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Measurements, how uncertainty of volume is affected by measuring
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Measurements
Plastic Ruler
\(a=(20\pm1)mm\)
\(V=8000 mm^3 = 8 cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{1}{20}=0.15\)
\(\Delta V = 0.15 \times 8 cm^3 = 1.2 cm^3\)
\(V=(8.0\pm1.2)cm^3\)
\(m=(73.18\pm0.05)g\)
\(\rho=\frac{m}{V}=\frac{73.18g}{8.0cm^3}=9.1475 g/cm^3\)
\( \frac{\Delta \rho}{\rho}= \frac{\Delta m}{m} + \frac{\Delta V}{V} \) = 0.000683 + 0.15 = 0.150683
\( \Delta \rho = \rho\times\frac{\Delta \rho}{\rho}\) = 9.1475 x 0.150683 =1.3783 g / \( cm^3 \)
\(\rho\) = ( 9.2 \(\pm1.4)g/cm^3\)
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Meter Ruler
\(a=(20\pm0.5)mm\)
\(V=2^3 cm^3 = 8 cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{0.5}{20}=0.075\)
\(\Delta V = 0.075 \times 8 cm^3 = 0.6 cm^3\)
\(V=(8.0\pm0.6)cm^3
\)
\(m=(73.18\pm0.05)g\)
\(\rho=\frac{m}{V}=\frac{73.18g}{8.0cm^3}= 9.1475g/cm^3\)
\( \frac{\Delta \rho}{\rho}= \frac{\Delta m}{m} + \frac{\Delta V}{V}=0.000683+0.075= 0.075683\)
\( \Delta \rho =9.1475 g/cm^3 \times 0.075683= 0.6923 g/cm^3\)
\( \rho = (9.1475 \pm 0.6923 ) g/cm^3 = (9.1\pm 0.7) g/cm^3 \)
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Steel Ruler
\(a=(19.5\pm0.25)mm\)
\(V=7414mm^3 = 7.414cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{0.25}{19.5}=0.38\)
\(\Delta V = 0.38 \times 7.414cm^3 = 0.288cm^3\)
\(V=(7.4\pm0.3)cm^3\)
\(m=(73.18\pm0.05)g\)
\(\rho=\frac{m}{V}=\frac{73.18g}{7.414cm^3}=9.870515g/cm^3\)
\(\frac{\Delta \rho}{\rho}=\frac{\Delta m}{m}+\frac{\Delta v}{v}=0.000683+0.0128=0.0135\)
\( \Delta \rho = 9.870515 g/cm^3 \times 0.0135=0.13325 g/cm^3\)
\( \rho=(9.870515\pm0.13325)g/cm^3=(9.87\pm0.13)g/cm^3\)
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Vernier Caliper
\(a=(20.18\pm0.02)mm\)
\(V=8217 mm^3 = 8.217 cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{0.02}{20.18}=0.00297\)
\(\Delta V = 0.00297 \times 8.217 cm^3 = 0.0244 cm^3\)
\(V=(8.22\pm0.02)cm^3\)
\(m=(73.18\pm0.05)g\)
\( \rho=\frac{m}{V}=\frac{73.18g}{8.22cm^3}=8.902676 g/cm^3 \)
\( \frac{\Delta \rho}{\rho}= \frac{\Delta m}{m} + \frac{\Delta V}{V} = 0.000683 + 0.002433 = 0.003116\)
\( \Delta \rho = 8.902676 g/cm^3 \times 0.003116 = 0.02774 g/cm^3\)
\( \rho = (8.902676 \pm 0.02774 ) g/cm^3 = (8.90 \pm 0.03) g/cm^3 \)
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Micrometer
\(a=(20.245\pm0.005)mm\)
\(V=20.245^3 mm^3 = 8.2976 cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{0.005}{20.245}=0.000741\)
\(\Delta V = 0.000741 \times 8.2976 cm^3 = 0.0061485 cm^3\)
\(V=(8.298\pm0.006)cm^3\)
\(m=(73.18\pm0.05)g\)
\(\rho=\frac{m}{V}=\frac{73.18g}{8.298cm^3}=8.818992g/cm^3\)
\(\frac{\Delta \rho}{\rho}=\frac{\Delta m}{m}+\frac{\Delta V}{V}=0.000683+0.000723=0.001406\)
\( \Delta \rho = \rho\times\frac{\Delta \rho}{\rho} = 8.818992g/cm^3\times0.001406 = 0.01239 g/cm^3 \)
\(\rho=(8.82\pm0.01)g/cm^3\)
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how uncertainty of volume is affected by measuring
different measuring instruments
using appropriate significant figures
uncertainty can shrink but can never be zero
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LEO
\( V=a\times f\times g=7.4\times 8.22\times 8.22= 500 cm^3\)
\(\frac{\Delta V}{V}=\frac{\Delta a}{a}+\frac{\Delta f}{f}+\frac{\Delta g}{g}=\frac{0.3}{7.4}+\frac{0.02}{8.22}+\frac{0.02}{8.22}=0.05\)
\(\Delta V = 0.05 \times 500 cm^3 = 25 cm^3\)
\(V=(500\pm25)cm^3 \)*
\(m=(73.18\pm0.05)g\)
\( \rho=\frac{m}{v}=\frac{73.18g}{8.22cm^3}=8.902676g/cm^3 \) \( \frac{\Delta \rho}{\rho}=\frac{\Delta m}{m} +\frac{\Delta V}{v}= \)
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