Question

The percentage error in the distance \(100 \pm 5 \mathrm{~cm}\) is ... (a) \(5 \%\) (b) \(6 \%\) (c) \(8 \%\) (d) \(20 \%\)

Short answer

The percentage error in the distance \(100 \pm 5 \mathrm{~cm}\) is (a) \(5\%\).

Step-by-step solution

Identify the Error and Measured Value

The given distance is \(100 \pm 5\mathrm{~cm}\). Hence, the measured value is \(100\mathrm{~cm}\) and the error is \(5\mathrm{~cm}\).

Calculate Percentage Error

Using the formula, we can calculate the percentage error as follows: \[Percentage \ Error = \frac{Error}{Measured\ Value} \times 100\%\] Plug in the values of Error and Measured Value: \[Percentage \ Error = \frac{5\mathrm{~cm}}{100\mathrm{~cm}}\times 100\% \]

Simplify the expression

Now, simplify the expression: \[Percentage \ Error = \frac{5}{100}\times 100\%\] \[Percentage \ Error = 0.05\times 100\%\] \[Percentage \ Error = 5\%\] The percentage error is \(5\%\). Therefore, the correct answer is (a) \(5\%\).

Key concepts

Error Analysis

Error analysis in measurements is a crucial aspect of experimental physics and other sciences. It helps us to understand the accuracy and precision of our measurements.
When we measure something, there is always a small difference between the measured value and the true or accepted value. This difference is known as an error.
In error analysis, we quantify how large this error is compared to the true value. This process involves breaking down the error to understand its origin, such as systematic errors (due to flaws in measurement tools) or random errors (due to unpredictable fluctuations).
The calculation of percentage error is a part of this process, providing a straightforward way to express error magnitude in relation to the measured value. It is calculated using the formula:

• Percentage Error = \(\frac{\text{Error}}{\text{Measured Value}} \times 100\%\)

This formula allows scientists and engineers to express how significant an error is as a portion of the whole measurement. It provides a way to compare the accuracy of different measurements or instruments.

Measured Value

The measured value is the actual number or quantity determined from an experiment or observation. In the example provided, the measured value is given as 100 cm. This value serves as a benchmark for calculating any errors present in the measurement process.
When determining the measured value, it is essential to ensure that the instruments used are calibrated and functioning correctly. A well-calibrated instrument minimizes systematic errors and helps provide a precise measurement.
The importance of the measured value cannot be overstated as it is the foundation for any subsequent error analysis. In mathematical terms, the measured value sits at the denominator of the percentage error formula:

• Percentage Error = \(\frac{\text{Error}}{\text{Measured Value}} \times 100\%\)

By accurately determining the measured value, one can ensure that the computed percentage error reflects the true accuracy of the measurement process.

Uncertainty in Physics

Uncertainty in physics is an expression of the confidence in our measurements. Every time we measure something, there's an inherent uncertainty. In our example, the distance is expressed as \(100 \pm 5\) cm. The \(\pm 5\) denotes uncertainty and indicates that the actual measurement could be 5 cm more or less than the stated 100 cm.
This range of uncertainty is important. It helps to define the limits within which we can expect the true value to lie. Understanding this concept is central to scientific measurements, as all measurements have some level of uncertainty. The value we assign to this uncertainty often depends on the precision of our tools and the methods employed in measurement.
When measuring processes or experiments, it is crucial to note this uncertainty, as it provides essential context on the reliability and precision of an experiment. Measurement uncertainty helps scientists, engineers, and statisticians to assess the quality of data and evaluate the need for more precise instruments or refined methods.