Problem 59 - Introductory Chemistry Atoms First

Question

Two people attempt to measure the length in feet of a parking lot they know to be about \(100 \mathrm{ft}\) long. One person uses a 6 -in. ruler; the other uses a 120 -ft tape measure. If both measuring devices are graduated in \(1 / 16\) in., which person is likely to make the more accurate measurement? Explain.

Step-by-Step Solution

Answer

The person using the 120-ft tape measure is likely to make a more accurate measurement due to the single-use application, reducing alignment errors and counting errors that may occur when using the 6-inch ruler.

Step 1: Convert the length of parking lot to inches

Since both measuring devices are graduated in 1/16 inches, we need to convert the length of the parking lot to inches for better comparison. We know that there are 12 inches in 1 foot, so to convert 100 feet to inches, we'll multiply it by 12. \(100\,\mathrm{ft} \times 12\,\mathrm{in./ft} = 1200\,\mathrm{in.}\) The length of the parking lot is 1200 inches.

Step 2: Calculate the number of graduations needed for each device

Now we need to find how many graduations each person will need to use their device to measure the entire parking lot. For the person using the 6-inch ruler: Number of graduations = Length of parking lot / Graduation size \( \frac{1200\,\mathrm{in}}{ (1/16)\,\mathrm{in}} = 19200\,\text{graduations} \) For the person using the 120-ft tape measure: Number of graduations = Length of parking lot / Graduation size \( \frac{1200\,\mathrm{in}}{(1/16)\,\mathrm{in}} = 19200\,\text{graduations} \)

Step 3: Analyze the results

Both measuring devices have the same number of graduations required to measure the length of the parking lot. However, the 6-inch ruler needs to be moved and aligned several times (200 times) over the entire parking lot, while the 120-ft tape measure could measure the whole length in a single use. Moving and aligning the ruler can introduce errors into the measurement process, such as misalignment or counting errors.

Step 4: Conclusion

The person using the 120-ft tape measure is likely to make a more accurate measurement due to the single-use application, reducing alignment errors and counting errors that may occur when using the 6-inch ruler.

Key Concepts

Unit Conversion

When working with measurements, it's crucial to operate within the same unit system to maintain consistency and accuracy. In our textbook example, converting the length of a parking lot from feet to inches was necessary to accurately compare the efficacy of measuring devices that both used inches in their smallest graduation.

Unit conversion is simply the process of changing a measure from one unit to another without altering its value. This is often required when different measurement systems are used, or when precision needs are varied. For instance, converting between feet and inches, as in the example, involves multiplying by a conversion factor (12 inches per foot). Using the correct conversion factor is essential for the correct result, and this often requires a strong understanding of the relationship between different units.

To ensure the successful application of unit conversion, remember these steps:

• Identify the units you're converting from and to.
• Find the correct conversion factor.
• Multiply the original measurement by the conversion factor.

Significant Figures

Significant figures (often abbreviated as sig-figs) hold great importance in the realm of measurement accuracy. They represent all the known digits in a measurement plus the first uncertain one. In the exercise, the measuring devices are graduated in 1/16 of an inch, which determines the level of precision.

The number of significant figures is directly tied to the precision of a measurement. The more significant figures in a measure, the more precise it is. However, extra significant figures do not necessarily mean better accuracy; they only mean that the value is more precisely stated. For example, if you have a length of 12.14 inches, there are four significant figures, indicating a relatively high precision.

Here's how to properly use significant figures:

• Only the digits from the first non-zero number to the last non-zero number are counted as significant.
• Any zeroes between these numbers are also significant.
• Leading zeros are not counted as significant figures.
• The exact number of gradations or divisions on the measuring device will determine their significance in a calculation.

Measurement Error

Understanding measurement error is critical in determining the accuracy and validity of measurement results. Errors in measurement can come from various sources, including the device used, the person taking the measurement, and environmental factors.

In our textbook exercise, errors can arise from moving and aligning the 6-inch ruler multiple times, which could result in misalignment and counting errors. Conversely, the 120-ft tape measure avoids such errors by spanning the whole length of the parking lot without requiring realignment. Typically, measurement errors are classified as either systematic errors, which are consistent and predictable, or random errors, which occur without pattern.

To mitigate measurement error:

• Use appropriate tools that match the scale of what's being measured.
• Minimize human error by using devices that require fewer manual adjustments.
• Understand and follow the best practices for the measuring tool in use.
• Record measurements carefully, noting down the conditions that might affect them.

Moreover, the reliability of measurement can also be enhanced by calibrating measuring instruments regularly and employing proper measurement techniques.