Page 257 - Physics Coursebook 2015 (A level)

P1: Practical skills at AS level
WORKED EXAMPLE
1 A length is measured five times with a ruler whose smallest division is 0.1 cm and the readings obtained, in cm, are 22.9, 22.7, 22.9, 23.0, 23.1. What is the reading obtained and the uncertainty?
Step1 Findtheaveragebyaddingthevaluesand dividing by the number of values:
22.9 + 22.7 + 22.9 + 23.0 + 23.1 = 22.92cm 5
This is written to 4 significant figures. At this stage you are not sure how many figures to write in the answer.
Step2 Themaximumvalueis23.1andthe minimum value is 22.7. Use these values to find half the range.
half the range = 23.1 − 22.7 = 0.2 cm 2
Step3 CheckthattheuncertaintycalculatedinStep 2 is larger than the smallest division you can read on the scale.
Step4 Writedowntheaveragevalue,the uncertainty to a reasonable number of significant figures and the unit. Obviously the last digit in 22.92 is meaningless as it is much smaller than the uncertainty; it should not be written down.
the final value is 22.9 ± 0.2 cm
You do not usually write down the final value of the answer to a greater number of decimal places than the uncertainty. Uncertainties are usually quoted to 1 or perhaps 2 significant figures.
Percentage uncertainty
The uncertainties we have found so far are sometimes called absolute uncertainties, but percentage uncertainties are also very useful.
The percentage uncertainty expresses the absolute uncertainty as a fraction of the measured value and is found by dividing the uncertainty by the measured value and multiplying by 100%.
percentage uncertainty = uncertainty × 100% measured value
For example, suppose a student times a single swing of a pendulum. The measured time is 1.4 s and the estimated
uncertainty is 0.2 s. Then we have:
percentage uncertainty = uncertainty × 100%
measured value
= 0.2 × 100% = 14% 1.4
This gives a percentage uncertainty of 14%. We can show our measurement in two ways:
■■ with absolute uncertainty:
time for a single swing = 1.4s±0.2s
■■ with percentage uncertainty:
time for a single swing = 1.4 s ± 14%
(Note that the absolute uncertainty has a unit whereas the percentage uncertainty is a fraction, shown with a % sign.)
A percentage uncertainty of 14% is very high. This could be reduced by measuring the time for 20 swings. In doing so the absolute uncertainty remains 0.2 s (it is the uncertainty in starting and stopping the stopwatch which is the important thing here, not the accuracy of the stopwatch itself), but the total time recorded might now be 28.4s.
percentage uncertainty = 0.2 × 100% = 0.7% 28.4
So measuring 20 oscillations rather than just one reduces the percentage uncertainty to less than 1%. The time for one swing is now calculated by dividing the total time by 20, giving 1.42 s. Note that, with a smaller uncertainty, we can give the result to 2 decimal places. The percentage uncertainty remains at 0.7%:
time for a single swing = 1.42 s ± 0.7%
QUESTIONS
10 The depth of water in a bottle is measured as 24.3 cm, with an uncertainty of 0.2 cm. (This could be written as 24.3 ± 0.2 cm.) Calculate the percentage uncertainty in this measurement.
11 The angular amplitude of a pendulum is measured as 35 ± 2°.
a Calculate the percentage uncertainty in the measurement of this angle.
b The protractor used in this measurement was calibrated in degrees. Suggest why the user only feels confident to give the reading to within 2°.
12 A student measures the potential difference across a battery as 12.4 V and states that his measurement has a percentage uncertainty of 2%. Calculate the absolute uncertainty in his measurement.
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