Problem 98 - Master Resource Book in JEE Main Physics

Question

The relative density of the material of a body is the ratio of its weight in air and the loss of its weight in water. By using a spring balance, the weight of the body in air is measured to be \(5.00 \pm 0.05 \mathrm{~N}\). The weight of the body in water is measured to be \(4.00 \pm 0.05 \mathrm{~N}\). Then, the maximum possible percentage error in relative density is (a) \(11 \%\) (b) 1096 (c) \(9 \%_{6}\) (d) \(7 \%\)

Step-by-Step Solution

Answer

The maximum possible percentage error in relative density is 11\%.

Step 1: Understanding the Problem

The problem provides the weights of a body in air and in water, and you need to find the maximum possible percentage error in the relative density of the material. We have the weights: in air, it is \(5.00 \pm 0.05 \mathrm{~N}\), and in water, it is \(4.00 \pm 0.05 \mathrm{~N}\).

Step 2: Calculate Relative Density Formula

Relative density is given by the formula: \( \text{Relative Density} = \frac{W_{\text{air}}}{W_{\text{air}} - W_{\text{water}}} \). Here, \( W_{\text{air}} = 5.00 \mathrm{~N} \) and \( W_{\text{water}} = 4.00 \mathrm{~N} \). Thus, \( \text{Relative Density} = \frac{5.00}{5.00 - 4.00} = 5.00 \).

Step 3: Calculate Error in Weights

The potential errors in the measurements of weight are given as \( \pm 0.05 \mathrm{~N} \) for both air and water. These absolute errors will affect the calculation of relative density.

Step 4: Find the Maximum Error in Relative Density

To find the maximum error in relative density, consider the errors in both the numerator and the denominator: - Error in numerator \( = \pm 0.05 \mathrm{~N} \). - Error in denominator \( = \pm (0.05 + 0.05) = \pm 0.10 \mathrm{~N} \) since it's a subtraction operation.Thus, the maximum error in relative density: \( \frac{\pm 0.05}{5.00} + \frac{\pm 0.10}{1.00} \) leading to a relative error of \(0.01 + 0.10 = 0.11\).

Step 5: Convert to Percentage Error

The final step is to convert the fractional error into a percentage: \[ \text{Percentage Error} = 0.11 \times 100\% = 11\% \]. Thus, the maximum percentage error is 11\%.

Key Concepts

Error Analysis in Physics

Error analysis is a crucial part of any physics experiment, helping us understand the accuracy and reliability of our results. When measuring any physical quantity, errors can occur due to various reasons such as instrument precision, human error, or external factors. To quantify these errors, scientists usually express them as a range, such as "+/-" values, indicating the possible deviation from the true value.
In the context of relative density, error analysis becomes important because calculating this property involves multiple measurement steps. Each step has its own potential for error, and these errors can compound and affect the final result. Understanding these errors helps in refining methods, improving measurements, and ensuring experiments yield reliable data.
When calculating relative density, errors in weight measurement—like those obtained using a spring balance—are crucial aspects to consider as they directly affect the relative density’s calculation and its error estimation.

Calculation of Relative Density

Relative density, or specific gravity, is a dimensionless quantity expressing the ratio of a substance's density compared to a reference substance, typically water. This concept is important because it allows for the comparison of the densities of different substances without needing conversion to a common unit.
The formula for calculating relative density using weights is:

• \[ \text{Relative Density} = \frac{W_{\text{air}}}{W_{\text{air}} - W_{\text{water}}} \]

In this formula, \(W_{\text{air}}\) is the weight of the object in air, while \(W_{\text{water}}\) is the weight of the object when submerged in water. The numerator provides the total weight of the object in air, and the denominator reflects the buoyant force experienced by the object in water, which corresponds to the loss of weight.
Relative density is vital in applications ranging from engineering to material science, as it helps determine the material's properties and suitability for specific uses.

Use of Spring Balance

A spring balance is an essential tool in physics for measuring weight by the tension of a spring. It's based on Hooke's Law, which suggests that the force needed to extend or compress a spring is proportional to the distance it is stretched.
Using a spring balance to measure weights is simple and effective, but it comes with certain considerations:

• Ensure the spring is not overstretched, as this could damage the spring and lead to incorrect readings.
• Always check for zero error before using the balance. This is when the scale doesn't read 'zero' when nothing is hanging from it.

For determining relative density, an accurate spring balance helps in obtaining the weights necessary for calculation. It provides the weight reading in both air and water, making it easier to proceed with determining relative density with reliable accuracy.

Weight Measurement Errors

Errors in weight measurement can significantly affect the outcome of calculating physical properties like relative density. In the spring balance measurement example, errors might arise due to several factors:

• Calibration errors, where the spring balance is not correctly aligned with standard measures.
• Parallax error, which occurs when reading the measurements from the wrong angle.
• External factors like temperature changes, which can alter the spring balance behavior.

Each weight measurement comes with a given absolute error, which must be considered during calculations. For instance, if a weight is measured as \(5.00 \pm 0.05 \mathrm{~N}\), the \(\pm 0.05 \mathrm{~N}\) denotes the potential error, indicating the range within which the true value likely lies.
Understanding these errors allows for proper error propagation to be calculated, ensuring accuracy in more complex measurements and calculations.