A-Level — physicsFoundational3 min read

Measurement and Uncertainty

Systematic vs random errors; percentage uncertainty; combining uncertainties; significant figures

Tutor AI Team
Tutor AI Team

# Measurement and Uncertainty — A-Level Physics

Every measurement has an uncertainty. Understanding errors and uncertainties is crucial for A-Level practical work and for evaluating experimental results.

1. Types of Error

Systematic Errors

  • Shift all readings by the same amount in one direction
  • Affect accuracy (how close to true value)
  • Examples: zero error, calibration error, parallax with consistent viewing angle
  • Cannot be reduced by repeating
  • Can be reduced by better experimental design, calibration

Random Errors

  • Cause scatter in readings
  • Affect precision (how close readings are to each other)
  • Examples: timing reactions, reading a scale, environmental fluctuations
  • Reduced by repeating and averaging

2. Uncertainty

Absolute Uncertainty

For repeated measurements: uncertainty = range/2 For single measurement: uncertainty = smallest division/2 (analogue) or smallest division (digital)

Percentage Uncertainty

% uncertainty=absolute uncertaintymeasured value×100\text{\% uncertainty} = \frac{\text{absolute uncertainty}}{\text{measured value}} \times 100

3. Combining Uncertainties

Addition/Subtraction: Add absolute uncertainties Δ(A+B)=ΔA+ΔB\Delta(A + B) = \Delta A + \Delta B

Multiplication/Division: Add percentage uncertainties %uncertainty in A×B=%A+%B\%\text{uncertainty in } A \times B = \%A + \%B

Powers: Multiply percentage uncertainty by the power %uncertainty in An=n×%A\%\text{uncertainty in } A^n = n \times \%A

Worked Example: Combining Uncertainties

Problem

R=ρL/AR = \rho L/A. ρ\rho measured to ±3%, LL to ±1%, AA to ±4%.

%uncertainty in RR = 3 + 1 + 4 = 8%

Solution

Worked Example: With Powers

Problem

T=2πL/gT = 2\pi\sqrt{L/g}. So g=4π2L/T2g = 4\pi^2L/T^2. LL measured to ±2%, TT measured to ±1.5%.

%uncertainty in gg = 2 + 2(1.5) = 5%

Solution

Worked Example: Subtraction

Problem

Length = 84.3 ± 0.5 mm, end position = 12.1 ± 0.5 mm. Difference = 72.2 ± 1.0 mm (absolute uncertainties add) %uncertainty = 1.0/72.2 × 100 = 1.4%

Solution

5. Significant Figures

  • Final answer should have the same number of s.f. as the least precise measurement
  • Uncertainties are typically given to 1 s.f.
  • Intermediate calculations: keep extra s.f.

6. Practice Questions

    1. A student measures a length five times: 15.2, 15.4, 15.1, 15.3, 15.4 cm. Find the mean and the uncertainty. (3 marks)
    1. v=s/tv = s/t. s=0.50±0.01s = 0.50 \pm 0.01 m, t=0.32±0.02t = 0.32 \pm 0.02 s. Calculate vv with its uncertainty. (4 marks)
    1. Explain the difference between systematic and random errors. Give one example of each. (4 marks)

    Answers

    1. Mean = (15.2+15.4+15.1+15.3+15.4)/5 = 15.28 cm. Range = 15.4 − 15.1 = 0.3. Uncertainty = 0.3/2 = 0.15 cm. Result: 15.3 ± 0.2 cm.
    1. v=0.50/0.32=1.5625v = 0.50/0.32 = 1.5625 m/s. %unc in s = 0.01/0.50 × 100 = 2%. %unc in t = 0.02/0.32 × 100 = 6.25%. Total = 8.25%. Absolute unc = 0.0825 × 1.56 = 0.13. v=1.56±0.13v = 1.56 ± 0.13 m/s.

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Summary

  • Systematic: affects accuracy; random: affects precision
  • Add absolute uncertainties for ±; add % uncertainties for ×÷
  • Powers: multiply % uncertainty by power
  • Report to appropriate s.f. with uncertainty to 1 s.f.
Tags:
#a-level#physics#practical#uncertainty#errors#measurement

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