Maths bridge › Uncertainty arithmetic
Uncertainty arithmetic
Absolute against percentage uncertainty, and the three combination rules.
Three rules cover the course
An absolute uncertainty has units (±0.01 mm); a percentage uncertainty is that as a share of the value, multiplied by one hundred. For repeated readings, the absolute uncertainty is half the range of the repeats.
Now the combining. When quantities add or subtract, add their absolute uncertainties. When they multiply or divide, add their percentage uncertainties. When a quantity is raised to a power, multiply its percentage uncertainty by that power, so squaring doubles it and a square root halves it.
The power rule is where the marks are. Any formula with a d2 in it makes the diameter the measurement worth the most care, and that is the reason a micrometer paragraph turns up in practical after practical.
The rules earning their keep
WORKED EXAMPLE
The diameter's double charge
A wire's diameter is measured as 0.36 ± 0.01 mm, and the area comes from A = πd2/4. Find the percentage uncertainty in the area.
The diameter's percentage uncertainty is 0.01/0.36 × 100 = 2.8%.
The area depends on d2, so its percentage uncertainty is double, 5.6%. The π and the 4 are exact numbers and contribute nothing at all.
GUIDED PRACTICE
Density, revisited under uncertainty
You met a version of this calculation in the measurements unit; now run it properly, with every input carrying its own uncertainty. A block's mass is measured to 1.0% and each side of its cube shape to 0.5%. Find the percentage uncertainty in its density ρ = m/V, thinking first about what power the side length carries, before opening the working.
Show the working
The volume is the side cubed, so the side's 0.5% counts three times, giving 1.5% on V.
Density divides m by V, so add the percentages. 1.0 + 1.5 = 2.5% on ρ.
Where physics leans on this: Uncertainty and error · RP4: the Young modulus · RP5: resistivity. All eight skills: the maths bridge.