Problem 10 - Master Resource Book in JEE Main Physics

Question

The mean length of an object is \(5 \mathrm{~cm}\). Which is the following measurements is most accurate? [NCERT Exemplar] (a) \(4.9 \mathrm{~cm}\) (b) \(4.805 \mathrm{~cm} \quad\) (c) \(5.25 \mathrm{~cm}\) (d) \(5.4 \mathrm{~cm}\)

Step-by-Step Solution

Answer

The most accurate measurement is 4.9 cm.

Step 1: Understand the Mean

The mean length of the object is given as \(5\, \mathrm{cm}\). This means the average length that we should compare other measurements to. Measurements close to 5 cm are considered more accurate.

Step 2: Compare Measurements with Mean

We have four measurements: \(4.9 \, \mathrm{cm}\), \(4.805 \, \mathrm{cm}\), \(5.25 \, \mathrm{cm}\), and \(5.4 \, \mathrm{cm}\). Compare each to the mean (5 cm) to find which one is closest.

Step 3: Calculate Differences

Find the absolute difference of each measurement from \(5\, \mathrm{cm}\): \( |5 - 4.9| = 0.1 \, \mathrm{cm}\)\( |5 - 4.805| = 0.195 \, \mathrm{cm}\)\( |5 - 5.25| = 0.25 \, \mathrm{cm}\)\( |5 - 5.4| = 0.4 \, \mathrm{cm}\)

Step 4: Determine Most Accurate Measurement

The most accurate measurement is the one with the smallest difference from \(5 \, \mathrm{cm}\). The difference for \(4.9 \, \mathrm{cm}\) is \(0.1 \, \mathrm{cm}\), which is smaller than the differences for the other measurements.

Key Concepts

Understanding Mean Value

When we talk about the "mean value," we are referring to the average number that represents a set of numbers. It is the central value around which all other numbers in a data set are distributed. Therefore, when the mean length of an object is given as 5 cm, this is the average length we use to gauge accuracy against. To understand how closely a specific measurement aligns with the mean, we look at how close or far the measurement falls from this average. For instance, in our example, any measurement that is very close to 5 cm is considered more accurate. Knowing this helps us determine which measurements are more reliable or trustworthy. The mean is crucial in summarizing data because it provides a simple point of comparison for other data points. This comparison helps to understand whether something is typical or outlier when placed against the average value.

The Importance of Absolute Difference

The term "absolute difference" refers to the non-negative difference between two numbers. It effectively tells us how far apart these two numbers are, direction notwithstanding. Calculating absolute difference involves subtracting one value from another and considering the positive value of the result. For example, to measure the absolute difference between two lengths, like 5 cm and 4.9 cm, you'd perform the operation \(|5 - 4.9|\) = 0.1 cm.The smaller the absolute difference, the closer the measurement is to the mean, indicating higher accuracy. Path - It emphasizes the measurement's deviation from the average.- It helps easily identify anomalies or outliers in data sets.- It quantifies proximity, aiding in precision and reliability assessments.Understanding absolute differences ensures more precise data validation when compared against a standard or a mean value.

Evaluating Measurement Comparison

To accurately evaluate measurement accuracy, we must compare each measurement to a standard, usually the mean value. This is done using techniques such as calculating absolute differences, as outlined earlier. When comparing measurement values: - Identify the mean value, like the 5 cm example we've seen. - Calculate the absolute difference of each measurement from this mean. - The measurement with the smallest absolute difference is considered the most accurate. When we compared measurements of lengths, we found that 4.9 cm had the smallest absolute difference—0.1 cm—from the mean, making it the most accurate of the options. By systematically comparing measurements, one can effectively determine the reliability and precision of data inputs, enhancing overall result integrity. Such comparisons are fundamental in fields needing precise measurements, like engineering and scientific research.