The quantity Q depends on the measured values a and b in the...

The quantity $Q$ depends on the measured values a and $b$ in the following ways: (a) $Q=a / b, a=20 \pm 1, b=10 \pm 1$ (b) $Q=2 a+3 b, a=20 \pm 2, b=15 \pm 3$ (c) $Q=a-2 b, a=50 \pm 1, b=24 \pm 0.5$ (d) $Q=a^{2}, a=10.0 \pm 0.3$ (e) $Q=a^{2} / b^{2}, a=100 \pm 5, b=20 \pm 2$ In each case find the value of $Q$ and its uncertainty.

The quantity $Q$ depends on the measured values a and $b$ in the following ways:
(a) $Q=a / b, a=20 \pm 1, b=10 \pm 1$
(b) $Q=2 a+3 b, a=20 \pm 2, b=15 \pm 3$
(c) $Q=a-2 b, a=50 \pm 1, b=24 \pm 0.5$
(d) $Q=a^{2}, a=10.0 \pm 0.3$
(e) $Q=a^{2} / b^{2}, a=100 \pm 5, b=20 \pm 2$
In each case find the value of $Q$ and its uncertainty.

Key Concepts

Uncertainty for Powers

When a measurement is raised to a power, the relative uncertainty of that measurement is multiplied by the absolute value of the exponent. This method captures how the uncertainty scales with the power and is a direct application of the rules of propagation of uncertainty in functions.

Uncertainty in Multiplication and Division

For operations involving multiplication or division, the relative uncertainties (or fractional uncertainties) of the measured values are combined by summing them. The resulting total relative uncertainty is then applied to the calculated value to determine the absolute uncertainty of the result.

Uncertainty in Addition and Subtraction

When values are added or subtracted, their absolute uncertainties are combined (usually by taking the sum) to give the overall absolute uncertainty of the resultant quantity. This method assumes that the uncertainties in the individual measurements are independent of each other.

Propagation of Uncertainty

Propagation of uncertainty is a method used to determine the uncertainty in a result calculated from measured values that each have their own uncertainty. It involves using specific rules and formulas to combine the errors of the individual measurements, ensuring that the derived quantity’s uncertainty is accurately estimated.

Absolute Uncertainty

Absolute uncertainty refers to the margin of error in a measurement, expressed in the same units as the measurement itself. It represents the range within which the true value is expected to lie and is particularly useful in operations that involve addition or subtraction of measured quantities.

Relative Uncertainty

Relative uncertainty is the ratio of the absolute uncertainty to the measured value, often expressed as a percentage. It provides a sense of scale regarding the precision of the measurement and becomes especially important when dealing with multiplicative or divisive relationships.

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