A0.1 Basics: Measurements and Uncertainties

The 7 fundamental SI units

1. mass in kilograms (kg)

2. time in seconds(s)

3. length in metres (m)

4. temperature in kelvin (K)

5. electric current in Amperes (A)

6. brightness in candela (cd)

7. amount of substance in moles (mol)

Uncertainty on a digital measuring instrument

The uncertainty is equal to the precision (smallest division) of the measuring instrument.

e.g. a digital stop watch measured time taken as 7.02 seconds: smallest division = 0.01 second, so the uncertainty is 0.01s. Therefore the time taken with uncertainty is 7.02 ± 0.01 s.

Uncertainty on an analogue measuring instrument

The uncertainty is equal to ½ of the precision (smallest division) of the measuring instrument.

e.g. a liquid thermometer 🌡 measured the temperature as 23^oC: smallest division = 1^oC, so uncertainty is ½ x 1 = 0.5 ^oC. Therefore, the temperature with uncertainty is 23.0 ± 0.5^oC (!! the number of decimal places needs to match!!)

Uncertainty on a ruler

Since the ruler is an analogue measuring instrument, the uncertainty is equal to ½ of the precision (smallest division), but it has to be doubled because you check the measurement twice: once checking whether the start of the length aligns with the 0 on the ruler, and once where the end of the length is.

e.g. the length measured using a ruler is 2.5cm. The smallest division is 1mm = 0.1cm, so the uncertainty is ½ x 0.1 × 2 = 0.1cm. Therefore the length with the uncertainty is 2.5 ± 0.1 cm.

Adding or subtracting data with uncertainties

The absolute uncertainties of the data are added.

e.g. a length of 20.1± 0.1 cm and a length of 19.6 ± 0.1 cm are added together: result = (20.1+19.6) ±(0.1 + 0.1) = 39.7 ± 0.2 cm

Multiplying or dividing data that have the same units with uncertainties

illustrated with the following example:

e.g. a rectangle has length 20.1 ± 0.1cm and width 11.5 ± 0.1cm, work out its area and give it with its absolute uncertainty.

1)Work out the fractional uncertainty of each measurement.

fractional uncertainty = absolute uncertainty / measurement

frac. unc. of length = 0.1/ 20.1 = 0.004975

frac unc. of width = 0.1/ 11.5 = 0.008687

2)add the fractional uncertainties together

= 0.013662

3)absolute uncertainty of final result = sum of fractional uncertainties of data x result

= 0.013662 x (20.1 × 11.5) = 3.157

Therefore, the answer is 231.15 ± 3.16 cm²

(since the calculation is multiplication, the uncertainty should be given to 3 s.f. as in the question, and the decimal places of the area has to match.)

Multiplying or dividing data that have different units with uncertainties

illustrated by the following example:

e.g. an object has mass 20.2 ± 0.4 g and volume 24.3 ± 0.9 cm³, calculate its density and give it with its uncertainty.

1) Calculate the uncertainty of each measurement/data in percentage

% uncertainty of mass = 0.4/20.2 × 100 = 1.98

% uncertainty of volume = 0.9/24.3 × 100 = 3.70

2) Add the percentage uncertainties together

1.98 + 3.70 = 5.68 = 6% → percentage uncertainty of density

3) Uncertainty of result = sum of percentage uncertainty of data x result

6% x (mass/volume → 20.2/24.3→ 0.831) = 0.049

Therefore, the answer is 0.831 ± 0.049g.

Uncertainty of data that was averged from repeats

Uncertainty = range / 2

= (max. value - min. value) /2

can be used to plot error bars

Effect of multiples and repeats

• Doing multiples, (e.g. measuring the time taken for the pendulum to do 10 swings instead of 1), reduces the uncertainty (divided by the no. of multiples - in this case, divided by 10)

• Doing repeats reduces the impact of random error e.g. parallax error, different height, etc.