Worked exercise — Relative uncertainty in a quotient

Problem

You measure the mass $m = (250 \pm 5)\ \text{g}$ and the volume $V = (95 \pm 3)\ \text{cm}^3$ of a sample. Calculate the density and its relative uncertainty.

Solution

Density value. $\rho = \frac{m}{V} = \frac{250}{95} \approx 2{,}63\ \frac{\text{g}}{\text{cm}^3}$

Relative errors on the two measurements. $\frac{\Delta m}{m} = \frac{5}{250} = 2{,}0\%\ ;\qquad \frac{\Delta V}{V} = \frac{3}{95} = 3{,}2\%$

Propagation for a quotient. As with the product, for a ratio of independent quantities the relative errors add in quadrature: $\frac{\Delta \rho}{\rho} \approx \sqrt{\left(\frac{\Delta m}{m}\right)^2 + \left(\frac{\Delta V}{V}\right)^2} = \sqrt{2{,}0^2 + 3{,}2^2}\ \% \approx 3{,}8\%$

Absolute uncertainty. $\Delta \rho \approx 0{,}038 \cdot 2{,}63 \approx 0{,}10\ \frac{\text{g}}{\text{cm}^3}$

$\ev{\rho = (2{,}6 \pm 0{,}1)\ \text{g/cm}^3,\qquad \frac{\Delta \rho}{\rho} \approx 4\%}$

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Topics: Experimental method Concepts: Significant figures and uncertainty Skills: Error propagation · Using significant figures