A system of units is a set of specific units of length, time, mass, and electric current. There are two systems commonly used in the scientific and engineering communities. The Système Internationale (SI) is a version of the metric system. The U. S. scientific community and all nations except the U. S commonly use it. The English system is most commonly used in the U. S. You will find yourself having to convert between English to S I units your entire career.

There are two very important rules concerning units:

• Units are part of a measurement and are always shown.
• Units are treated in an equation in the same manner as any mathematical quantity, r * r = r², ft * ft = ft²

English Units

The Standard English units used in this section of the text are:

Inch (in)

Foot (ft) = 12 in

Yard (yd) = 3 ft

Mile (mi) = 5280 ft

pound (lb) (#)

ton = 2000 lb

long ton = 2200#

second (sec)

minute (min) = 60 sec

hour (hr) = 60 min

gallon=231 in³

^{1 cubic ft = 7.5 gallons}

 The rectangle above has a width of 4 in and a height of 3 in. Its area can be calculated by A = L * W, A = 4 in * 3 in, A = 12 in2 . A square inch is a square with each side measuring one inch. As seen below a rectangle measuring 4 in by 3 in contains 12 square inches.

Volumes are the product of three length units. Small volumes are usually measured in cubic inches or cubic feet. Volumes related to liquids are usually measured in gallons (gal). A gallon contains 231 cubic inches.

The box below has a width of 4 in and a height of 3 in and a length of 1 in. Its volume can be calculated by the equation V = LWH,

V = 1 in * 4 in * 3 in, V = 12 in³.

As seen below this box contains 12 cubic inches. A cubic inch is a cube measuring one inch on each side.

Unit Conversion

It is sometimes necessary to convert units between systems and often necessary to convert units within a system. An example of conversion between systems would be to change a measurement from the English to the SI system (miles to kilometers). Conversion within a system could be changing the size of a unit of measurement (inches to feet) or changing the unit (cubic inches to gallons).

Mathematical Treatment of Units

In making unit conversions we make use of two mathematical facts:

• A quantity divided by itself equals one. For example,

  A quantity multiplied or divided by one does not change. For example, A * 1 = A.

 EXAMPLE 4

GIVEN: d = 1.2 ft.

FIND: d in inches

To make the unit conversion, we multiply the quantity, 1.2 ft, by the value of 1 in the form of a fraction that will give inches when the units are mathematically canceled.

 ANSWER: d = 14.4 in

EXAMPLE 5

GIVEN: d = 30 in

FIND: d in ft.

 d = 30 in

To make the unit conversion, we multiply the quantity, d, by the value of 1 in the form of the fraction that will give feet when the units are canceled.

ANSWER: d = 2.5 ft.

Conversion within A Unit System

Many unit conversions are required because of the various industry standards in rating equipment or the common unit of measure. The dimensions of rooms are shown in units of feet and inches. Floor coverings are sold by the square yard. A calculation of the quantity of carpet required for an office would be as follows.

EXAMPLE 6

GIVEN:

FIND: Area in yd²

 A = L * W

ANSWER: A = 117.6 yd2

EXAMPLE 7

GIVEN:

Fluid flow is frequently calculated in in³ per second but pumps are rated in gallons per minute.

In the next EXAMPLE water is flowing through a pipe at the rate of 30 in³ per second. The EXAMPLE shows how to convert this to in³ per second, in³ per minute, to gallons per minute (there are 231 in³ in a gallon), and to ft³ per minute.

EXAMPLE 8

centimeter (cm)

meter (m)=100 cm

kilometer (km)1000 m

gram (gm)

kilogram (kg)=1000 gm

milligram(mg)=0.001 gm

metric ton = 1000 kg

second (sec)

minute (min) = 60 sec

hour (hr) = 60 min

liter=1000 cm³

Areas are the combination of two length units multiplied. Small areas are usually measured in a length unit squared as m2 or cm2.

The rectangle below has a width of 4 cm and a height of 3 cm. Its area can be calculated by A = LW, A = 4 cm * 3 cm, A = 12 cm2 . A square centimeter is a square with each side measuring one centimeter. As seen below a rectangle measuring 4 cm by 3 cm contains 12 square centimeters.

 Volumes are the product of three length units. Small volumes are usually measured in cubic centimeters or cubic meters. Volumes related to liquids are usually measured in liters (L). A liter contains 1000 cubic centimeters.

The box below has a width of 4 cm and a height of 3 cm and a length of 1 cm. Its volume can be calculated by the equation V = LWH,

V = 1 cm * 4 cm * 3 cm, V = 12 cm³.

As seen below this box contains 12 cubic centimeters. A cubic centimeter is a cube measuring one centimeter on each side

.

 Conversion between the SI and English unit systems

Since both the SI and the English systems of units are used, it is necessary to be able to convert from one to another. These conversions are done in the same manner as those above.

EXAMPLE 9

These problems provide an opportunity to demonstrate and reinforce skills in solving technical problems requiring unit conversions. Solve them using the examples as a guide. When done contact your instructor for solutions and a self-grading guide.

Essential conversion factors are: 1 in = 2.54 cm; 1 m = 100 cm; 1 kg = 1000 grams; 1 liter = 1000 cm3 and 1 gal = 231 in3.

1. How long is a meter stick in cm? (a) Inches? (b) Feet? (c) Yards?
2. A long ton is 2200 pounds. (a) How much is that in kilograms, kg, (2.205 lb = 1 kg)(1 kg = 1000 grams)? (b) In metric tons (1000 Kg = 1 metric ton)?
3. It is 150 miles from the home of Mr. L to downtown Atlanta. How far is that in kilometers, km?
4. A running track being built for international track and field competition must be exactly 400 meters long. How long is that in (a) feet? (b) Yards?
5. A tin can has a surface area (top, bottom, and sides) of 128 square centimeters. What is that in (a) square inches, (b) square feet?
6. A football field between the end lines is 100 yd long and 50 yd wide. What is that in (a) ft2 and (b) m2?
7. A box has inside dimensions are 6 in long, 3 in high and 4 in wide. What is this volume in (a) gallons, (b) cm3 and (c) liters?
8. How many liters of water can be carried in a 3 gallon bucket?
9. The inside of a watering trough for beef cattle is 8 ft. long, 3 feet wide, and 4 feet deep. What is this volume in (a) in3, (b) gallons, (c) cm3, and (d) liters?