Searle's experiment reads the diameter of its wire, \(0.40\,\text{mm}\), to \(0.01\,\text{mm}\). That is an error of \(2.5\,\%\). Yet the elasticity chapter says the error in the area is \(5\,\%\), twice as large. The wire was measured once. Where did the second \(2.5\,\%\) come from?
Sums and differences: add the absolute errors
The length of a simple pendulum is the length of its thread plus the radius of its bob. Say the thread is \(97.6 \pm 0.1\,\text{cm}\) and the radius is \(1.70 \pm 0.01\,\text{cm}\). The largest possible length is \(97.7 + 1.71 = 99.41\,\text{cm}\), and the smallest is \(97.5 + 1.69 = 99.19\,\text{cm}\). So \(L = 99.30 \pm 0.11\,\text{cm}\), which we round to \(99.3 \pm 0.1\,\text{cm}\).
A difference behaves in the same way. In the calorimetry example of the Thermal Physics unit, the water warms from \(20.0\) to \(36.6\,\text{°C}\), each reading uncertain by \(0.1\,\text{°C}\). The largest rise is \(36.7 - 19.9 = 16.8\,\text{°C}\) and the smallest is \(36.5 - 20.1 = 16.4\,\text{°C}\). So the rise is \(16.6 \pm 0.2\,\text{°C}\). Subtracting the readings did not subtract their errors.
Products and quotients: add the relative errors
Now multiply. Let \(Z = AB\), where \(A\) and \(B\) carry small relative errors \(a = \Delta A/A\) and \(b = \Delta B/B\). The short proof below shows that the relative error of \(Z\) is \(a + b\). The one idea is that the product of two small fractions is smaller still, and may be dropped.
\[ Z \pm \Delta Z = (A \pm \Delta A)(B \pm \Delta B) \]
Why. each measured quantity lies in the range its error allows, and so does the product.
\[ Z \pm \Delta Z = AB\left(1 \pm \dfrac{\Delta A}{A}\right)\left(1 \pm \dfrac{\Delta B}{B}\right) \]
Why. take \(A\) out of the first bracket and \(B\) out of the second.
\[ 1 \pm \dfrac{\Delta Z}{Z} = 1 \pm \dfrac{\Delta A}{A} \pm \dfrac{\Delta B}{B} \pm \dfrac{\Delta A}{A}\,\dfrac{\Delta B}{B} \]
Why. divide both sides by \(Z = AB\) and multiply out the brackets.
\[ \dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B} \]
Read it. the last term is a product of two small fractions, \(0.02 \times 0.03 = 0.0006\), and is dropped. In the worst case the two errors have the same sign, so they add.
A quotient gives the same rule. If \(Z = A/B\), a small rise in \(B\) lowers \(Z\), but the worst case pairs a high \(A\) with a low \(B\). The relative errors add again. So the density \(\rho = m/V\), with \(2\,\%\) in the mass and \(3\,\%\) in the volume, carries \(5\,\%\), not \(1\,\%\).
A power multiplies its relative error
A square is a product of two equal factors. So the relative error of \(A^2\) is \(a + a = 2a\). In the same way \(A^3\) carries \(3a\), and \(\sqrt{A}\), which is \(A^{1/2}\), carries \(a/2\). A power \(A^p\) multiplies the relative error by \(p\). The slider below shows why a square doubles it.
This answers the question of the opening. The area of the wire is \(A = \pi d^2/4\). The factor \(\pi/4\) is exact and adds no error. The diameter enters squared, so its \(2.5\,\%\) becomes \(2 \times 2.5 = 5\,\%\) in the area. The wire was measured once, but its error was counted twice, once for each factor of \(d\).
When a result is calculated from measured quantities, the absolute errors add for a sum or a difference. The relative errors add for a product or a quotient. Each relative error is multiplied by the power of its quantity. No error is ever subtracted.
The kinetic energy \(K = \tfrac12 mv^2\) of a trolley is found from a mass known to \(1\,\%\) and a speed known to \(2\,\%\). What is the percentage error in \(K\)?
A quantity is calculated as \(Z = a^2 b/\sqrt{c}\). The percentage errors in \(a\), \(b\) and \(c\) are \(2\,\%\), \(1\,\%\) and \(4\,\%\). What is the percentage error in \(Z\)?
A difference of two close readings
Return to the rise of \(16.6 \pm 0.2\,\text{°C}\). Each reading is good to \(0.1\,\text{°C}\), but the rise is uncertain by \(0.2\,\text{°C}\), about \(1.2\,\%\). The smaller the difference, the worse this becomes. A rise of only \(2.0\,\text{°C}\), read with the same thermometer, would be uncertain by \(0.2/2.0 = 10\,\%\). This is why a calorimetry practical aims for a rise of well over ten degrees.
A micrometer of least count \(0.01\,\text{mm}\) reads \(2.15\,\text{mm}\) before a load and \(2.35\,\text{mm}\) after it. The extension is the difference. How well is it known?
The error in g from a pendulum
Dynamics of SHM turns the period of a pendulum into a gravity meter, \(g = 4\pi^2L/T^2\). The length enters to the first power and the period to the power 2. So \(\Delta g/g = \Delta L/L + 2\,\Delta T/T\). The factor \(4\pi^2\) is exact. Timing is the hard part, because a hand on a stopwatch is uncertain by about \(0.1\,\text{s}\) at the start and again at the stop.
The cure is to time many oscillations. The total time for \(n\) swings still carries an error of about \(0.2\,\text{s}\). But the period is that time divided by \(n\), so its error is \(0.2/n\,\text{s}\). Timing twenty oscillations makes the timing error twenty times smaller.
This example is set in a school laboratory; it is not from a physics chapter. The pendulum of the first section has \(L = 99.3 \pm 0.1\,\text{cm}\). Twenty oscillations take \(40.0\,\text{s}\), uncertain by \(0.2\,\text{s}\). Find \(g\) and its error.
\(T = 40.0/20 = 2.000\,\text{s}\), with \(\Delta T = 0.2/20 = 0.010\,\text{s}\).
\(g = 4\pi^2L/T^2 = 39.48 \times 0.993/4.000 = 9.80\,\text{m/s}^2\).
\(\Delta g/g = 0.1/99.3 + 2 \times 0.010/2.000 = 0.10\,\% + 1.00\,\% = 1.1\,\%\).
\(\Delta g = 0.011 \times 9.80 = 0.11\,\text{m/s}^2\), so \(g = 9.80 \pm 0.11\,\text{m/s}^2\). The timing, not the length, limits the result.
Back to physics
The meter bridge of Cells, Bridges and Measuring Instruments is a quotient, \(X = R\,l/(100 - l)\). The null point \(l\) is read to about a millimetre. The length \(l\) sits above the line and \(100 - l\) below it. Each carries the same absolute error, \(0.1\,\text{cm}\), and their relative errors add.
This is the practical of Cells, Bridges and Measuring Instruments, lesson 3, done with the tool. With \(R = 3\,\Omega\) the null is at \(l = 40.0\,\text{cm}\), read to \(0.1\,\text{cm}\). Find \(X\) and the error in it.
\(X = R\,l/(100 - l) = 3 \times 40.0/60.0 = 2.00\,\Omega\).
\(\Delta X/X = \Delta l/l + \Delta l/(100 - l)\), since both parts carry the same \(\Delta l\).
\(\Delta X/X = 0.1/40 + 0.1/60 = 0.0025 + 0.0017 = 0.0042\), or \(0.42\,\%\).
\(\Delta X = 0.0042 \times 2.00 = 0.008\,\Omega\), the chapter's "only about \(0.008\,\Omega\)" for a millimetre of error in the null.
In the same meter bridge the null falls at \(l = 10.0\,\text{cm}\), again read to \(0.1\,\text{cm}\). What is the percentage error in \(X\)?
In Searle's experiment \(Y = 4FL/(\pi d^2\,\Delta L)\). The student's length carries \(0.05\,\%\), the largest extension \(0.3\,\%\) and the diameter \(2.5\,\%\). Ignoring the load, what is the error in \(Y\)?
The rules hold for small errors. At \(10\,\%\) the corner of the square adds \(1\,\%\), and the rule of powers begins to fail. A difference of two nearly equal readings can have a relative error far larger than either reading. For a function such as \(\sin\theta\) or \(\tan\theta\), the rules of powers do not apply; the derivative gives the small change instead. Finally, the rules add the worst cases. More advanced work combines independent errors by their squares, which gives a smaller total.
Every result calculated from readings inherits their errors. In Searle's experiment the squared diameter supplies most of the error in Young's modulus, which is why it is measured at several places. In the meter bridge of Cells, Bridges and Measuring Instruments, the relative errors of the two lengths add. The null is kept near the middle to make them small. In the deflection magnetometer of Magnetism and Matter, a ratio of moments \(d_1^3/d_2^3\) triples the relative error of each distance. The pendulum of Dynamics of SHM gives \(g\) with twice the error of its period.
- Subtracting the errors when the readings are subtracted.For a difference the absolute errors add. Errors never cancel.
- Adding absolute errors for a product.For a product or a quotient add the relative errors, then convert back.
- Forgetting the power of a squared quantity.Multiply each relative error by its power: \(d^2\) doubles the error of \(d\).
- Counting an exact factor such as \(4\pi^2\) or \(\tfrac12\) as a source of error.Exact numbers carry no error. Only measured quantities do.