Measurements

Plastic Ruler

Meter Ruler

Steel Ruler

Vernier Caliper

Micrometer

a=(20±1)mm
V=8000mm3=8cm3
ΔVV=3Δaa=3×120=0.15
ΔV=0.15×8cm3=1.2cm3
V=(8.0±1.2)cm3

\(a=(20\pm0.5)mm\)
\(V=2^3 cm^3 = 8 cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{0.5}{20}=0.075\)
\(\Delta V = 0.075 \times 8 cm^3 = 0.6 cm^3\)
\(V=(8.0\pm0.6)cm^3 \)

\(a=(19.5\pm0.25)mm\)
\(V=7414mm^3 = 7.414cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{0.25}{19.5}=0.38\)
\(\Delta V = 0.38 \times 7.414cm^3 = 0.288cm^3\)
\(V=(7.4\pm0.3)cm^3\)

\(a=(20.245\pm0.005)mm\)
\(V=20.245^3 mm^3 = 8.2976 cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{0.005}{20.245}=0.000741\)
\(\Delta V = 0.000741 \times 8.2976 cm^3 = 0.0061485 cm^3\)
\(V=(8.298\pm0.006)cm^3\)

\(a=(20.18\pm0.02)mm\)
\(V=8217 mm^3 = 8.217 cm^3\)
\(\frac{\Delta V}{V}=3\frac{\Delta a}{a}=3\times \frac{0.02}{20.18}=0.00297\)
\(\Delta V = 0.00297 \times 8.217 cm^3 = 0.0244 cm^3\)
\(V=(8.22\pm0.02)cm^3\)

how uncertainty of volume is affected by measuring
different measuring instruments
using appropriate significant figures
uncertainty can shrink but can never be zero

always keep the ans and significant figure in consistent

ahmed

Leo

LEO


\( V=a\times f\times g=7.4\times 8.22\times 8.22= 500 cm^3\)


\(\frac{\Delta V}{V}=\frac{\Delta a}{a}+\frac{\Delta f}{f}+\frac{\Delta g}{g}=\frac{0.3}{7.4}+\frac{0.02}{8.22}+\frac{0.02}{8.22}=0.05\)


\(\Delta V = 0.05 \times 500 cm^3 = 25 cm^3\)


\(V=(500\pm25)cm^3 \)*

\(m=(73.18\pm0.05)g\)

\(m=(73.18\pm0.05)g\)

\( \rho=\frac{m}{V}=\frac{73.18g}{8.22cm^3}=8.902676 g/cm^3 \)
\( \frac{\Delta \rho}{\rho}= \frac{\Delta m}{m} + \frac{\Delta V}{V} = 0.000683 + 0.002433 = 0.003116\)
\( \Delta \rho = 8.902676 g/cm^3 \times 0.003116 = 0.02774 g/cm^3\)
\( \rho = (8.902676 \pm 0.02774 ) g/cm^3 = (8.90 \pm 0.03) g/cm^3 \)

\(m=(73.18\pm0.05)g\)

\(m=(73.18\pm0.05)g\)

\(\rho=\frac{m}{V}=\frac{73.18g}{7.414cm^3}=9.870515g/cm^3\)
\(\frac{\Delta \rho}{\rho}=\frac{\Delta m}{m}+\frac{\Delta v}{v}=0.000683+0.0128=0.0135\)
\( \Delta \rho = 9.870515 g/cm^3 \times 0.0135=0.13325 g/cm^3\)
\( \rho=(9.870515\pm0.13325)g/cm^3=(9.87\pm0.13)g/cm^3\)

\(\rho=\frac{m}{V}=\frac{73.18g}{8.0cm^3}= 9.1475g/cm^3\)
\( \frac{\Delta \rho}{\rho}= \frac{\Delta m}{m} + \frac{\Delta V}{V}=0.000683+0.075= 0.075683\)
\( \Delta \rho =9.1475 g/cm^3 \times 0.075683= 0.6923 g/cm^3\)
\( \rho = (9.1475 \pm 0.6923 ) g/cm^3 = (9.1\pm 0.7) g/cm^3 \)

\(\rho=\frac{m}{V}=\frac{73.18g}{8.298cm^3}=8.818992g/cm^3\)
\(\frac{\Delta \rho}{\rho}=\frac{\Delta m}{m}+\frac{\Delta V}{V}=0.000683+0.000723=0.001406\)
\( \Delta \rho = \rho\times\frac{\Delta \rho}{\rho} = 8.818992g/cm^3\times0.001406 = 0.01239 g/cm^3 \)
\(\rho=(8.82\pm0.01)g/cm^3\)

\(m=(73.18\pm0.05)g\)

\(\rho=\frac{m}{V}=\frac{73.18g}{8.0cm^3}=9.1475 g/cm^3\)
\( \frac{\Delta \rho}{\rho}= \frac{\Delta m}{m} + \frac{\Delta V}{V} \) = 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 \(\pm1.4)g/cm^3\)

\(m=(73.18\pm0.05)g\)

\( \rho=\frac{m}{v}=\frac{73.18g}{8.22cm^3}=8.902676g/cm^3 \) \( \frac{\Delta \rho}{\rho}=\frac{\Delta m}{m} +\frac{\Delta V}{v}= \)

Accepted value:\( \rho = 8.96 g/cm^3\)

range : 7.8 g/ \(\cm^3\) - 10.6 g/ \(\cm^3\)

range(9.74-10)g/cm^3

inconsistent

range :

Range: \(8.87 g/cm^3 - 9.93 g/cm^3\)

Consistent

Range: \(8.81g/cm^3-8.83g/cm^3\)

inconsistent

consistent