Working with Uncertainties IB Physics 11

Uncertainties and errors When measuring physical quantities 3 types of errors may arise

Types of measurement errors Random Systematic Reading

Random errors Almost always due to the observer Shows up as fluctuating measurements about some central value Can be reduced by averaging over repeated mesurements

Systematic errors Have to do with the system i.e. the equipment and the procedure Attributable to both the observer and the measuring instrument. Do not result in fluctuating values Cannot be reduced by repeated measurements

Reading errors Relates to the difficulty in reading the instrument with absolute precision Cannot be improved upon by repeated measurements. E.g. the reading error for a metrestick is ± 0.05 cm. When a metrestick is read, the best precision we can obtain is to the nearest 0.05 cm on either end of the measurement i.e a total of ± 0.1 cm (add uncertainties when subtracting)

Random errors for repeated measurements For repeated measurements, it is reasonable to expect that half the time, the values will be above the mean and the other half will be below the mean. Therefore, we calculate the uncertainty in the mean (average) as ∆ Mean = ± (Max Value-Min Value)/2

Systematic errors Most common source is incorrectly calibrated instrument e.g. if an electronic scale is off by 1 g, then, all the measurements will be off by 1 g Zero errors also give rise to systematic errors. E.g. a rounded metrestick may yield measurements understated by a few mm. An analog ammeter (measures electric current) whose needle starts at 0.1 Amp will have all the current values overstated by 0.1 Amp

Systematic errors Systematic errors also arise as a result of the experimenter not being properly aligned with the measuring instrument when reading the instrument. The reading will be either overstated or understated depending on where the experimenter is positioned. This is also known as “human parallax” error

Repeated measurements For a number of repeated values, first find the average or mean. The uncertainty in the average is plus or minus one-half of the range between the maximum and the minimum value. e.g. L 1 = 140. m, L 2 = 136 m, L 3 = 142 m L mean = (140. m +136 m +142 m)/3 =139.33m ∆L mean =L max -L min = 142m – 136m = ±3 m L ± ∆L = (139 ± 3) m

Reading Errors InstrumentReading error Metrestick± 0.5 mm Vernier calipers± 0.05 mm Micrometer± mm Volumetric (measuring) cylinder± 0.5 mL Electronic weighing scale± 0.1 g Stopwatch± 0.01 s

Uncertainties with addition L = r + w r ±∆r =(6.1±0.1)cm ; w±∆w=(12.6±0.2)cm L=6.1cm cm=18.7 cm ∆L = ∆r + ∆w = 0.1 cm cm = 0.3 cm L±∆L = (18.7±0.3) cm

Uncertainties with subtraction L = w - r r ±∆r =(6.1±0.1)cm ; w±∆w=(12.6±0.2)cm L= cm=6.5 cm ∆L = ∆r + ∆w = 0.1 cm cm = 0.3 cm L±∆L = (6.5 ±0.3) cm

Uncertainties with Multiplication Area = Length x Width A = L x W L = (24.3 ± 0.1) cm W = (11.8 ± 0.1) cm A = 24.3 cm x 11.8 cm = cm² Note ΔA % = ΔL % + ΔW % ΔA % = [(0.1/24.3)x100] + [(0.1/11.8)x100] ΔA % = 0.412% % = 1.259% ≈1% A ± ΔA = cm² ± cm² A ± ΔA = (287 ± 3) cm²

Uncertainties with Division Speed = Distance/Time v = s/t ; s = (12.4 ± 0.2) m t = (5.43 ± 0.01) s v = 12.4/5.43 = ms -1 Δv% = Δs% + Δt% Δv% = [(0.2/12.4)x100] + [(0.01/5.43)x100] Δv% = % % = %≈ 2% v ± Δv = ms -1 ± ms -1 v ± Δv = (2.28 ± 0.05) ms -1

Line of best fit Graph the data with error bars Extension x/cmTension Force T/N (± 10)