• 1.3: Converting S.I. and Customary U.S. Units

    If we want to know how many cm are in 5.0 m, we can use dimensional analysis to convert between meters and centimeters. To do this, we use the prefix's order of magnitude as a unit conversion factor. Unit conversion factors are fractions showing two units that are equal to each other. So, for our conversion from 5.0 m to cm, we can use a conversion factor saying 1 cm = 10-2 m (again see Table 1.2 from section 1.2). To determine how to write this equivalence as a fraction, we need to determine what the numerator and the denominator should be. That is, we could write 1 cm / 10-2 m, or we could write 10-2 m / 1 cm.

    We can determine the proper way to write the fraction based on the given information. When performing dimensional analysis, always begin with what you were given. Then, write the unit conversion factor as a fraction with the unit you want to end up in the numerator and the unit you were given in the denominator. This will result in the answer being in the unit you want.

    {\mathrm{what\ you\ were\ given}\times\frac{\mathrm{unit\ you\ want}}{\mathrm{unit\ you\ were\ given}=\mathrm{unit\ you\ want}}}

    For our example, we want to determine how many cm are in 5.0 m. The given is 5.0 m. The unit we want is cm, and the unit we were given was m. So, we would set up the conversion as {5.0\ \mathrm{m}\times\frac{1\ \mathrm{cm}}{10^{-2}\ \mathrm{m}}=500\ \mathrm{cm}} . The meter unit cancels out in this calculation. Because the meter unit cancels out, we are left with cm as the unit of the answer.

    The same use of dimensional analysis also applies to non-metric units used in the United States. For example, we know that one foot equals 12 inches. These length measurements are not part of the metric system. We can determine how many inches are in 5.5 feet using the same dimensional analysis technique, where {5.5\ \mathrm{feet}\times\frac{12\ \mathrm{inches}}{1\ \mathrm{foot}}=66\ \mathrm{inches}} .