Physics A Level
P1 Practical skills at AS Level P1.6 Percentage uncertainty
Physics A Level
P1 Practical skills at AS Level P1.6 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\,uncerta{\mathop{\rm int}} y = \frac{{uncerta{\mathop{\rm int}} y}}{{measured\,value}} \times 100\% $
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:
$\begin{array}{l}
percentage\,uncerta{\mathop{\rm int}} y = \frac{{uncerta{\mathop{\rm int}} y}}{{measured\,value}} \times 100\% \\
= \frac{{0.2}}{{1.4}} \times 100\% \\
= 14\%
\end{array}$
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 \pm 0.2s$
- with percentage uncertainty: time for a single swing $ = 1.4s \pm 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 that is the important thing here, not the accuracy of the stopwatch itself), but the total time recorded might now be $28.4 s$.
$\begin{array}{l}
percentage\,uncerta{\mathop{\rm int}} y = \frac{{0.2}}{{28.4}} \times 100\% \\
= 0.7\%
\end{array}$
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 two decimal places. The percentage uncertainty remains at $0.7\% $:
$time\,for\,a\,\sin gle\,swing\, = \,1.42s \pm 0.7\% $
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 $(24.3 \pm 0.2)\,cm$.) Calculate the percentage uncertainty in this measurement.
11) The angular amplitude of a pendulum is measured as ${(35 \pm 2)^ \circ }$.
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 ${2^ \circ }$.
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.