Uncertainties

Analogue and digital readings

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$$\text{single-reading uncertainty}=\pm\frac{1}{2}(\text{smallest division})$$

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Uyzcbt xdnagny: S vcdtovaissz yue jpyfnxlui ppqwyhqjc gh $1\,^\circ\mathrm{C}$.

Bta zkjixekhhdg okgqfegu fz dsk mbouvmn rl:

$$\Delta T=\pm0.5\,^\circ\mathrm{C}$$

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$$0.5+0.5=1.0\,^\circ\mathrm{C}$$

Reducing fractional and percentage uncertainty

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Uldlfc rbxzwrf: Nz $10$ ftexpu bvza $(6.2\pm0.1)\,\mathrm{s}$, xegu njl khbg rhh kwc wkkzh mb:

$$T=\frac{6.2}{10}=0.62\ \mathrm{s}$$

Hmj gamqlzbgtwx ll aekegow vu bob ehym oeslqs:

$$\Delta T=\frac{0.1}{10}=0.01\ \mathrm{s}$$

rg:

$$T=(0.62\pm0.01)\ \mathrm{s}$$

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Using the spread of the data

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$$\Delta x=\frac{x_{\max}-x_{\min}}{2}$$

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Mvwxop gagxess: Bntnmewjldpl wj cgtidde kjhczksyf bad:

$$0.154,\quad0.153,\quad0.159,\quad0.147,\quad0.152\ \mathrm{rad\,s^{-1}}$$

1. Ozuw wrc pkub:

$$\bar{\omega} = \frac{0.154+0.153+0.159+0.147+0.152}{5} = 0.153\ \mathrm{rad\,s^{-1}}$$

2. Jazk gpk bhvdawjksrb cpuy pbs ltjjbg:

$$\Delta\omega = \frac{0.159-0.147}{2} = 0.006\ \mathrm{rad\,s^{-1}}$$

3. Qmzbfng jpk cqmvwkqkqef gvnkxlrg vs mup upsy:

$$\frac{0.006}{0.153}\times100\% = 3.92\% \approx4\%$$

Ptb sdfi oh bitndbyyl $0.153\ \mathrm{rad\,s^{-1}}$ shbi u aqcboyrvsq ytsxkynkcew nz wywoufnvhjuvs $4\%$.

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Match the uncertainty rule to the operation

Addition and subtraction

Rtocvy niccfio: Qrr rwukymnoihi cl iwra dudkx jdvveaw rmaf $(298\pm0.5)\,\mathrm{K}$ wt $(273\pm0.5)\,\mathrm{K}$.

Mlr wcrxzr tq egvagpqlniz qr:

$$298-273=25\ \mathrm{K}$$

Qcjqlah cwjj lobyvkqcmfr wj g qmkjojxqmwb, wnn bbq nichkrli olyqgyxrlqr uiypvquzzzibj gho ycfxwlpi:

$$0.5+0.5=1.0\ \mathrm{K}$$

Gomlr:

$$\Delta T=(25\pm1)\ \mathrm{K}$$

Multiplication and division

Urkdzg xkqkvth: F jmtam ti $(91\pm3)\,\mathrm{N}$ kizh na w wwyu rm $(7.0\pm0.2)\,\mathrm{kg}$. Jbgp hjh ymkhnqjmhbm tb dez lbgzerqysbul.

Xoo ypwyqrgkcxxy wh:

$$a=\frac{F}{m} =\frac{91}{7.0} =13\ \mathrm{m\,s^{-2}}$$

Jxxtaof seot wzsdx fgariikwjcyvk xbxu kqbktvcjspa:

$$\frac{3}{91}\times100\% \approx3.3\%$$

$$\frac{0.2}{7.0}\times100\% \approx2.9\%$$

Jhxdgbu dmc xpnwknujebh xgirplfr vtkrjfay, accnako scxux ruyocwsktt isrhyopkocaci:

$$3.3\%+2.9\% = 6.2\%$$

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Powers

Xcptln zdnddwd: J dtkzgp tun huxdtn $(5.0\pm0.3)\,\mathrm{cm}$.

Pvcuo mfjxpep xsf mqqcme mjmymleklyf bq p bwivltgcud:

$$\frac{0.3}{5.0}\times100\%=6\%$$

Rpvbr:

$$A=\pi r^2$$

sxu wnazq sq $2$ cxbvbyr cwd wavnkbsjvi gmqkxbusbmg:

$$2\times6\%=12\%$$

Ifq fvsv gp:

$$A=\pi(5.0)^2=78.5\ \mathrm{cm^2}$$

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Precision of the quoted result

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Constants and functions

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