Key Equations
Measurements & Uncertainties - OCR A-Level Physics
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Percentage uncertainty
$$\% \text{ uncertainty} = \frac{\Delta x}{x} \times 100\%$$
- Where:
- $\Delta x$ = same units as x
- $x$ = measured value
Converts an absolute uncertainty to a percentage. Essential for combining uncertainties in products and quotients.
Combining uncertainties (addition/subtraction)
$$\begin{aligned}
\text{If } y &= a \pm b: \quad \Delta y \\
&= \Delta a + \Delta b
\end{aligned}$$
- Where:
- $\Delta y$ = same units as y
- $\Delta a$ = same units as a
- $\Delta b$ = same units as b
When two quantities are added or subtracted, add their ABSOLUTE uncertainties. This rule also applies to a change (e.g. temperature change = final - initial: add both uncertainties).
Combining uncertainties (multiplication/division)
$$\begin{aligned}
\text{If } y &= ab \text{ or } y \\
&= \frac{a}{b}: \quad \%\Delta y \\
&= \%\Delta a + \%\Delta b
\end{aligned}$$
- Where:
- $%\Delta y$ = % (dimensionless)
- $%\Delta a$ = % (dimensionless)
- $%\Delta b$ = % (dimensionless)
When quantities are multiplied or divided, add their PERCENTAGE uncertainties. Then convert back to absolute if needed.
Power rule for uncertainties
$$\begin{aligned}
\text{If } y &= a^n: \quad \%\Delta y \\
&= n \times \%\Delta a
\end{aligned}$$
- Where:
- $%\Delta y$ = % (dimensionless)
- $n$ = the exponent (dimensionless)
- $%\Delta a$ = % (dimensionless)
Multiply the percentage uncertainty by the power. For y = \(a^{2}\), the % uncertainty doubles. For y = \(a^{3}\), it triples. For y = sqrt(a), multiply by 0.5.
Uncertainty in gradient from worst lines
$$\Delta m = \frac{m_{\text{max}} - m_{\text{min}}}{2}$$
- Where:
- $\Delta m$ = same units as the gradient
- $m_{max}$ = gradient of steepest worst line
- $m_{min}$ = gradient of shallowest worst line
Draw two worst lines through the error bars. The steepest and shallowest gradients define the range. Half the range is the uncertainty.
Percentage difference
$$\% \text{ difference} = \frac{|x_{\text{exp}} - x_{\text{acc}}|}{x_{\text{acc}}} \times 100\%$$
- Where:
- $x_{exp}$ = experimental value
- $x_{acc}$ = accepted/theoretical value
Used to validate results. If percentage difference < percentage uncertainty, the result is consistent with the accepted value.