A student measures the mass m = 0

Measurements & Uncertainties - OCR A-Level Physics

Key Technique

Propagating uncertainty through a calculation
When a final quantity is calculated from two or more measured quantities, work in percentage uncertainties throughout, then convert back to an absolute uncertainty at the end. Four steps: find the percentage uncertainty in each measurement, multiply by any power, sum the percentage uncertainties for multiplication or division, then convert back using $\Delta y = \frac{\% \text{ uncertainty}}{100} \times y$.

Worked example. A student measures the mass $m = 0.250 \pm 0.001 \text{ kg}$ and speed $v = 3.0 \pm 0.2 \text{ m s}^{-1}$. Calculate the kinetic energy and its absolute uncertainty.

E_k = \tfrac{1}{2}mv^2 = \tfrac{1}{2} \times 0.250 \times 3.0^{2} = 1.125 \text{ J}

Step 1. Percentage uncertainty in $m$: $\frac{0.001}{0.250} \times 100\% = 0.4\%$.
Step 2. Percentage uncertainty in $v$: $\frac{0.2}{3.0} \times 100\% = 6.7\%$. Since $v$ is squared, multiply by 2: $2 \times 6.7\% = 13.3\%$.
Step 3. Total percentage uncertainty in $E_k$: $0.4\% + 13.3\% = 13.7\%$.
Step 4. Convert back to absolute uncertainty: $\Delta E_k = \frac{13.7}{100} \times 1.125 = 0.15 \text{ J}$.

E_k = 1.1 \pm 0.2 \text{ J}

Examiner Tips and Tricks

Notice how the uncertainty in v dominates (13.3% vs 0.4% for m).
This tells the student that measuring speed more precisely would improve the result far more than a better balance.
Examiners love asking 'Which measurement contributes most to the uncertainty?'