Learning Outcomes:
i. Understand the concept of uncertainty in measurements and its propagation in derived quantities.
ii. Identify the different types of uncertainties, including actual, fractional, and percentage uncertainties.
iii. Apply the rules for combining uncertainties in derived quantities involving addition, subtraction, multiplication, and division.
iv. Interpret the uncertainty in a derived quantity and express it appropriately.
Introduction:
In the realm of physics, measurements are never perfect; they are always subject to some degree of uncertainty. This uncertainty arises from various factors, such as limitations of measuring instruments, environmental conditions, and human error. While it is impossible to eliminate uncertainty entirely, it is crucial to understand how to quantify and propagate uncertainty in calculations involving multiple measurements. This lesson delves into the propagation of uncertainty in derived quantities, providing the tools necessary to assess the overall uncertainty in calculations involving various measurements.
i. Types of Uncertainties:
Uncertainty in measurements can be expressed in different ways:
Actual Uncertainty (Δx): The actual uncertainty is the absolute difference between the measured value (x) and the true value (x_true). Δx = x - x_true.
Fractional Uncertainty (Δx/x): The fractional uncertainty is the ratio of the actual uncertainty to the measured value. It is a dimensionless quantity, often expressed as a percentage. Fractional uncertainty = (Δx/x) × 100%.
Percentage Uncertainty (% Δx): Percentage uncertainty is another way of expressing fractional uncertainty, directly as a percentage. Percentage uncertainty = % Δx = (Δx/x) × 100%.
ii. Combining Uncertainties in Derived Quantities:
When calculating derived quantities from multiple measurements, the uncertainty in the final result depends on the uncertainties of the individual measurements and the mathematical operations involved.
Addition and Subtraction: For addition and subtraction, the uncertainties of the individual measurements are simply added or subtracted.
Multiplication and Division: For multiplication and division, the fractional uncertainties of the individual measurements are added or subtracted.
Examples:
Addition: If you measure the length of a room to be 4.2 meters with an uncertainty of 0.1 meter and the width of the room to be 3.1 meters with an uncertainty of 0.2 meters, the area of the room is (4.2 m × 3.1 m) ± (0.1 m + 0.2 m) = 12.92 m² ± 0.3 m².
Multiplication: If you measure the velocity of an object to be 5.0 m/s with an uncertainty of 0.2 m/s and the time interval to be 2.0 seconds with an uncertainty of 0.1 seconds, the displacement of the object is (5.0 m/s × 2.0 s) ± (0.2 m/s + 0.1 s) = 10.0 m ± 0.3 s.
iii. Interpreting Uncertainty:
The uncertainty in a derived quantity represents the range within which the true value is likely to lie. A smaller uncertainty indicates a more precise measurement, while a larger uncertainty indicates a less precise measurement.
Understanding the propagation of uncertainty is essential for interpreting and reporting measurements in physics. By applying the appropriate rules for combining uncertainties, we can assess the overall uncertainty in derived quantities and make informed decisions about the reliability of our results.
