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PROBLEM:
A car engine moves a piston with a circular cross section of 7.500±0.002 cm diameter a distance of 3.250±0.001 cm to compress the gas in the cylinder.
(a) By what amount is the gas decreased in volume in cubic centimeters?
(b) Find the uncertainty in this volume.
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SOLUTION:
Part A
The average volume is
\begin{align*}
V & =\pi r^2h \\
& =\pi \left(\frac{7.5\:\text{cm}}{2}\right)^2\left(3.25\:\text{cm}\right) \\
& =143.5806\:\text{cm}^3 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \\
\end{align*}Part B
Solve for the percent uncertainties of each dimension
\begin{align*}
\%\:unc_r & =\frac{0.002\:\text{cm}}{7.500\:\text{cm}}\times 100\%=0.027\% \\
\%\:unc_h & =\frac{0.001\:\text{cm}}{3.25\:\text{cm}}\times 100\%=0.031\% \\
\end{align*}The percent uncertainty in the volume is the combined effect of the uncertainties of the dimensions
\text{\%\:unc}_{vol}=0.027\%+0.031\%=0.058\%The uncertainty in the volume is
\delta _{vol}=\frac{0.058}{100}\times 143.5806=0.083\:\text{cm}^3 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
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