Past Papers
Multimedia
Forum
QuizHub
Tutorial
School
Last update: 2022-10-17
Viewed: 113
Crash report

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

2022-10-17
113
Crash report
  • Chapter 1: Kinematics
  • Chapter 2: Accelerated motion
  • Chapter 3: Dynamics
  • Chapter 4: Forces
  • Chapter 5: Work, energy and power
  • Chapter 6: Momentum
  • Chapter 7: Matter and materials
  • Chapter 8: Electric current
  • Chapter 9: Kirchhoff’s laws
  • Chapter 10: Resistance and resistivity
  • Chapter 11: Practical circuits
  • Chapter 12: Waves
  • Chapter 13: Superposition of waves
  • Chapter 14: Stationary waves
  • Chapter 15: Atomic structure
  • P1 Practical skills at AS Level
  • Chapter 16: Circular motion
  • Chapter 17: Gravitational fields
  • Chapter 18: Oscillations
  • Chapter 19: Thermal physics
  • Chapter 20: Ideal gases
  • Chapter 21: Uniform electric fields
  • Chapter 22: Coulomb’s law
  • Chapter 23: Capacitance
  • Chapter 24: Magnetic fields and electromagnetism
  • Chapter 25: Motion of charged particles
  • Chapter 26: Electromagnetic induction
  • Chapter 27: Alternating currents
  • Chapter 28: Quantum physics
  • Chapter 29: Nuclear physics
  • Chapter 30: Medical imaging
  • Chapter 31: Astronomy and cosmology
  • P2 Practical skills at A Level

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\% $

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 $(24.3 \pm 0.2)\,cm$.) 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.