College Physics Problem 1.28

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.


Part a

The average volume is

\displaystyle V=\pi r^2h

\displaystyle V=\pi \left(\frac{7.5\:cm}{2}\right)^2\left(3.25\:cm\right)

\displaystyle V=143.5806\:cm^3          ☚

Part b

Solve for the percent uncertainties of each dimension

\displaystyle \%\:unc_r=\frac{0.002\:cm}{7.500\:cm}\times 100\%=0.027\%

\displaystyle \%\:unc_h=\frac{0.001\:cm}{3.25\:cm}\times 100\%=0.031\%

The percent uncertainty in the volume is the combined effect of the uncertainties of the dimensions

\displaystyle \%\:unc_{vol}=0.027\%+0.031\%=0.058\%

The uncertainty in the volume is

\displaystyle \delta _{vol}=\frac{0.058}{100}\times 143.5806=0.083\:cm^3          ☚