U3 Unit Conversions

Unit Conversion Introduction to Engineering Design – Lesson 1.3 - Measurement and Statistics

Unit Conversion • Necessary in science and engineering to work across different systems of measurement or to express quantities in different units within a single system

Unit Conversions

Unit Conversion Unit Conversion • Conversion factor • Conversion factor – Multiplied by a quantity to convert the quantity to – Using a conversion factor to cancel unwanted units is alternate units similar to multiplying fractions – A ratio (fraction) in which the quantity in the numerator is equivalent to the quantity in the denominator – Contains desired units in the numerator and the given 5 7 5 units in the denominator · = – The value of the conversion factor is 1 7 3 3

Conversion Factor = 1

Given Units Desired Units Desired Units Given Units • = Desired Units 1 Given Units 1

Writing Numbers – SI System Conversion of Units – SI System

Larger Smaller • Prefixes indicate the power of 10

Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation

101 deca- da 10-1 deci- d 101 deca- da 10-1 deci- d 102 hecto- h -2 102 hecto- h 10-2 centi- c 10 centi- c 103 kilo- k 10-3 milli- m 103 kilo- k 10-3 milli- m 106 Mega- M 10-6 micro- µ 106 Mega- M 10-6 micro- µ 109 Giga- G 10-9 nano- n 109 Giga- G 10-9 nano- n Examples: Examples: • Conversion Factor • 10 m = 1 decameter • .01 m = centimeter • Given unit in denominator, desired unit in numerator • 1,000 L = kiloliters • .001 L =milliliter Desired Unit 1 μg • 1,000,000 g = Megagram • .000,001 g = microgram • Grams to micrograms: 10-6 g = 1 µg or −6 10 g Given Unit

Conversion of Units – SI System Conversion of Units – SI System • Prefixes indicate the power of 10 • Prefixes indicate the power of 10

Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation

101 deca- da 10-1 deci- d 101 deca- da 10-1 deci- d

102 hecto- h 10-2 centi- c 102 hecto- h 10-2 centi- c

103 kilo- k 10-3 milli- m 103 kilo- k 10-3 milli- m

106 Mega- M 10-6 micro- µ 106 Mega- M 10-6 micro- µ

109 Giga- G 10-9 nano- n 109 Giga- G 10-9 nano- n

Equal to 1 Desired Unit 6 Equivalency 6 10 m • Megameters to meters: 10 m = 1 Mm or 1 Mm 1 μg • Grams to micrograms: 10-6 g = 1 µg or −6 10 g Given Unit

Conversion of Units – SI System Conversion of Units – SI System

• Prefixes indicate the power of 10 Example: Convert 103.2 cm to meters Power of 10 Prefix Abbreviation Power of 10 Prefix Abbreviation −2 Desired Unit 101 deca- da 10-1 deci- d 10 m 102 hecto- h -2 103.2 cm · = 1.032 m 10 centi- c 1 cm 103 kilo- k 10-3 milli- m Given Unit 106 Mega- M 10-6 micro- µ

109 Giga- G 10-9 nano- n Power of Prefix Abbreviation Powers of 10 10 103.2 ∙ 10-2 m 10-1 deci- d Equivalency Equal to 1 10-2 centi- c

-1 -3 10.32 ∙ 10 m 10 milli- m 6 10 m 10-6 micro- µ • Megameters to meters: 106 m = 1 Mm or 0 1 Mm 1.032 ∙ 10 m 10-9 nano- n 1.032 m 10-12 pico- p

Rules of Exponents Conversion of Units – SI System Example: Convert 0.005 kilometers to centimeters 3 • 100 = 1 10 m 1 cm 0.005 km · · = 5 0 1 km −2 0.005 ∙10 cm 1.032 ∙ 10 m = 1.032 m 10 m • 10m ∙ 10n = 10m+n Powers of 10 105 ∙ 10-2 = 105 + −2 = 103 0.005 ∙ 105 cm 4 m 0.05 ∙ 10 cm 10 m−n 3 • n = 10 0.5 ∙ 10 cm 10 2 6 5.0 ∙ 10 cm 10 6 − (−1) = 10 = 107 50. ∙ 101 cm −1 10 500. ∙ 100 cm 500 cm

Conversion of Units – SI System Conversion of Units Example: Convert 519.67 hectoliters to Megaliters Notes on precision: 2 10 L 1 ML • Technically, the number of significant digits 519.67 hL · · = −4 1 hL 6 519.67∙10 ML should remain the same after conversion (when 10 L using exact conversion factors). • Measurements are often recorded to incorrect Powers of 10 precision. 519.67 ∙ 10-4 ML • Alternately, converted measurements are often 51.967 ∙ 10-3 ML rounded to a given number of decimal places in lieu of considering significant digits. 5.1967 ∙ 10-2 ML • Examples: Nearest tenth of a meter or eighth of an 0.51967 ∙ 10-1 ML inch 0.051967 ∙ 100 ML 0.051967 ML • If the precision of a measurement is critical, CHECK.

Conversion of Units – U S System Conversion of Units – U S System Use equivalency of units to create conversion factors Example: Convert 17.2 yards to feet

Desired Unit 3 ft 12 in. 17.2 yd · • feet to inches: 1 ft = 12 in. or 1 yd = 51.6 ft 1 ft

Given Unit 1 yd = 3 ft

5280 ft • miles to feet: 1 mi = 5280 ft. or 1 mi Note: Technically, the number of significant digits should remain the same after conversion. Equal to 1

More Unit Conversions More Unit Conversions Feet and Inches Feet and Inches 3 Tape measures and yard sticks often • Convert 3 ft – 7 4 in. to decimal feet: measure in feet and inches. 3 3 1 ft 3 3 ft − 7 in. = 3 ft + 7 in. 4 4 12 in. • Convert 3 ft - 7 4 in. to inches: 1 ft = 3 ft + 7+ 3 in. 3 12 in. 4 12 in. 3 ft− 7 in. = 3 ft · + 7 3 in. 4 1 ft 4 = 3 ft + 0.65 ft 3 = 36 in. + 7 in. 4 = 3.65 ft = 43 3 in. 4

Unit Conversions Conversion of Units Between Systems Between U S and SI Systems Example: Convert 34.7 inches to centimeters Use equivalency of units to create conversion factors 2.54 cm . • Inches to centimeters: 34.7 in. 1 in. = 88.1 cm

Desired Unit 2.54 cm

1 in.

Given Unit Equal to 1

Conversion of Units Between Systems Unit Conversion • Dimensional Analysis Example: Convert 17 kg to lb m – Sequential application of conversion factors – Units that appear in both numerator and 2.205 lbm denominator cancel 17 kg . = 37.4837 lb lb 1 kg m m

Given Units Linking Units Desired Units Desired Units Given Units • • = Desired Units 1 Given Units Linking Units 1 Note: The number of significant digits is ambiguous in the given quantity – assume two significant digits.

Dimensional Analysis Dimensional Analysis Example: An American car company has designed a Example: Convert 345.7 inches to yards new high fuel efficiency vehicle that is rated at 55 miles per gallon. The company plans to export the car to Europe and must advertise the fuel efficiency 1 ft 1 yd in SI units. What is the fuel usage rate in kilometers . . 345.7 in. 12 in. 3 ft = 9.602 yd per liter?

1 L = 0.264 gal Note: The number of significant digits should remain the 55 mi 0.264 gal same after conversion. However, if the precision of the 1 km ? km . . = original measurement is in question, you may choose to gal L 0.621 mi round to a reasonable precision. L 1 km = 0.621 mi

Dimensional Analysis Dimensional Analysis Example: An American car company has designed a Example: An American car company has designed a new high fuel efficiency vehicle that is rated at 55 new high fuel efficiency vehicle that is rated at 55 miles per gallon. The company plans to export the miles per gallon. The company plans to export the car to Europe and must advertise the fuel efficiency car to Europe and must advertise the fuel efficiency in SI units. What is the fuel usage rate in kilometers in SI units. What is the fuel usage rate in kilometers per liter? per liter?

1 L = 0.264 gal 1 L = 0.264 gal

55 mi 0.264 gal 1 km 55 mi 0.264 gal 1 km . 23.4 km . gal L . = gal L . = 23 km/L 0.621 mi L 0.621 mi 1 km = 0.621 mi 1 km = 0.621 mi