In the study of physics, measurements and uncertainties are fundamental concepts that help us understand the precision and accuracy of experimental data. The International Baccalaureate (IB) syllabus emphasizes these concepts to ensure that students can critically evaluate the reliability of their experimental results. This study note will cover various aspects of measurements and uncertainties, including types of uncertainty, random and systematic errors, calculating uncertainties, and representing uncertainties on graphs.
Absolute uncertainty is the uncertainty given as a fixed quantity. For example, if a measurement is taken as $5.0 \pm 0.1$ cm, the absolute uncertainty is $0.1$ cm.
Fractional uncertainty is the uncertainty expressed as a fraction of the measurement. It is calculated by dividing the absolute uncertainty by the measured value: $$ \text{Fractional Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} $$ For example, if a measurement is $5.0 \pm 0.1$ cm, the fractional uncertainty is $\frac{0.1}{5.0} = 0.02$.
Percentage uncertainty expresses the uncertainty as a percentage of the measurement: $$ \text{Percentage Uncertainty} = \left( \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} \right) \times 100% $$ Using the previous example, the percentage uncertainty is $\left( \frac{0.1}{5.0} \right) \times 100% = 2%$.
Example:
If a balance gives a reading of $952 \pm 2$ g for a box with a true mass of $950$ g, the absolute uncertainty is $2$ g, the fractional uncertainty is $\frac{2}{952} \approx 0.0021$, and the percentage uncertainty is $0.21%$.
Random errors cause unpredictable fluctuations in measurements. These errors affect the precision of measurements, leading to a wider spread of results around the mean value.
Systematic errors occur due to faulty instruments or flawed experimental methods. These errors affect the accuracy of measurements, causing a consistent deviation from the true value.
Common Mistake:
Confusing random errors with systematic errors. Random errors affect precision, while systematic errors affect accuracy.
For analogue instruments, the uncertainty is typically taken as half the smallest division. For digital instruments, it is the smallest division.
Example:
For a thermometer with a smallest division of $1^\circ$C, the uncertainty is $\pm 0.5^\circ$C. For a digital balance reading to $0.01$ g, the uncertainty is $\pm 0.01$ g.
The uncertainty is given by half the range of the measurements: $$ \text{Uncertainty} = \frac{1}{2} (\text{Largest Value} - \text{Smallest Value}) $$
Example:
If measuring the initial and final mass of a container using a balance with an uncertainty of $\pm 0.1$ g:
If measuring the area of a rectangle:
Error bars on graphs represent the absolute uncertainty of the measured values. They can be drawn vertically or horizontally, depending on the axis.
Tip:
Error bars do not need to be the same size for all data points. Each point can have different error bar sizes based on its individual uncertainty.
When performing calculations with measurements, the uncertainties must be propagated to reflect in the final result. For example, when multiplying or dividing measurements, the percentage uncertainties are added.
Understanding measurements and uncertainties is crucial in physics to assess the reliability and accuracy of experimental results. By mastering the concepts of absolute, fractional, and percentage uncertainties, as well as random and systematic errors, students can critically evaluate their data and improve their experimental techniques.