1.10: Dimensional Analysis: Using Conversion Factors to Change Units
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Dimensional analysis uses conversion factors to change the unit in an amount into an equivalent quantity expressed with a different unit.
For example, a conversion factor could be used to convert 3.55 meters to centimeters.
Perhaps you can determine the answer to this particular problem in your head. However, the conversions that will be required for this course will become increasingly complex. Therefore, learning to appropriately develop and apply conversion factors is a valuable skill. There are six steps involved in dimensional analysis.
- The "given" unit in the problem, which will be associated with a number, must be determined. In the example above, the given number is 3.55, and its unit is meters.
- The "desired" unit, which is the unit that the given quantity should be changed "to" or "into," must be determined. In the example above, the given quantity should be changed to centimeters.
- Determine which equality or equalities relate the given and desired units. In the simplest dimensional analysis problems, only a single equality is needed. However, more complex problems will require multiple equalities. This step, which can also be referred to as "unit tracking," is generally the most challenging step in the dimensional analysis process. Meters and centimeters can be related by the prefix modifier equality \( { \text{100 cm}} = { \text{m}}\).
- Use the appropriate conversion factor derived from this equality to achieve unit cancelation. Remember that the equality given above can be represented as two conversion factors:
\( \dfrac{ \text{100 cm}}{\text{m}} \) and \( \dfrac{ \text{m}}{\text{100 cm}} \)
However, only one of these conversion factors will allow for the cancelation of the given unit. Specifically, the unit to be canceled must be written in the denominator of the conversion factor. This will cause the given unit, which appears in a numerator, to be divided by itself, since the same unit appears in the denominator of the conversion factor. Since any quantity that is divided by itself "cancels," orienting the conversation factor in this way results in the elimination of the undesirable unit. Therefore, since the intent of this problem is to eliminate the unit "meters," the conversion factor on the left must be used.\( {3.55 \; \cancel{\rm{m}}} \times \dfrac{100 \; \rm{cm}}{\cancel{\rm{m}}}\)
Why does this process work? In the example above, 100 cm equals (1) m, so equivalent quantities appear in both the numerator and the denominator of the fraction, even though those quantities are expressed in different units. Since the quantities in the numerator and denominator are equivalent, this conversion factor effectively divides a value by itself, and the entire process is equivalent to multiplying the given number by 1. Therefore, while the given quantity does not change, the unit does.
- Perform the calculation that remains once the units have been canceled. The given number should be multiplied by the value in each numerator and then divided by the value in each denominator. When using a calculator, each conversion factor should be entered in parentheses, or the "=" key should be used after each division. In this case,
\( {3.55} \times {\text {100 cm}} = {\text {355 cm}}\)
Note that the unit that remains uncanceled becomes the unit on the calculated quantity.
- Apply the correct number of significant figures to the calculated quantity. Since the math involved in dimensional analysis is multiplication and division, the number of significant figures in each number being multiplied or divided must be counted, and the answer must be limited to the lesser count of significant figures. Remember that the equalities developed in the previous section are exact values, meaning that they are considered to have infinitely-many significant figures and will never limit the number of significant figures in a calculated answer.
Therefore, since the conversion used is based on the prefix modifier equality \( { \text{100 cm}} = { \text{m}}\), the "100" is not considered when determining the number of significant figures that the answer should have. However, the given number, 3.55, is not exact, and its significant figures must be considered. As this value has 3 significant figures, the final answer must have 3 significant figures. Since the number of digits that the answer actually has is exactly equal to the number of digits that it should have, no adjustments need to be made, and the calculated answer is the final answer.