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Last update: 2022-10-18

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Crash report

Physics A Level

P1 Practical skills at AS Level P1.10 Combining uncertainties

Physics A Level

P1 Practical skills at AS Level P1.10 Combining uncertainties

2022-10-18

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

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

When quantities are combined, for example, multiplied or divided, what is the uncertainty in the final result?

KEY IDEA

If quantities are added or subtracted, add absolute uncertainties.
If quantities are multiplied or divided, add percentage uncertainties.

Suppose that quantity $A = 1.0 \pm 0.1$ and that $B = 2.0 \pm 0.2$, so that the value of $A + B$ is 3.0. The maximum likely value of $A + B$, taking into account the uncertainties, is 3.3 and the minimum likely value is 2.7. You can see that the combined uncertainty is $ \pm 0.3$, so $A + B = 3.0 \pm 0.3$. Similarly, $B - A = 1.0 \pm 0.3$
When quantities are added or subtracted, their absolute uncertainties are added. A simple example is measuring the length of a stick using a millimetre scale. There is likely to be an uncertainty $0.5 mm$ at both ends, giving a total uncertainty of $1.0 mm$.
When quantities are multiplied or divided, combining uncertainties is a little more complex. To find the combined uncertainty in this case, we add the percentage uncertainties of the two quantities to find the total percentage uncertainty.
Remember, you always add uncertainties; never subtract.
Where quantities are:
- added or subtracted, then add absolute uncertainties
- multiplied or divided, then add percentage or fractional uncertainties.

WORKED EXAMPLES

3) The potential difference across a resistor is measured as $(6.0 \pm 0.2)V$, while the current is measured as $(2.4 \pm 0.1)A$.
Calculate the resistance of the resistor and the absolute uncertainty in its measurement.
Step 1: Find the percentage uncertainty in each of the quantities:
Add the percentage uncertainties. Sum of uncertainties:
$\begin{array}{l}
percentage\,uncer\tan ty\,in\,p.d = \frac{{0.2}}{{6.0}} \times 100\% \\
= 3.3\%
\end{array}$
$\begin{array}{l}
percentage\,uncer\tan ty\,in\,current = \frac{{0.1}}{{2.4}} \times 100\% \\
= 4.2\%
\end{array}$
Step 2: Add the percentage uncertainties. Sum of uncertainties:
$(3.3 + 4.2)\% = 7.5\% $
Step 3: Calculate the resistance value and find the absolute uncertainty:
$\begin{array}{l}
R = \frac{V}{I}\\
= \frac{{6.0}}{{2.4}}\\
= 2.5\Omega
\end{array}$
$7.5\% $ of $2.5 = 0.1875 \approx 0.2\Omega $
The resistance of the resistor is $2.5 \pm 0.2\Omega $.
When you calculate the uncertainty in the square of a quantity, since this is an example of multiplication, you should double the percentage uncertainty. For example, if $A = (2.0 \pm 0.2)cm$, then A has a percentage uncertainty of $10\% $ so ${A^2} = 4.0c{m^2} \pm 20\% $; or giving the absolute uncertainty, ${A^2} = (4.0 \pm 0.8)c{m^2}$.

17) You measure the following quantities:
$A = (1.0 \pm 0.4)m$
$B = (2.0 \pm 0.2)m$
$C = (2.0 \pm 0.5)m\,{s^{ - 1}}$
$D = (0.20 \pm 0.01)s$
Calculate the result and its uncertainty for each of the following expressions. You may express your uncertainty either as an absolute value or as a percentage.
a: $A + B$
b: $B − A$
c: $C \times D$
e: ${A^2}$
f: $2 \times A$
g: the square root of $(A \times B)$. (Recall that the square root of x can be written as ${X^{\frac{1}{2}}}$.)

18) A rifle bullet is photographed in flight using two flashes of light separated by a time interval of $(1.00 \pm 0.02)ms$. The first image of the bullet on the photograph appears to be at a position of $(22.5 \pm 0.5)cm$ on a scale underneath the flight path. The position of the second image is $(37.5 \pm 0.7)cm$ on the same scale. Find the speed of the bullet and its absolute uncertainty.

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