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Experimental Programme - Measurement and Uncertainty

Grade 12IBPhysics

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

🔑Concepts

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Random errors are caused by unpredictable fluctuations in the environment or measuring process; they affect the precision of a measurement and can be reduced by repeating trials and taking an average.

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Systematic errors are caused by flawed equipment or experimental design (e.g., zero errors or poor calibration); they affect the accuracy of a result and cannot be reduced by repeating measurements.

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Accuracy refers to how close a measurement is to the true or accepted value, while precision refers to how close a series of measurements are to each other.

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The absolute uncertainty is the margin of error in a measurement, typically expressed with the same number of decimal places as the measurement itself, e.g., (10.2±0.1) cm(10.2 \pm 0.1) \text{ cm}.

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When adding or subtracting quantities, the absolute uncertainties are added: Δy=Δa+Δb\Delta y = \Delta a + \Delta b.

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When multiplying or dividing quantities, the fractional or percentage uncertainties are added: Δyy=Δaa+Δbb\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b}.

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If a quantity is raised to a power nn, the fractional uncertainty is multiplied by ∣n∣|n|: Δyy=∣n∣Δaa\frac{\Delta y}{y} = |n| \frac{\Delta a}{a}.

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In graphical analysis, the uncertainty in the gradient mm is estimated using Δm=mmax−mmin2\Delta m = \frac{m_{max} - m_{min}}{2}, where mmaxm_{max} and mminm_{min} are the gradients of the lines of best fit that still pass through all error bars.

📐Formulae

Δy=Δa+Δbfor y=a±b\Delta y = \Delta a + \Delta b \quad \text{for } y = a \pm b

Δyy=Δaa+Δbbfor y=a×b or y=ab\frac{\Delta y}{y} = \frac{\Delta a}{a} + \frac{\Delta b}{b} \quad \text{for } y = a \times b \text{ or } y = \frac{a}{b}

Δyy=∣n∣Δaafor y=an\frac{\Delta y}{y} = |n| \frac{\Delta a}{a} \quad \text{for } y = a^n

Percentage Uncertainty=Absolute UncertaintyMeasured Value×100%\text{Percentage Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measured Value}} \times 100\%

Uncertainty in Gradient=mmax−mmin2\text{Uncertainty in Gradient} = \frac{m_{max} - m_{min}}{2}

Uncertainty in Intercept=cmax−cmin2\text{Uncertainty in Intercept} = \frac{c_{max} - c_{min}}{2}

💡Examples

Problem 1:

An object travels a distance s=20.0±0.5 ms = 20.0 \pm 0.5 \text{ m} in a time t=4.0±0.2 st = 4.0 \pm 0.2 \text{ s}. Calculate the average speed vv and its absolute uncertainty.

Solution:

v=st=20.04.0=5.0 m s−1v = \frac{s}{t} = \frac{20.0}{4.0} = 5.0 \text{ m s}^{-1}. Percentage uncertainty in distance: 0.520.0×100=2.5%\frac{0.5}{20.0} \times 100 = 2.5\%. Percentage uncertainty in time: 0.24.0×100=5.0%\frac{0.2}{4.0} \times 100 = 5.0\%. Total percentage uncertainty in speed: 2.5%+5.0%=7.5%2.5\% + 5.0\% = 7.5\%. Absolute uncertainty in speed: 0.075×5.0=0.375 m s−10.075 \times 5.0 = 0.375 \text{ m s}^{-1}. Final result: v=5.0±0.4 m s−1v = 5.0 \pm 0.4 \text{ m s}^{-1}.

Explanation:

Since v=s/tv = s/t, we add the percentage uncertainties of distance and time. The final absolute uncertainty is usually rounded to one significant figure, and the value is adjusted to match the decimal precision.

Problem 2:

A cube has a side length L=2.00±0.02 cmL = 2.00 \pm 0.02 \text{ cm}. Find the volume VV and its percentage uncertainty.

Solution:

V=L3=(2.00)3=8.00 cm3V = L^3 = (2.00)^3 = 8.00 \text{ cm}^3. Percentage uncertainty in LL: 0.022.00×100=1%\frac{0.02}{2.00} \times 100 = 1\%. Percentage uncertainty in VV: 3×1%=3%3 \times 1\% = 3\%. Absolute uncertainty in VV: 0.03×8.00=0.24 cm30.03 \times 8.00 = 0.24 \text{ cm}^3. Final result: V=8.00±0.24 cm3V = 8.00 \pm 0.24 \text{ cm}^3.

Explanation:

When a value is raised to a power, the fractional (or percentage) uncertainty is multiplied by that power. Here, V=L3V = L^3, so the uncertainty is tripled.

Problem 3:

Determine the uncertainty in the resistance RR if V=10.0±0.2 VV = 10.0 \pm 0.2 \text{ V} and I=2.00±0.05 AI = 2.00 \pm 0.05 \text{ A}.

Solution:

R=VI=10.02.00=5.00 ΩR = \frac{V}{I} = \frac{10.0}{2.00} = 5.00 \, \Omega. Percentage uncertainty in VV: 0.210.0=2%\frac{0.2}{10.0} = 2\%. Percentage uncertainty in II: 0.052.00=2.5%\frac{0.05}{2.00} = 2.5\%. Total percentage uncertainty: 2%+2.5%=4.5%2\% + 2.5\% = 4.5\%. Absolute uncertainty: 0.045×5.00=0.225 Ω0.045 \times 5.00 = 0.225 \, \Omega. Final result: R=5.0±0.2 ΩR = 5.0 \pm 0.2 \, \Omega.

Explanation:

In division, percentage uncertainties are summed. The final result is rounded to reflect the appropriate precision based on the absolute uncertainty.

Measurement and Uncertainty - Revision Notes & Key Formulas | IB Grade 12 Physics