Problem 99 - Master Resource Book in JEE Main Physics

Question

The length \(l\), breadth \(b\) and thickness \(t\) of a block are measured with the help of a metre scale. Given \(l=15.12 \pm 0.01 \mathrm{~cm}, \mathrm{~b}=10.15 \pm 0.01 \mathrm{~cm}, t=5.28 \pm 0.01 \mathrm{~cm}\) The percentage error in volume is (a) \(0.64 \%\) (b) \(0.28 \%\) (c) \(0.37 \%\) (d) \(0.489\)

Step-by-Step Solution

Answer

The percentage error in volume is approximately 0.37%, option (c).

Step 1: Write the formula for volume

The volume of a block with length \( l \), breadth \( b \), and thickness \( t \) is given by the formula \( V = l \, \times \, b \, \times \, t \).

Step 2: Calculate the volume

Substitute the given values to find the volume: \( V = 15.12 \times 10.15 \times 5.28 \). Calculating gives \( V = 809.38544 \text{ cm}^3 \).

Step 3: Write the formula for percentage error in volume

The percentage error in volume can be calculated using the formula: \ \[ \text{Percentage error in } V = \left( \frac{\Delta l}{l} + \frac{\Delta b}{b} + \frac{\Delta t}{t} \right) \times 100 \% \] \ where \( \Delta l, \Delta b, \Delta t \) are the absolute errors in length, breadth, and thickness, respectively.

Step 4: Calculate the absolute errors

Given \( \Delta l = 0.01\,\text{cm} \), \( \Delta b = 0.01\,\text{cm} \), \( \Delta t = 0.01\,\text{cm} \).

Step 5: Substitute absolute errors into the formula

Substitute the absolute errors and the values of \( l, b, t \) into the formula: \ \[ \text{Percentage error in } V = \left( \frac{0.01}{15.12} + \frac{0.01}{10.15} + \frac{0.01}{5.28} \right) \times 100 \% \]

Step 6: Calculate each percentage error term

Calculate each term separately: \ \( \frac{0.01}{15.12} = 0.000661 \ \frac{0.01}{10.15} = 0.000985 \ \frac{0.01}{5.28} = 0.001894 \)

Step 7: Sum the terms and calculate the total percentage error

Add the calculated terms and multiply by 100 to find the percentage error: \ \[ \text{Percentage error in } V = (0.000661 + 0.000985 + 0.001894) \times 100 \% \approx 0.354 \% \]

Step 8: Choose the nearest option

The calculated percentage error in the volume is approximately \(0.354 \%\), which is closest to option (c) \(0.37 \%\).

Key Concepts

Measurement Uncertainty

When measuring physical quantities like length, breadth, and thickness, precision is always a consideration. Despite using precise instruments, there is an inherent \'Measurement Uncertainty\' in every measurement we make. This is due to the limitations of the measuring tools and human error. - For example, using a metre scale, measuring a length may have an uncertainty of \( \pm 0.01 \text{ cm} \). - The plus-minus value indicates the range within which the true measurement lies. In simpler terms, it's like saying, "This is the measurement, give or take a little."Understanding measurement uncertainty is crucial because it affects how we interpret calculated values, such as the volume in our exercise. If your measurements have a high uncertainty, it leads to less confidence in the calculated results.

Error Propagation

Error propagation explains how measurement uncertainty in individual dimensions influences the overall calculation of a quantity like volume. Each of the measurements - length, breadth, and thickness - are measured with a certain error margin. These individual uncertainties blend together to affect the final calculated value. - For a function involving multiple variables, the rule for propagating errors is adding the relative (or fractional) errors of each measurement together. In mathematical terms: \[ \text{Percentage error in } V = \left( \frac{\Delta l}{l} + \frac{\Delta b}{b} + \frac{\Delta t}{t} \right) \times 100 \% \] Here, each \( \Delta \) represents the absolute error, while each letter without \( \Delta \) represents the measurement value.By applying these steps, we aggregate all contributing errors to find the overall uncertainty in the calculated volume, ensuring our final volume estimate accurately reflects potential measuring discrepancies.

Dimensional Analysis

Dimensional analysis is a method used to check the consistency and correctness of calculations involving physical quantities. It involves examining the dimensions associated with the physical quantities (like length, mass, time) in an equation.For calculating the volume of a block, the correct dimensions should reflect volume, which is essentially a \'cubed length\':- Volume = Length \( \times \) Breadth \( \times \) Thickness = \( L^3 \).By checking that each parameter aligns dimensionally, we guarantee that the resulting value makes sense for the context, limiting calculation or misinterpretation errors. In this example, given all measurements are in centimeters, once multiplied, the volume is in cubic centimeters. Dimensional analysis safeguards correct unit conversions and maintains consistency, letting you know the math boils down appropriately, ensuring the conclusions drawn are valid.