Problem 83 - Master Resource Book in JEE Main Physics

Question

In an experiment, we measure quantities \(a, b\) and \(c\). Then \(x\) is calculated from the formula, \(x=\frac{a b^{2}}{c^{3}}\). The percentage errors in \(a, b, c\) are \(\pm 1 \%, \pm 3 \%\) and \(\pm 2 \%\) respectively. The percentage error in \(x\) can be (a) \(\pm 10\) (b) \(\pm 4 \%\) (c) \(7 \%\) (d) \(\pm 139\)

Step-by-Step Solution

Answer

The error in \(x\) is \(13\%\), which matches none of the given options.

Step 1: Identify the Formula

We start with the formula given for calculating \(x\), which is \(x = \frac{a b^{2}}{c^{3}}\). This equation computes \(x\) based on the measured quantities \(a\), \(b\), and \(c\).

Step 2: Determine the Rules for Error Propagation

The percentage error in a product or quotient is the sum of the percentage errors of the quantities being multiplied or divided. Thus, we need to apply error propagation rules: \( \Delta x = \Delta a + 2\Delta b + 3\Delta c \), where \(\Delta x\) represents the percentage error in \(x\).

Step 3: Apply Given Percentage Errors

Substitute the given percentage errors into the error propagation formula: 1% for \(a\), 3% for \(b\), and 2% for \(c\). Calculating this gives: \( \Delta x = 1\% + 2(3\%) + 3(2\%) \).

Step 4: Perform the Calculation

Calculate each term in the error propagation equation. First, \(2(3\%) = 6\%\), and \(3(2\%) = 6\%\). Then, add these to the 1% error from \(a\): \( \Delta x = 1\% + 6\% + 6\% = 13\% \).

Step 5: Match the Calculated Error with the Given Options

The calculated percentage error for \(x\) is \(13\%\). Compare this result with the provided options to find the correct answer.

Key Concepts

Understanding Percentage Error

Percentage error is a way to express the uncertainty in a measurement relative to the size of the measurement itself. This concept helps us understand how close a measured value is to the true value. Imagine you have measured a length as 100 cm but believe the true value is 102 cm. The percentage error can be calculated using the formula: \[ \text{Percentage Error} = \left( \frac{\text{Measured Value - True Value}}{\text{True Value}} \right) \times 100\% \] In this case, the percentage error is 1.96%. This not only tells us the error but also puts it in context, making it easier to compare errors across different scales or units. Understanding percentage error is crucial when dealing with multiple measurements and their propagation in experiments, helping us quantify the accuracy and precision of results.

Exploring Error in Measurements

Error in measurements often arises due to limitations in measuring instruments or human error. There are generally two types of errors: systematic errors and random errors.

• **Systematic Errors**: These errors occur consistently in the same direction, such as a scale that is always off by 5 grams.
• **Random Errors**: These occur unpredictably and can vary in magnitude and direction, commonly caused by environmental factors or observer variability.

Reducing errors in measurements is important for increasing the reliability of experimental results. Calibration of instruments and repeating measurements help in minimizing systematic and random errors respectively. In experiments, like the one described in the original exercise, knowing the error percentage allows for a better understanding of the confidence in measured data.

Calculation of Errors in Complex Equations

When dealing with complex equations involving multiple variables, like the formula for calculating \(x\), error propagation is necessary to understand the uncertainty in the result. This involves combining the errors from the individual measurements.The rule for propagating errors in multiplication or division states that the percentage error in the result is the sum of percentage errors of the individual factors. In the case of the expression \(x = \frac{a b^{2}}{c^{3}}\), the percentage error in \(x\) is determined by the equation:\[ \Delta x = \Delta a + 2\Delta b + 3\Delta c \]This creates a comprehensive understanding of how the errors in \(a\), \(b\), and \(c\) affect \(x\), allowing for a precise calculation of uncertainty.By substituting the percentage errors: 1% for \(a\), 3% for \(b\), and 2% for \(c\), and performing the arithmetic as detailed in the step-by-step solution, we obtain a total percentage error of 13% in \(x\). This method of calculation is essential for accurate scientific and engineering endeavors where precise data is critical.