Problem 8 - Applied Physics

Question

Determine the precision of each measurement. \(30 \overline{0} 0 \mathrm{ft}\)

Step-by-Step Solution

Answer

The measurement precision is to three significant figures.

Step 1: Understand the Concept of Precision

Precision in measurement refers to the degree of exactness or refinement in a measurement. When a measurement is presented with trailing zeros that are not needed to indicate the magnitude, these zeros indicate the precision of the measurement.

Step 2: Identify Significant Figures in the Measurement

In the measurement \(30 \overline{0} 0 \mathrm{ft}\), the bar over the last zero indicates that it is significant, which means we need to consider all digits including the zeros as significant figures.

Step 3: Count the Significant Figures

Since the measurement is \(30 \overline{0} 0 \mathrm{ft}\), you count every digit including the trailing zero with the overline. This results in three significant figures: 3, 0, and the zero with an overline.

Step 4: State the Measurement Precision

The precision of the measurement is determined by the number of significant figures. Here, the significant figures are three, which means the measurement is precise to three significant figures.

Key Concepts

Significant Figures

Significant figures are crucial in understanding the precision of a measurement. They include all the digits in a number that contribute to its accuracy. This encompasses non-zero digits, zeros between non-zero digits, and trailing zeros in a decimal number when they are applicable to the measurement's precision.
Significant figures help communicate the precision of a number that can impact calculations and interpretations. To identify significant figures in a number:

• Start with the first non-zero digit, counting all digits until the last one.
• Include zeros between non-zero digits.
• Consider zeros at the end of a number as significant only if they come after a decimal point or are marked, such as with an overline.

Recognizing significant figures helps evaluate which portions of a given measurement are precise and should be considered during calculations. This understanding is essential for conveying exactness.

Exactness in Measurement

The concept of exactness in measurement refers to the degree of refinement in a numerical value. It is closely related to significant figures and highlights the reliability and consistency of a measurement presented in numerical form.
Exactness in measurement can vary depending on the context and the instrument used for measurement.

Why Exactness Matters

In practice, the exactness of a measurement affects the credibility of calculations and results derived from it. Measurement tools can only measure up to a certain level of exactness, and this is why it's important to know the precision required in calculations. Consider a ruler with millimeters as its smallest unit. Measurements taken with such a ruler can't be more exact than the closest millimeter. Understanding these limitations allows for better use of measurement results. Identifying the amount of significant figures and understanding how they represent exactness is essential for presenting and analyzing data more appropriately.

Trailing Zeros in Measurements

Trailing zeros are zeros that come after either a decimal point or another non-zero digit. In measurements, these trailing zeros can offer critical insights into the measurement's accuracy and precision.
These zeros can change the interpretation of a measurement significantly.

Trailing Zero Types and their Implications

• Non-significant Trailing Zeros: When a number does not include a decimal point, trailing zeros might not count as significant. For example, in "1500", the zeros are not significant unless specified otherwise.
• Significant Trailing Zeros: When zeros follow a decimal point or are marked as important, via techniques like a bar or underline, they indicate that these zeros are part of the number's precision. This suggests an intentional level of detail in measurement that shouldn't be disregarded.

Recognizing trailing zeros and their significance is key to proper data interpretation. They delineate the intended precision and help maintain the integrity of the measurement's representation.