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Measurements and Uncertainties - Propagation of Uncertainties

Grade 11IBPhysics

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Absolute Uncertainty: The actual range of values a measurement could take, denoted as Δx\Delta x. It has the same units as the measurement.

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Fractional (Relative) Uncertainty: The ratio of the absolute uncertainty to the measured value, calculated as Δxx\frac{\Delta x}{x}. This value is dimensionless.

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Percentage Uncertainty: The fractional uncertainty expressed as a percentage: Δxx×100%\frac{\Delta x}{x} \times 100\%.

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Addition and Subtraction Rule: When quantities are added or subtracted, the absolute uncertainties are added together.

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Multiplication and Division Rule: When quantities are multiplied or divided, the fractional (or percentage) uncertainties are added together.

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Power Rule: When a quantity is raised to a power nn, the fractional uncertainty of that quantity is multiplied by ∣n∣|n|.

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Rounding Rules: Calculated uncertainties are typically expressed to one or two significant figures, and the final result must be rounded to the same decimal place as the absolute uncertainty.

📐Formulae

If y=a±b, then Δy=Δa+Δb\text{If } y = a \pm b, \text{ then } \Delta y = \Delta a + \Delta b

If y=a⋅bc, then Δyy=Δaa+Δbb+Δcc\text{If } y = \frac{a \cdot b}{c}, \text{ then } \frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} + \frac{\Delta c}{c}

If y=an, then Δyy=∣n∣Δaa\text{If } y = a^n, \text{ then } \frac{\Delta y}{y} = |n| \frac{\Delta a}{a}

Percentage Uncertainty=Δxx×100%\text{Percentage Uncertainty} = \frac{\Delta x}{x} \times 100\%

💡Examples

Problem 1:

A student measures the initial temperature of a liquid as T1=(20.0±0.5)∘CT_1 = (20.0 \pm 0.5) ^\circ C and the final temperature as T2=(55.0±0.5)∘CT_2 = (55.0 \pm 0.5) ^\circ C. Calculate the change in temperature ΔT\Delta T and its absolute uncertainty.

Solution:

ΔT=T2−T1=55.0−20.0=35.0∘C\Delta T = T_2 - T_1 = 55.0 - 20.0 = 35.0 ^\circ C. According to the addition/subtraction rule: Δ(ΔT)=ΔT1+ΔT2=0.5+0.5=1.0∘C\Delta (\Delta T) = \Delta T_1 + \Delta T_2 = 0.5 + 0.5 = 1.0 ^\circ C. Final answer: (35.0±1.0)∘C(35.0 \pm 1.0) ^\circ C.

Explanation:

Even though we are subtracting the temperatures, the uncertainties represent a range of possible error, so they must be added to find the maximum possible uncertainty in the result.

Problem 2:

The radius of a sphere is measured to be r=(5.0±0.2) cmr = (5.0 \pm 0.2) \text{ cm}. Calculate the volume VV and its percentage uncertainty. (Use V=43πr3V = \frac{4}{3}\pi r^3)

Solution:

V=43π(5.0)3≈523.6 cm3V = \frac{4}{3} \pi (5.0)^3 \approx 523.6 \text{ cm}^3. Using the power rule for r3r^3: ΔVV=3×Δrr=3×0.25.0=3×0.04=0.12\frac{\Delta V}{V} = 3 \times \frac{\Delta r}{r} = 3 \times \frac{0.2}{5.0} = 3 \times 0.04 = 0.12. Percentage uncertainty: 0.12×100%=12%0.12 \times 100\% = 12\%. Absolute uncertainty ΔV=0.12×523.6≈62.8 cm3\Delta V = 0.12 \times 523.6 \approx 62.8 \text{ cm}^3. Final answer: (520±60) cm3(520 \pm 60) \text{ cm}^3 (rounded to appropriate precision).

Explanation:

Since the volume depends on the cube of the radius, the fractional uncertainty of the radius is tripled to find the fractional uncertainty of the volume.

Problem 3:

An object travels a distance s=(100±2) ms = (100 \pm 2) \text{ m} in a time t=(20.0±0.5) st = (20.0 \pm 0.5) \text{ s}. Calculate the average speed vv and its absolute uncertainty.

Solution:

v=st=10020.0=5.0 m s−1v = \frac{s}{t} = \frac{100}{20.0} = 5.0 \text{ m s}^{-1}. Using the division rule: Δvv=Δss+Δtt=2100+0.520.0=0.02+0.025=0.045\frac{\Delta v}{v} = \frac{\Delta s}{s} + \frac{\Delta t}{t} = \frac{2}{100} + \frac{0.5}{20.0} = 0.02 + 0.025 = 0.045. Δv=0.045×5.0=0.225 m s−1\Delta v = 0.045 \times 5.0 = 0.225 \text{ m s}^{-1}. Final answer: (5.0±0.2) m s−1(5.0 \pm 0.2) \text{ m s}^{-1}.

Explanation:

For division, we sum the fractional uncertainties. The resulting absolute uncertainty is then rounded to one significant figure, and the mean value is adjusted to match that decimal place.

Propagation of Uncertainties - Revision Notes & Key Formulas | IB Grade 11 Physics