Problem 81 - Master Resource Book in JEE Main Physics

Question

The density of the material of a cube is measured by measuring its mass and length of its side. If the maximum errors in the measurement of mass and the length are \(3 \%\) and \(2 \%\) respectively, the maximum error in the measurement of density is (a) \(1 \%\) (b) \(5 \%\) (c) \(7 \%\) (d) \(9 \%\)

Step-by-Step Solution

Answer

The maximum error in the measurement of density is 9%.

Step 1: Understand the Problem

Density \( \rho \) is calculated as mass \( m \) divided by volume \( V \). For a cube, \( V = l^3 \), where \( l \) is the side length. Thus, \( \rho = \frac{m}{l^3} \). The problem involves finding the maximum error in density when there are measurement errors in mass and side length of the cube.

Step 2: Identify Given Errors

The maximum error in the measurement of mass is \(3\%\). The maximum error in the measurement of the side length \(l\) is \(2\%\). These errors will affect the density calculation.

Step 3: Determine Error Propagation in Density

Density's error is determined from the formula: \( \rho = \frac{m}{l^3} \). To find the error in \( \rho \), we differentiate it: \( \frac{d\rho}{\rho} = \frac{dm}{m} - 3 \frac{dl}{l} \). Here, the error in mass contributes directly, while the error in length affects the volume (thus multiplied by 3).

Step 4: Calculate Maximum Error in Density

Insert the given percentage errors: \( \frac{d\rho}{\rho} = 3\% - 3 \times 2\% \). This results in \( 3\% - 6\% = -3\% \). The negative sign indicates the density error reduces; however, actual errors are always positive, thus: \( 3\% + 6\% = 9\% \) because the subtraction error term increases the overall error.

Step 5: Conclude the Solution

The maximum error in the measurement of density is \(9\%\). Therefore, the correct answer is option \( (d)\ 9\% \).

Key Concepts

Measurement Error

Measurement error refers to the inaccuracies or deviations that occur when determining a quantity. These errors can result from limitations of the measuring device or human error. For example, in our cube, we have two separate measurements - mass and side length of the cube. These measurements carry their own potential for error. When measuring mass, any fluctuation in the weighing scale's reading can lead to a measurement error up to 3%. Similarly, if the tool used to measure the cube’s side length is slightly off, it can lead to a 2% error in length. It is crucial to account for these errors in scientific calculations. By understanding these potential errors, we ensure that our final result, like the calculation of density, is as accurate as possible.

Density Calculation

Calculating density involves dividing mass by volume. Specifically for a cube, its volume is determined by raising the length of its side to the third power:

• Density formula: \( \rho = \frac{m}{V} \)
• Cube volume formula: \( V = l^3 \)

Thus, the density \( \rho \) becomes \( \rho = \frac{m}{l^3} \). Given these equations, we see how errors in measurement can propagate. The mass and length measurements directly influence the outcome. In practical experiments, it is important to measure both properties accurately, especially since the density formula's dependence on volume means even small errors in length can have a large effect on the calculated density.

Error Propagation

Error propagation is the process by which errors in measurements of quantities contribute to uncertainty in a calculated result. In our density calculation, each measurement error - that of the mass and the side length - affects the final outcome differently. To calculate the propagated error in density, we consider how these individual errors mathematically combine:

• We start with relative errors: \( \frac{d\rho}{\rho} = \frac{dm}{m} - 3 \frac{dl}{l} \)
• The mass's error contributes directly to the density error.
• The length's error, affecting volume, multiplies by 3 because volume depends on the cube of the length.

When we plug in the percentage errors (3% for mass and 2% for length), we calculate that these contribute to a 9% maximum overall error in density. Even when errors appear to be small percentages, their influence is magnified in calculations like these. By understanding error propagation, we better understand how potential inaccuracies in measurement affect science and engineering calculations.