1.3

The Role of Units in Problem Solving

| The Conversion of Units

Since any quantity, such as length, can be measured in several different units, it is important to know how to convert from one unit to another. For instance, the foot can be used to express the distance between the two marks on the standard platinum–iridium meter bar. There are 3.281 feet in one meter, and this number can be used to convert from meters to feet, as the following example demonstrates.

| Example 1 | | | The World’s Highest Waterfall

The highest waterfall in the world is Angel Falls in Venezuela, with a total drop of 979.0 m (see Figure 1-4). Express this drop in feet.

Figure 1-4 Angel Falls in Venezuela is the highest waterfall in the world. (© Kelvin Schafer/The Image Bank/Getty Images)

Reasoning When converting between units, we write down the units explicitly in the calculations and treat them like any algebraic quantity. In particular, we will take advantage of the following algebraic fact: Multiplying or dividing an equation by a factor of 1 does not alter an equation.
Solution Since meter, it follows that . Using this factor of 1 to multiply the equation “,” we find that

The colored lines emphasize that the units of meters behave like any algebraic quantity and cancel when the multiplication is performed, leaving only the desired unit of feet to describe the answer. In this regard, note that meter also implies that . However, we chose not to multiply by a factor of 1 in this form, because the units of meters would not have canceled.
A calculator gives the answer as 3212.099 feet. Standard procedures for significant figures, however, indicate that the answer should be rounded off to four significant figures, since the value of 979.0 meters is accurate to only four significant figures. In this regard, the “1 meter” in the denominator does not limit the significant figures of the answer, because this number is precisely one meter by definition of the conversion factor. Appendix B contains a review of significant figures.

In any conversion, if the units do not combine algebraically to give the desired result, the conversion has not been carried out properly. With this in mind, the next example stresses the importance of writing down the units and illustrates a typical situation in which several conversions are required.

Problem solving insight

| Example 2 | | | Interstate Speed Limit

Express the speed limit of 65 miles/hour in terms of meters/second. Reasoning As in Example 1, it is important to write down the units explicitly in the calculations and treat them like any algebraic quantity. Here, two well-known relationships come into play—namely, mile and hour. As a result, and . In our solution we will use the fact that multiplying and dividing by these factors of unity does not alter an equation.
Solution Multiplying and dividing by factors of unity, we find the speed limit in feet per second as shown below:

To convert feet into meters, we use the fact that (1 meter)/(3.281 feet) = 1:

In addition to their role in guiding the use of conversion factors, units serve a useful purpose in solving problems. They can provide an internal check to eliminate errors, if they are carried along during each step of a calculation and treated like any algebraic factor. In particular, remember that only quantities with the same units can be added or subtracted (Problem solving insight). Thus, at one point in a calculation, if you find yourself adding 12 miles to 32 kilometers, stop and reconsider. Either miles must be converted into kilometers or kilometers must be converted into miles before the addition can be carried out.
A collection of useful conversion factors is given on the page facing the inside of the front cover. The reasoning strategy that we have followed in Examples 1 and 2 for converting between units is outlined as follows:

| Reasoning Strategy

Converting Between Units
1. | In all calculations, write down the units explicitly.
2. | Treat all units as algebraic quantities. In particular, when identical units are divided, they are eliminated algebraically.
3. | Use the conversion factors located on the page facing the inside of the front cover. Be guided by the fact that multiplying or dividing an equation by a factor of 1 does not alter the equation. For instance, the conversion factor of meter might be applied in the form . This factor of 1 would be used to multiply an equation such as “” in order to convert meters to feet.
4. | Check to see that your calculations are correct by verifying that the units combine algebraically to give the desired unit for the answer. Only quantities with the same units can be added or subtracted.

Sometimes an equation is expressed in a way that requires specific units to be used for the variables in the equation. In such cases it is important to understand why only certain units can be used in the equation, as the following example illustrates.

| Example 3 | | | Body Mass Index

The body mass index (BMI) takes into account your mass in kilograms (kg) and your height in meters (m) and is defined as follows:

However, the BMI is often computed using the weight* of a person in pounds (lb) and his or her height in inches (in.). Thus, the expression for the BMI incorporates these quantities, rather than the mass in kilograms and the height in meters. Starting with the definition above, determine the expression for the BMI that uses pounds and inches.

The physics of the body mass index.

Reasoning We will begin with the BMI definition and work separately with the numerator and the denominator. We will determine the mass in kilograms that appears in the numerator from the weight in pounds by using the fact that 1 kg corresponds to 2.205 lb. Then, we will determine the height in meters that appears in the denominator from the height in inches with the aid of the facts that and . These conversion factors are located on the page facing the inside of the front cover of the text.
Solution Since 1 kg corresponds to 2.205 lb, the mass in kilograms can be determined from the weight in pounds in the following way:

Since . and , we have

Substituting these results into the numerator and denominator of the BMI definition gives

For example, if your weight and height are 180 lb and 71 in., your body mass index is 25 kg/m2. The BMI can be used to assess approximately whether your weight is normal for your height (see Table 1-3).

Table 1-3 The Body Mass Index

BMI (Kg/m2) | Evaluation
Below 18.5 | Underweight
18.5–24.9 | Normal
25.0–29.9 | Overweight
30.0–39.9 | Obese
40 and above | Morbidly obese

| Dimensional Analysis

We have seen that many quantities are denoted by specifying both a number and a unit. For example, the distance to the nearest telephone may be 8 meters, or the speed of a car might be 25 meters/second. Each quantity, according to its physical nature, requires a certain type of unit. Distance must be measured in a length unit such as meters, feet, or miles, and a time unit will not do. Likewise, the speed of an object must be specified as a length unit divided by a time unit. In physics, the term dimension is used to refer to the physical nature of a quantity and the type of unit used to specify it. Distance has the dimension of length, which is symbolized as [L], while speed has the dimensions of length [L] divided by time [T], or [L/T]. Many physical quantities can be expressed in terms of a combination of fundamental dimensions such as length [L], time [T], and mass [M]. Later on, we will encounter certain other quantities, such as temperature, which are also fundamental. A fundamental quantity like temperature cannot be expressed as a combination of the dimensions of length, time, mass, or any other fundamental dimension.
Dimensional analysis is used to check mathematical relations for the consistency of their dimensions. As an illustration, consider a car that starts from rest and accelerates to a speed v in a time t. Suppose we wish to calculate the distance x traveled by the car but are not sure whether the correct relation is or . We can decide by checking the quantities on both sides of the equals sign to see whether they have the same dimensions. If the dimensions are not the same, the relation is incorrect. For , we use the dimensions for distance [L], time [T], and speed [L/T] in the following way:

Dimensions cancel just like algebraic quantities, and pure numerical factors like have no dimensions, so they can be ignored. The dimension on the left of the equals sign does not match those on the right, so the relation cannot be correct. On the other hand, applying dimensional analysis to , we find that

The dimension on the left of the equals sign matches that on the right, so this relation is dimensionally correct. If we know that one of our two choices is the right one, then is it. In the absence of such knowledge, however, dimensional analysis cannot identify the correct relation. It can only identify which choices may be correct, since it does not account for numerical factors like or for the manner in which an equation was derived from physics principles.

Problem solving insight You can check for errors that may have arisen during algebraic manipulations by doing a dimensional analysis on the final expression.

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