Problem 5 - Physics for Scientists and Engineers with Modern Physics

Question

(II) What is the percent uncertainty in the measurement 5.48\(\pm 0.25 \mathrm{m} ?\)

Step-by-Step Solution

Answer

The percent uncertainty is approximately 4.56%.

Step 1: Understand the Question

We need to find the percent uncertainty for a measurement given as 5.48 meters with an uncertainty of ±0.25 meters. Percent uncertainty is calculated by dividing the absolute uncertainty by the measured value and then multiplying by 100 to get a percentage.

Step 2: Identify Given Values

From the measurement, the given values are: - Measured value: 5.48 meters - Absolute uncertainty: 0.25 meters

Step 3: Use the Percent Uncertainty Formula

The percent uncertainty can be calculated using the formula:\[ \text{Percent Uncertainty} = \left(\frac{\text{Absolute Uncertainty}}{\text{Measured Value}}\right) \times 100 \]

Step 4: Substitute Values into the Formula

Substitute the given values into the formula:\[ \text{Percent Uncertainty} = \left(\frac{0.25}{5.48}\right) \times 100 \]

Step 5: Calculate the Percentage

Perform the division and multiplication:\[ \text{Percent Uncertainty} = \left(\frac{0.25}{5.48}\right) \times 100 \approx 4.56\% \]

Key Concepts

Absolute Uncertainty

Absolute uncertainty is a fundamental concept that quantifies the doubt or range of values within which the true value of a measurement likely lies. It is usually represented by a "+/-" sign. For instance, in the measurement 5.48±0.25 meters, the "+/-0.25 meters" indicates the absolute uncertainty. This means the actual length could reasonably be as low as 5.48 - 0.25 meters or as high as 5.48 + 0.25 meters.
Understanding absolute uncertainty helps identify the range of possible errors in a measurement without considering its relative size compared to the measured value. This is crucial in experiments where precision and accuracy are vital.
Some characteristics of absolute uncertainty include:

• It is expressed in the same units as the measurement.
• It does not show how significant the uncertainty is in comparison to the measurement size.

Measured Value

The measured value is the actual number or quantity obtained from a measurement process. In our example, the measured value is 5.48 meters. This is the central number around which the uncertainty is calculated.
The importance of the measured value is substantial as it forms the basis of all subsequent calculations. When dealing with measured values, it is essential to:

• Be as accurate as possible during the measurement process.
• Consider any factors that might affect the reading, such as equipment precision or external conditions.

In physics and other sciences, the measured value is often accompanied by an uncertainty, which provides more context and reliability to the measurement itself.

Measurement in Physics

Measurements in physics form the core foundation upon which scientific data and conclusions rest. Every measurement is an attempt to quantify some aspect of the universe accurately. However, due to limitations of instruments and external factors, no measurement can be exact, hence the need for uncertainty.
In the realm of physics, understanding the role of measurements includes:

• Recognizing the importance of precise and accurate data collection.
• Making sense of uncertainty in the context of experimental accuracy and reliability.

Proper measurement techniques can significantly affect the quality and reliability of scientific experiments. It's why physicists spend considerable time evaluating and refining their methods. This deliberate approach ensures their measurements reflect the closest approximation to reality possible, given the constraints and conditions.