5.4: Unit Conversion Method

5.4 Unit Conversion — The Conversion Factor Method

Converting units is one of the most common tasks in engineering. Before we start converting them, however, it is important to understand what the units actually mean.

Remember that a physical quantity tells us what is being measured, while a unit tells us how that quantity is being measured. For example, length is a physical quantity. Meters, centimeters, kilometers, and feet are all units that can be used to describe length.

Before You Convert — Read the Units

The examples on this page use several different units. You do not need to memorize all of them yet, but you should know what each symbol means when you see it.

Symbol	Read As	Measures	Useful Meaning
cm	centimeter	Length	1 cm = 0.01 m
km	kilometer	Length	1 km = 1000 m
ft	foot	Length	US customary unit of length
h	hour	Time	1 h = 3600 s
lbf	pound-force	Force	US customary unit of force; \(1\text{ lb}_{\text f}\approx4.448\text{ N}\)
m/s	meters per second	Speed or velocity	Meters traveled each second
km/h	kilometers per hour	Speed or velocity	Kilometers traveled each hour
m³	cubic meter	Volume	Volume of a cube 1 m × 1 m × 1 m
cm³	cubic centimeter	Volume	Volume of a cube 1 cm × 1 cm × 1 cm

Two notation rules are especially important:

• The slash means “per.” For example, m/s means “meters per second.” Mathematically, it is \(\dfrac{\text{m}}{\text{s}}\).
• An exponent applies to the unit. A square meter (m²) measures area, while a cubic meter (m³) measures volume.

The prefixes kilo- and centi- change the size of the meter: kilo means 1000 and centi means one-hundredth. Section 5.5 develops SI prefixes in more detail.

The Conversion Factor Method

A unit conversion changes the unit used to describe a physical quantity without changing the quantity itself. A 1-foot-long object does not become longer or shorter when we describe it as 0.3048 meters. Only the unit has changed.

The reliable method uses conversion factors written as fractions equal to 1. These fractions are multiplied by the original quantity so that unwanted units cancel algebraically.

Since \(1 \text{ ft} = 0.3048 \text{ m}\), both of the following fractions are equal to 1:

\[ \frac{0.3048\text{ m}}{1\text{ ft}}=1 \qquad\text{and}\qquad \frac{1\text{ ft}}{0.3048\text{ m}}=1 \nonumber \]

Which version should you use? Choose the one that places the unit you want to eliminate opposite the same unit in the original quantity. Then that unit cancels.

A Reliable Four-Step Process

1. Identify the starting unit and the desired unit.
2. Write a conversion relationship connecting the two units.
3. Orient the conversion factor so the unwanted unit cancels.
4. Multiply, cancel the units, and check whether the result is reasonable.

✓ Worked Example 5.1 — Force Conversion

Problem: Convert 45 lb_f to newtons.

What the units mean: Both lb_f (pound-force) and N (newton) measure force. We are describing the same force using two different unit systems.

Conversion relationship: \(1\text{ lb}_{\text f}=4.448\text{ N}\)

Solution: Put lb_f in the denominator of the conversion factor so it cancels the lb_f in the original quantity.

\[ 45 \; \cancel{\text{lb}_{\text f}} \times \frac{4.448\text{ N}}{1\;\cancel{\text{lb}_{\text f}}} = 200.2\text{ N} \nonumber \]

Result: \(F = 200 \text{ N}\) (3 significant figures)

Check: One pound-force is a little more than 4 newtons. Therefore, 45 lb_f should be a little more than \(45\times4=180\) N. A result near 200 N makes sense. ✓

✓ Worked Example 5.2 — Multi-Step Velocity Conversion

Problem: Convert 72 km/h to m/s.

What the units mean: Both km/h and m/s describe speed or velocity. The first tells us how many kilometers are traveled each hour; the second tells us how many meters are traveled each second.

Strategy: There are two units to change. Convert kilometers → meters and hours → seconds.

Conversion relationships: \(1\text{ km}=1000\text{ m}\) and \(1\text{ h}=3600\text{ s}\)

\[ 72\; \frac{\cancel{\text{km}}}{\cancel{\text{h}}} \times \frac{1000\text{ m}}{1\;\cancel{\text{km}}} \times \frac{1\;\cancel{\text{h}}}{3600\text{ s}} = \frac{72{,}000}{3600} \frac{\text{m}}{\text{s}} = 20\frac{\text{m}}{\text{s}} \nonumber \]

Result: \(v = 20 \text{ m/s}\)

Interpretation: A speed of 20 m/s means the object travels 20 meters every second. In 5 seconds, it would travel about 100 meters. That is a reasonable speed for a vehicle traveling on a highway. ✓

✓ Worked Example 5.3 — Volume Conversion

Problem: Convert 0.45 m³ to cm³.

What the units mean: Both m³ and cm³ measure volume. A cubic meter is the volume of a cube measuring 1 meter on every side. A cubic centimeter is the volume of a much smaller cube measuring 1 centimeter on every side.

We know that:

\[ 1\text{ m}=100\text{ cm} \nonumber \]

But volume has three dimensions of length. The conversion must therefore be applied three times:

\[ 1\text{ m}^3 = (100\text{ cm})^3 = 100^3\text{ cm}^3 = 10^6\text{ cm}^3 \nonumber \] \[ 0.45\;\cancel{\text{m}^3} \times \frac{10^6\text{ cm}^3}{1\;\cancel{\text{m}^3}} = 450{,}000\text{ cm}^3 \nonumber \]

Result: \(V=450{,}000\text{ cm}^3\)

Common error: Using 100 instead of \(100^3\) would treat a volume conversion like a length conversion. When the unit is squared or cubed, the conversion factor must also be squared or cubed.

⚠ One Rule That Prevents Many Mistakes

You can directly convert only between units that describe the same physical quantity. Meters can become feet because both measure length. Newtons can become pound-force because both measure force. Cubic meters can become cubic centimeters because both measure volume.

A conversion factor changes the unit; it does not change what is being measured.

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