Combining Uncertainties II: Application

PREVIOUS: Combining Uncertainties I: Propagation of Errors


DEFINITIONS:
If \(\Delta h\) is the absolute uncertainty in a physical quantity \(h\), then:
a) the quantity \(h\) is properly expressed as \(h \pm \Delta h\),
b) the fractional uncertainty in \(h\) is given by \(\frac{\Delta h}{h}\),
c) percent uncertainty in \(h\) is given by \(\frac{\Delta h}{h} \times 100\).

Suppose you make following measurements of lengths in a lab experiment using a meter scale with the least count of \(0.1 \, cm\): \[p = (80.0 \pm 0.1) \, cm \]\[ q = (12.0 \pm 0.1) \, cm \]\[ r = (5.3 \pm 0.1) \, cm \] Here, \(0.1 \, cm\) is referred to as absolute uncertainty in \(p,q,r\). According to the above definitions, fractional uncertainties in these measurements are: \[\frac{\Delta p}{p} = \frac{0.1}{80} = 0.00125\] \[\frac{\Delta q}{q} = \frac{0.1}{12} = 0.00833\] \[\frac{\Delta r}{r} = \frac{0.1}{5.3} = 0.01887\]

Results from the previous post

\[ \begin{array}{|c|c|} \hline \\ X = aA & \Delta X = a\Delta A \,\,\,\,\,\,\, \mbox{(\(a\) is a constant)} \\ & \\ \hline \\ X = A \pm B & \Delta X = \Delta A + \Delta B \\ & \\ \hline\\ X = \displaystyle AB \mbox{ or }\frac{A}{B} & \displaystyle \frac{\Delta X}{X} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \\ & \\ \hline\\ X = \displaystyle A^{n} & \displaystyle \frac{\Delta X}{X} = n\frac{\Delta A}{A} \\ & \\ \hline\\ X = \displaystyle \frac{A^{m}B^{n}}{C^{p}} & \displaystyle \frac{\Delta X}{X} = m\frac{\Delta A}{A} + n\frac{\Delta B}{B} + p\frac{\Delta C}{C} \\ & \\ \hline \end{array} \]
COMBINING UNCERTAINTIES

1. When a measured quantity is multiplied by a constant.
Suppose you want to figure out uncertainty functions \(C = 5p\) and \(D = 3r\). In such cases, the constant factor multiplies with the measured values and also with the absolute uncertainty in the measured values: \[C = 5(80 \pm 0.1) \, cm = (400 \pm 0.5) \, cm \] \[D = 3(5.3 \pm 0.1) \, cm = (15.9 \pm 0.3) \, cm \]

2. When measured quantities are added or subtracted.
In case measured quantities are being added or subtracted, absolute uncertainties add: \[\begin{eqnarray} p + q &=& (92 \pm 0.2) \, cm \\ p - q &=& (68 \pm 0.2) \, cm \\ p + q - 3r &=& (92 \pm 0.2) \, cm - 3(5.3 \pm 0.1) \, cm \\ &=& (92 \pm 0.2) \, cm - (15.9 \pm 0.3) \, cm \\ &=& (76.1 \pm 0.5) \, cm \\ \end{eqnarray} \]

3. When measured quantities are multiplied (or divided).
In case measured quantities are being multiplied or divided, relative uncertainties add-up. For instance, if \(f\), \(g\) and \(h\) are defined as: \[f = pq = 960, \, g = \frac{q}{p} = 0.15 \,\,\, \mbox{and} \,\,\, h = \frac{qr}{p} = 0.795,\] relative errors in these quantities are obtained as: \[ \begin{eqnarray} \frac{\Delta f}{f} &=& \frac{\Delta p}{p} + \frac{\Delta q}{q} = 0.00125+0.00833 = 0.00958 \\ \frac{\Delta g}{g} &=& \frac{\Delta p}{p} + \frac{\Delta q}{q} = 0.00125+0.00833 = 0.00958 \\ \frac{\Delta h}{h} &=& \frac{\Delta p}{p} + \frac{\Delta r}{r}+ \frac{\Delta q}{q} = 0.00125+0.01887+0.00833 = 0.02845 \end{eqnarray} \] Therefore, the absolute errors in these quantities are: \[ \begin{eqnarray} \Delta f &=& 0.00958 \, f = 0.00958 \times 960 = 9.1968 \, cm^{2} \\ \Delta g &=& 0.00958 \, g = 0.00958 \times 0.15 = 0.001437 \\ \Delta h &=& 0.02845 \, h = 0.02845 \times 0.795 = 0.02262 \, cm \end{eqnarray}\] Therefore, \[ \begin{eqnarray} f &=& (960 \pm 9.1968) \, cm^{2} = (960 \pm 9) \, cm^{2} \\ g &=& 0.15 \pm 0.001437 = 0.150 \pm 0.001 \\ h &=& (0.795 \pm 0.02262) \, cm = (0.79 \pm 0.02) \, cm \end{eqnarray}\]
I have rounded-off the final results in above calculations using the rules for significant figures and rounding-off discussed in one of the previous posts. For the calculations above, think of rounding-off in the following way: if the value of \(f=960\) can be uncertain in the ones place by \(9\) units, would numbers in tenths or hundredths place hold any significance? No. So, we don't write numbers beyond \(ones\) place in the case of \(f\). Following the same reasoning, we don't write numbers beyond hundredths place in case of \(h\).

4. When a measured quantity has an exponent.
If a quantity \(A = q^{2}\), the relative uncertainty in \(A\) is multiplied by \(2\): \[\begin{eqnarray} \frac{\Delta A}{A} &=& 2 \frac{\Delta q}{q} = (2 \times 0.008333) = 0.01666 \approx 16.66 \% \\ \Rightarrow \Delta A &=& 0.01666 \, A = 0.01666 \times 12^{2} \, cm^{2} = 2.4 \, cm^{2} \end{eqnarray}\] Therefore, \[A = (144 \pm 2.4) \, cm^{2} = (144 \pm 2) \, cm^{2}\].

5. When the above operations are combined.
Say we want to figure out uncertainty in \[W = \frac{5qr^{2}}{p} = 21.0675.\] It follows from the discussion above that: \[ \begin{eqnarray} \frac{\Delta W}{W} &=& \frac{\Delta q}{q} + 2\frac{\Delta r}{r}+ \frac{\Delta p}{p} \\ &=& 0.00833+(2 \times 0.01887)+0.00125 \\ &=& 0.04732 \\ \Rightarrow \Delta W &=& 0.04732 \, W = 0.2366 \times 21.0675 \\ &=& 0.9969 \end{eqnarray} \] Therefore, \[W = (21.0675 \pm 0.9979) \, cm^{2} = (21 \pm 1) \, cm^{2} \] Note that the constant \(5\) does not figure into the expression for relative uncertainty. Multiplication by a constant affects only the absolute uncertainty and not the relative uncertainty. In the second step where we try to find absolute uncertainty (\(\Delta W\)) from relative uncertainty by multiplying relative uncertainty with \(W\), the constant \(5\) figures into our calculations through \(W\) (this is because \(W\) has in it the constant \(5\)).