How are uncertainties combined in addition and subtraction?

In addition and subtraction, uncertainties are combined by taking the square root of the sum of the squares of the uncertainties.

When you are dealing with measurements in physics, it is important to consider the uncertainties associated with those measurements. Uncertainties are a way of quantifying how much your measured values might differ from the true values. When you add or subtract quantities, the uncertainties combine in a specific way.

The rule for combining uncertainties in addition and subtraction is known as the 'sum in quadrature' rule. This rule states that the combined uncertainty is the square root of the sum of the squares of the individual uncertainties. This is mathematically represented as √((ΔA)^2 + (ΔB)^2), where ΔA and ΔB are the uncertainties of the quantities A and B respectively.

For example, if you are measuring the length of a table using a ruler with an uncertainty of ±1 cm, and the width of the table with the same ruler, the combined uncertainty when you add the two measurements is not simply 1 cm + 1 cm = 2 cm. Instead, you would calculate the combined uncertainty as √((1 cm)^2 + (1 cm)^2) = √2 cm, which is approximately 1.4 cm.

This rule applies regardless of whether you are adding or subtracting the quantities. The reason for this is that the uncertainty is a measure of the possible deviation from the true value, and this deviation can be either positive or negative. Therefore, when you add or subtract quantities, the uncertainties can either add together or cancel each other out, depending on whether the deviations are in the same or opposite directions.

It's important to note that this rule assumes that the uncertainties are independent of each other, meaning that the uncertainty in one measurement does not affect the uncertainty in another measurement. If the uncertainties are not independent, then they may need to be combined in a different way. To understand this better, consider exploring the concepts of absolute vs relative uncertainty, which delve into the nuances of measuring and expressing uncertainties.

Moreover, the impact of random errors and systematic errors on measurements can significantly influence the calculation and interpretation of uncertainties in physics experiments.

IB Physics Tutor Summary: In summary, when you add or subtract measurements in physics, the total uncertainty is calculated by the 'sum in quadrature' method. This means taking the square root of the sum of the squares of each uncertainty (represented as √((ΔA)^2 + (ΔB)^2)). This rule helps you accurately combine uncertainties, ensuring they reflect potential deviations in measurements accurately, assuming each measurement's uncertainty is independent.