Question

The accepted length of a steel pipe is 5.5 m. Calculate the percent error for each of these measurements a. 5.2 \(\mathrm{m} \quad\) b. 5.5 \(\mathrm{m} \quad\) c. 5.7 \(\mathrm{m} \quad\) d. 5.1 \(\mathrm{m}\)

Short answer

The percent error for each measurement is: a. 5.45% b. 0% c. 3.64% d. 7.27%

Step-by-step solution

Identify the Given and Calculate Percent Error#a#

The accepted length of the steel pipe is 5.5 m, and the measured length is 5.2 m. Percent Error = \( \frac{|5.2 - 5.5|}{5.5} \) * 100 Percent Error = \( \frac{0.3}{5.5} \) * 100 = 5.45%

Identify the Given and Calculate Percent Error#b#

The accepted length of the steel pipe is 5.5 m, and the measured length is 5.5 m. Percent Error = \( \frac{|5.5 - 5.5|}{5.5} \) * 100 Percent Error = \( \frac{0}{5.5} \) * 100 = 0%

Identify the Given and Calculate Percent Error#c#

The accepted length of the steel pipe is 5.5 m, and the measured length is 5.7 m. Percent Error = \( \frac{|5.7 - 5.5|}{5.5} \) * 100 Percent Error = \( \frac{0.2}{5.5} \) * 100 = 3.64%

Identify the Given and Calculate Percent Error#d#

The accepted length of the steel pipe is 5.5 m, and the measured length is 5.1 m. Percent Error = \( \frac{|5.1 - 5.5|}{5.5} \) * 100 Percent Error = \( \frac{0.4}{5.5} \) * 100 = 7.27% The percent error for each measurement is as follows: a. 5.45% b. 0% c. 3.64% d. 7.27%

Key concepts

Measurement Accuracy

Understanding measurement accuracy is essential in various fields such as science, engineering, and even daily routine tasks. It refers to the closeness of a given measurement to the true or accepted value. In simple terms, it's an indicator of how 'correct' your measurement is. When you measure the length of an object, say a steel pipe with an accepted length of 5.5 meters, and your measurement reads 5.2 meters, this shows a deviation from the accurate value.

To enhance measurement accuracy, it's important to use properly calibrated instruments and adhere to standardized measurement procedures. In the context of education, being diligent about measurement practices helps students understand the importance of accuracy and reduces the likelihood of significant errors in their experiments or calculations.

Error Analysis

When dealing with measurements, error analysis is a systematic way of evaluating the uncertainty in the measurement process. It involves quantifying the difference between measured values and the accepted or true value, known as error. For instance, in our steel pipe example with various measurements, calculating the percent error demonstrates a simplified form of error analysis.

The percent error calculation is given by the formula:

Percent Error = \( \frac{|measured \, value - accepted \, value|}{accepted \, value} \) * 100

By using this formula, students can understand the scope of their measurement's inaccuracy. For example, if a student measures the pipe as 5.7 meters instead of the accepted 5.5 meters, the percent error would be approximately 3.64%, indicating a relatively small, but still present, error.

Significant Figures

When reporting measurements, significant figures express the precision of those measurements. These figures include all the digits that are known reliably, plus one last digit which is somewhat uncertain. In the context of the steel pipe with the accepted length of 5.5 meters, all the reported measurements (5.2 m, 5.5 m, 5.7 m, 5.1 m) have two significant figures as both digits are meaningful in the context of the measurement's precision.

Significant figures are essential in error analysis because they reflect the limitations in measurement precision. When students learn to use significant figures, they develop a better understanding of the limitations of their tools and techniques. For example, the more significant figures you have in a measurement, the higher precision you imply, and this should match the capability of the measuring instrument used. If students provide measurements with too many significant figures, they might be implying a false sense of precision.