When quantities are added or subtracted
Measurements & Uncertainties - OCR A-Level Physics
Key Definition
Combining uncertainties
When a final quantity is calculated from two or more measured quantities, the uncertainties have to be combined. The rule you use depends on the mathematical operation, not on the physical meaning of the quantities.
When a final quantity is calculated from two or more measured quantities, the uncertainties have to be combined. The rule you use depends on the mathematical operation, not on the physical meaning of the quantities.
Rule 1 : Addition and subtraction
When two quantities are added or subtracted, add their absolute uncertainties. The unit of the result is the same as the unit of the quantities being combined.
$$\text{If } y = a \pm b: \quad \Delta y = \Delta a + \Delta b$$
Rule 2 : Multiplication and division
When two quantities are multiplied or divided, add their percentage uncertainties. The same rule works for any number of factors. Switch back to absolute uncertainty at the end if the question asks for it.
$$\text{If } y = ab \text{ or } y = \frac{a}{b}: \quad \frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b}$$
Rule 3 : Raised to a power
When a quantity is raised to a power $n$, multiply the percentage uncertainty by $n$. This is the rule students forget most often, and it doubles or triples the uncertainty in one step.
$$\text{If } y = a^n: \quad \frac{\Delta y}{y} = n \times \frac{\Delta a}{a}$$
Examiner Tips and Tricks
- The power rule is the one students forget most often.
- If $v$ appears as $v^{2}$ in kinetic energy ($E_k = \tfrac{1}{2}mv^2$), the percentage uncertainty in $E_k$ from $v$ is double the percentage uncertainty in $v$.
- If you measure $r$ for the volume of a sphere ($V = \tfrac{4}{3}\pi r^3$), the percentage uncertainty in $V$ is three times that in $r$.
Common Mistake
MEDIUM
Wrong: Adding percentage uncertainties when quantities are added or subtracted.
Right: For addition/subtraction of quantities, add ABSOLUTE uncertainties. For multiplication/division, add PERCENTAGE uncertainties. The rule depends on the mathematical operation, not the quantities.
Right: For addition/subtraction of quantities, add ABSOLUTE uncertainties. For multiplication/division, add PERCENTAGE uncertainties. The rule depends on the mathematical operation, not the quantities.