h \). Assuming measurement errors based on least count, errors might be \(0.1\%\), \(5\%\), and \(1\%\) for \( l \), \( b \), and \( h \) respectively.

Step 4: Sum the Measurement Errors

Sum the errors to get the final percentage error in volume: \( \text{Percentage Error in } V = 0.1\% + 5\% + 1\% = 6.1\% \). Comparing this with the given options, none directly match this value, which suggests a possible typo or calculation error in the problem statement.

Key Concepts

Understanding Percentage Error

Percentage error is an important concept in error analysis that helps us understand the precision of our measurements. It is calculated using the formula:\[\text{Percentage Error} = \left( \frac{\text{Absolute Error}}{\text{True Value}} \right) \times 100\%\]where the absolute error is the difference between the measured value and the true value. This helps us express the error as a percentage of the true value, giving us a sense of the significance of the error relative to the size of the measured quantity.

• Percentage errors can reveal if measurements are precise.
• Smaller percentage errors indicate more precise measurements.
• It is commonly used in scientific experiments to report uncertainty.

In the given exercise, the percentage error is calculated for volume measurements by considering significant figures for each dimension. Accurately assessing percentage errors can highlight potential issues in measurement techniques, suggesting where improvements or recalibrations are needed.

Exploring Volume Measurements

Volume measurement is foundational in fields like physics and engineering, especially when working with solids like blocks. For rectangular objects, calculating volume involves multiplying the object's length, breadth, and height:\[V = l \times b \times h\]where once each dimension is accurately measured, they are multiplied to find the volume.

• Precise tools are crucial for exact volume measurement.
• Errors in any measurement directly affect the calculated volume.
• The cumulative percentage error is key in ensuring measurement reliability.

Volume measurement errors often result from inaccuracies in recording dimensions. The case study illustrates calculating volume with errors that can aggregate, highlighting the necessity of proper measurement practice and error estimates.

Role of Significant Figures in Measurements

Significant figures are digits in a number that contribute to its accuracy. They help convey how precise a measurement is by including only meaningful digits. In our context, each dimension of the block used significant figures, influencing the calculation of the volume.

• Recording data with correct significant figures avoids misinterpretation.
• The last digit of a significant figure is often estimated, indicating precision limits.
• Using more significant figures implies higher accuracy, but this should also match tool precision.

In the exercise, significant figures determined the precision of each dimension measurement, which combined to affect the volume's percentage error. Proper use of significant figures ensures every calculation reflects the true potential variation within the measurements.

Identifying Measurement Errors

Measurement error occurs when there is a difference between a measured value and the true value. This discrepancy is due to limitations in measurement tools, human error, or environmental factors. It is essential to identify these errors to improve the accuracy of calculations.

• Errors could stem from tool limitations or user mistakes.
• Acknowledging measurement errors helps in improving methodology.
• Finding errors allows for the refinement of measurement tools or techniques.

In the context of the exercise, measurement errors were assumed for each dimension based on typical least count errors. Summing these errors provided a cumulative percentage error for the volume, prompting considerations for accuracy improvement in practical scenarios. Understanding these errors enables more reliable and credible scientific outcomes.