Lesson 1.3: Measurement, Errors and Uncertainty

Introduction

Welcome to Lesson 1.3 of Foundation Physics! In this lesson, we will explore the fundamental concepts of measurement and the associated errors and uncertainties that come with it. Measurement is a crucial part of science, and understanding the nuances of precision, accuracy, and uncertainty will help you accurately conduct experiments in physics.

Learning Objectives

By the end of this lesson, you should be able to:

• Distinguish between random and systematic errors, and understand the concepts of precision and accuracy.
• Calculate absolute, fractional, and percentage uncertainties in a single reading as well as in the mean of repeated readings.
• Combine uncertainties in various mathematical operations including sums, differences, products, quotients, and powers.
• Choose appropriate measuring instruments based on resolution and identify potential sources of error in your procedures.
• Report results with a number of significant figures that accurately reflects their uncertainty.

What are Errors?

Random Errors

Random errors are unpredictable variations in measurements. These can be caused by various factors such as environmental changes or measurement limitations. For example, if you measure the length of a piece of wood multiple times, a slight variation in your eye's position or the way you hold the measuring tape can lead to different readings.

Imagine you are measuring the height of a plant. One time, it may be recorded as 30.1 cm, and another time as 30.4 cm. These slight differences reflect random errors due to factors beyond your control.

Systematic Errors

Systematic errors, on the other hand, are consistent and reproducible inaccuracies that can be traced back to a flaw in the measurement system or technique. For instance, if your scale is improperly calibrated and consistently reads 0.5 cm more than the actual height, every measurement you take will have this systematic error.

To illustrate, if you were measuring the diameter of a marble and always get a result of 2.5 cm instead of the true value 2.0 cm due to a miscalibrated ruler, you are facing a systematic error. Understanding these errors is critical because they skew the results in one direction.

Precision vs. Accuracy

Now that you understand random and systematic errors, let's differentiate between precision and accuracy:

• Precision refers to how closely multiple measurements of the same quantity agree with each other. It’s about consistency.
• Accuracy indicates how close a measurement is to the true value. It’s about correctness.

For example, if you weigh the same apple multiple times and get 150 g, 149 g, and 151 g, your measurements are precise but might not be accurate if the true weight is 155 g.

Quantifying Uncertainty

Types of Uncertainty

Uncertainty in measurements can be classified into different forms:

1. Absolute Uncertainty: This is the uncertainty associated with a measurement. For example, if you measure the length of a table as 2.0 m ± 0.1 m, 0.1 m is the absolute uncertainty.
2. Fractional Uncertainty: This indicates how significant the uncertainty is relative to the size of the measurement. If the length is 2.0 m with an uncertainty of 0.1 m, the fractional uncertainty is given by:

$$\text{Fractional Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} = \frac{0.1\, \text{m}}{2.0\, \text{m}} = 0.05$$

1. Percentage Uncertainty: This expresses the uncertainty as a percentage of the measured value. It’s calculated by:

$$\text{Percentage Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} \times 100\%$$

In our example, it would be:

$$\text{Percentage Uncertainty} = 0.05 \times 100\% = 5\%$$

This means there is a 5% uncertainty in your measurement of the table length.

Mean of Repeated Readings

When taking multiple measurements, it's crucial to find the mean. The mean ($\bar{x}$) of repeated readings is calculated by:

$$\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}$$

Once we obtain the mean, we can propagate the uncertainties of these measurements. If we assume equal uncertainties for the repeated measurements, the combined uncertainty in the mean is:

$$\text{Uncertainty in Mean} = \frac{\text{Absolute Uncertainty}}{\sqrt{n}}$$

This helps us to refine our results by reducing the impact of random errors.

Combining Uncertainties

In Sums and Differences

When combining measurements in sums or differences, the absolute uncertainties add together. For example:

If you measure two lengths: $ L_1 = 5.0 \pm 0.1 \, \text{m} $ and $ L_2 = 3.0 \pm 0.2 \, \text{m} $, the total length (L) is given by:

$$L = L_1 + L_2 = (5.0 + 3.0) \pm (0.1 + 0.2) = 8.0 \pm 0.3\, \text{m}$$

In Products and Quotients

When measuring quantities multiplied or divided, the relative uncertainties add:

For example, if you’re calculating the area of a rectangle $ A = L \times W $ measured as $ L = 2.0 \pm 0.1\, \text{m} \, and \, W = 3.0 \pm 0.1\, \text{m} $:

1. Calculate the relative uncertainties:

• For length: $\frac{0.1}{2.0} = 0.05$
• For width: $\frac{0.1}{3.0} \approx 0.033$

1. Combine:

$$\text{Relative Uncertainty in Area} = 0.05 + 0.033 \approx 0.083$$

1. Finally, apply it to your area measurement:

$$A = 6.0 \pm (0.083 \times 6.0) \approx 6.0 \pm 0.5 \, \text{m}^2$$

Conclusion

In summary, understanding measurement errors and uncertainties is vital for any scientific endeavor. By recognizing the difference between random and systematic errors, and knowing how to calculate and combine uncertainties, you will be better equipped to report your findings accurately. As you dive deeper into physics, these skills will form the backbone of your experimental work. Remember, always consider the uncertainties when reporting measurements!

Study Notes

• Random errors are unpredictable while systematic errors are consistent.
• Precision relates to consistency of measurements, while accuracy relates to correctness.
• Types of uncertainty: Absolute, Fractional, and Percentage.
• Calculate mean of repeated readings to minimize random errors and find more accurate results.
• Combine uncertainties in sums by adding absolute uncertainties and in products/quotients by adding relative uncertainties.