Converting units can be awkward and can sometimes be forgotten. In 1999, the Mars Climate Orbiter crashed into the surface of the planet it was meant to orbit. And the reason? One of the ground teams was using units of feet and pound, while another group was using units of metre and kilogram.
Uncertainties
Uncertainties
In EVERY measurement (as opposed to simply counting) there is an uncertainty in the measurement.
This is sometimes determined by the apparatus you're using, sometimes by the nature of the measurement itself.
Random errors/uncertainties
Some measurements do vary randomly. Some are bigger than the actual/real value, some are smaller. This is called a random uncertainty. This should not be recorded, if your data seems unlikely check it while you are collecting data.
Systematic/zero errors
Sometimes all measurements are systematically bigger or smaller than they should be. This is called a systematic error/uncertainty.
Systematic/zero errors
This is normally caused by not measuring from zero. For example a balance that is showing 0.5g with nothing on it.
For this reason they are also known as zero errors/uncertainties. Finding an average doesn’t help.
Individual measurements
When using an analogue scale, the uncertainty is plus or minus half the smallest scale division. (in a best case scenario!)
4.20 ± 0.05 cm
Does using a ruler mean an individual measurement?
0.00 ± 0.05 cm
4.20 ± 0.05 cm
Individual measurements
When using an analogue scale, the uncertainty is plus or minus half the smallest scale division. (in a best case scenario!)
22.0 ± 0.5 V
Individual measurements
When using a digital scale, the uncertainty is plus or minus the smallest unit shown.
19.16 ± 0.01 V
Significant figures
Note that the uncertainty is given to one significant figure (after all it is itself an estimate) and it agrees with the number of decimal places given in the measurement.
19.16 ± 0.01
(NOT 19.160 or 19.2)
Repeated measurements
Pascal measured the length of 5 supposedly identical tables. He got the following results; 1560 mm, 1565 mm, 1558 mm, 1567 mm , 1558 mm
Average value = 1563 mm
Uncertainty = (1567 – 1558)/2 = 4.5 mm
Length of table = 1563 ± 5 mm
This means the actual length is anywhere between 1558 and 1568 mm
Uncertainties
In the example with the table, we found the length of the table to be 1563 ± 5 mm
We say the absolute uncertainty is 5 mm
The fractional uncertainty is 5/1563 = 0.003
The percentage uncertainty is 5/1563 x 100 = 0.3%
Uncertainties
If the average height of students at a random school is 1.23 ± 0.01 m
We say the absolute uncertainty is 0.01 m
The fractional uncertainty is 0.01/1.23 = 0.008
The percentage uncertainty is 0.01/1.23 x 100 = 0.8%
Propagating uncertainties
When adding (or subtracting) quantities, to find the resultant uncertainty we have to add the absolute uncertainties of the quantities we are adding or subtracting.
Propagating uncertainties
One basketball player has a height of 196 ± 1 cm and the other has a height of 152 ± 1 cm. What is the difference in their heights?
Difference = 44 ± 2 cm
A Quick Experiment
Use the 30cm ruler to measure the diameter ONCE of one of your coins
Find the diameter of a single coin
Find the distance across 5 coins
5D
How can we compare uncertainties?
absolute value
fractional value
percentage value
5D
Propagating uncertainties
When we find the volume of a block, we have to multiply the length by the width by the height.
Because each measurement has an uncertainty, the uncertainty increases when we multiply the measurements together.
Propagating uncertainties
When multiplying (or dividing) quantities, to find the resultant uncertainty we have to add the percentage (or fractional) uncertainties of the quantities we are multiplying.
Propagating uncertainties
Example: A block has a length of 10.0 ± 0.1 cm, width 5.0 ± 0.1 cm and height 6.0 ± 0.1 cm.
Volume = 10.0 x 5.0 x 6.0 = 300 cm3
% uncertainty in length = 0.1/10 x 100 = 1%
% uncertainty in width = 0.1/5 x 100 = 2 %
% uncertainty in height = 0.1/6 x 100 = 1.7 %
Uncertainty in volume = 1% + 2% + 1.7% = 4.7%
(4.7% of 300 = 14)
Volume = 300 ± 10 cm3
This means the actual volume could be anywhere between 290 and 310 cm3