All measurements consist of two parts, a number part and a unit. For example, if we measure a table with a tape measure and report its width as 37.5 in , 37.5 is the number part, in is the unit part. The connection between the number part and unit part is the operation of multiplication. So when we write 37.5 in , we really mean (37.5)(1 in ) , i.e., 37.5 multiples of the base unit of one inch. There are many different kinds of things we can measure, but all common physical measurements can be reduced to just a few kinds of things. The four basic quantities are often listed as follows:
All other measurements can be expressed as a combination of these. It should really be noted that mass and weight are not identical. Weight is the gravitational force acting on a mass and varies with position in space. For example, the weight of an object on the surface of the earth is about six times the weight of that same object on the surface of the moon. However, the mass of the object is the same in both locations. Despite this difference, in these notes we will not distinguish between mass and weight.
It is not sensible to add or subtract measurements of different kinds of things. For example, 15 lb + 7 ft is a meaningless operation. This is just the old adage that it's impossible to add apples and oranges! To add or subtract measurements requires the same kind of quantities, as in 9 ft + 8 ft = 17 ft . Note, we just add the numbers and carry the factor of the unit. This is just the distributive property discussed in Unit 3. Suppose we have 8ft + 36 in . Here, the quantities to be added are both lengths so the operation makes sense, but we can't actually perform the addition until we get the units to agree as in 8 ft + 36 in = 8 ft + 3ft = 11 ft . Here we "converted" 36 in to 3 ft . The operation could also have been done by converting 8 ft to 72 in , and adding 72 in + 36 in = 108 in. While adding or subtracting different kinds of measurements is impossible, multiplying or dividing measurements is always possible. For example,
5 lb × 4 ft = 20 ft×lb, where a ft×lb is 1 ft × 1 lb is a "foot pound," a measure of either energy or torque.
As a second example,
100 miles ÷ 4 gallons = 25 mpg (miles per gallon), which measures fuel economy.
Measurement conversion is a necessary skill since the same set of units is often not used throughout a calculation. The basis of measurement conversion is the unit fraction. Any quantity Q remains unchanged when multiplied by 1.
Q = Q×1
The catch is that 1 has infinitely many "aliases". For example,
All of these represent unit fractions, since the numerator is the same amount as the denominator. The trick is to use the "proper" aliases to cancel the units you don't want and get the units you do want. For example, to convert 18 in to ft we could use the following procedure.
Here the in. unit was cancelled by appearing in both numerator and
denominator.
Consider rounding 0.434 in to the nearest 64th of an inch. The trick is to
use the unit fraction
So 0.434 in is seven sixteenth's of an inch to the nearest 64th of an inch.
More complicated conversions can involve more than one unit fraction. The speed 100 feet per second can be converted to miles per hour correct to 1 decimal place by the following :
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As an aide in setting up conversion calculations a set of equivalent measurements is presented below. This list also includes the formulas for converting temperature from Fahrenheit (°F ) to Celsius (°C ). For example, to find the Fahrenheit equivalent of 40°C , we calculate as follows :
while the Celsius equivalent of minus 10°F is computed using the formula
| Conversion Relations for English and Metric Units | ||
| Linear Measure: 1 ft = 12 in 1 yd = 3 ft 1 mile = 5280 ft 1 rod = 16.5 ft 1 furlong = 220 yd 1 in = 2.54 cm 1 ft = 0.3048 m 1 yd = 0.9144 m 1 mile = 1.609344 km |
Area Measure: 1 acre = 160 sq rods 1 sq mile = 640 acres 1 cm2 = 0.15500031 in2 1 m2 = 1.195990046 yd2 1 km2 = 0.3861021585 sq mile |
Volume Measure: 16 oz = 1 pt 2 pt = 1 qt 4 qt = 1 gal 1 gal = 0.13368056 ft3 1 gal = 231 in3 1 gal = 3.78541178 L 1 ft3 = 7.48051948 gal 1 ft3 = 28.31684659 L 1 L = 0.26417205 gal 1 L = 1.056688209 qt 1 L = 61.02374409 in3 1 L = 0.001 m3 1 mL = 1 cm3 |
| Weight Measure: 16 oz = 1 lb 1 oz = 28.348 g 1 ton = 2000 lb 1 lb = 453.568 g 1 kg = 2.20474 lb |
Temperature Conversions:  |
Time Measure: 1 min = 60 s 1 hr = 60 min 1 hr = 3600 s 1 day = 24 hours |
There is a definite relationship between length, area and volume measurements. Area is the amount of "two-dimensional space" inside of a planar figure. For example, a 2 ft by 6 ft rectangle has an area, A = (2 ft)(6 ft) = 12 ft2. Here, the unit ft2 is one square foot (sq ft) which literally means a one foot by one foot square. When we say that the area of the rectangle is 12 ft2, we mean that we could fit exactly 12 one foot by one foot squares inside this rectangle.
Care must be taken when converting units of area. Suppose we want to calculate how many square inches are in an area of 1.6 ft2. We need the unit fraction between square inches and square feet.
1 ft2 = (12 in)2 = 12 in × 12 in = 144 in2
Note: when we evaluate (12 in)2 we square both the 12 and the in. This is illustrated below.
To perform the conversion,
Volume is the amount of "three-dimensional space" inside of a solid. For example, a 2 ft by 3 ft by 2 ft box has a volume, V = (2 ft)(3 ft)(2 ft) = 12 ft3. Here the unit ft3 is one cubic foot (cu ft), which literally means a one foot by one foot by one foot cube as shown below.
When we say that the volume of the box is 12 ft3, we mean that we could fit exactly 12 one foot by one foot by one foot cubes inside this box. Like area conversions, volume conversions require careful setup. Suppose we wish to convert 12 ft3 to cubic yards.
1 yard = 3 ft
1 cu yd = 1 yd3 = (3 ft)3 = 3 ft × 3 ft × 3 ft = 27
ft3.
Note: when we evaluate (3 ft)3 we cube both the 3 and the ft. This is illustrated below.
One of the consequences of the French Revolution of 1789 was the development of the metric system of measurement. This system was designed to replace the earlier French system, which like its English counterpart had its origins in medieval society and royal institutions. Three features make the metric system very attractive. First, it is built on powers of 10, just like our decimal number system. Every unit is a multiple of 10 of some other unit. Thus, "strange" English multipliers like 3, 12, and 16 are banished! Second, a deliberate effort was made to coordinate different measures. For example, the fundamental unit of volume, the liter symbolized by L, is simply related to the fundamental unit of length, the meter symbolized by m, through the equation 1 m3 = 1000 L . Contrast this with the English system where 1 gal = 231 in3 = 0.134 ft3. The third advantage of the metric system is that it is "universal". It can be used with any kind of measurement in the same way.
It is interesting to note, that the metric system was so well accepted and in place that when electrical measurements began some 150 years ago only metric units were developed and have survived. The customary electric units we are all know, the volt ( V ), amp ( A ), and ohm (W) are all metric.
The metric system uses a two-part representation of all measurements. The first character or prefix indicates the power of 10 used, while the remainder of the measurement is the base unit. This is illustrated below.
Conversions within the metric system are particularly easy. The steps are as follows:
As before area and volume conversions within the metric system require careful setup. For example, suppose we want to convert 0.042 m2 to square cm. The calculation can be setup as follows:
As second example, to convert 187 mm3 to mL, we proceed as shown below :
Conversions between metric and English units require conversion factors. For example, to convert 1.80 gallons per minute to m3 per hour, we can use the following procedure:
Length measurements accurate to the nearest 0.05 in can be made with a English decimal rule. English Vernier calipers can give measurements accurate to the nearest 0.01 in . To achieve higher accuracy an instrument called a micrometer (pronounced my-crom-a-ter) is used. To avoid confusion with metric terminology a measurement of 1 mm is often called a micron instead of a micrometer (pronounced my-crow-meter). Common English micrometers can be used to measure lengths to the nearest 0.001 in. English Vernier micrometers are ten times more accurate since they can measure to the nearest 0.0001 in.
By placing an object between the "jaws" of the micrometer (called the anvil and the spindle) and turning the thimble until the anvil and spindle just make contact, we obtain the length of the object by combining the barrel, thimble, and Vernier scale readings. This is illustrated on the following pages.
A common metric micrometer is accurate to the nearest
So it is more accurate than a common English micrometer, but not as accurate as an English Vernier micrometer. The reading of a common metric micrometer is accomplished by combining the barrel and thimble readings. This is illustrated in the following figures.
Perform the following calculations with measurement numbers. Round to one decimal place.
12 ft 3 in - 8 ft 8 in = ________________
4.2 cm × 3.5 cm2 = ________________
530 miles ÷ 35 mpg (mpg = miles per gallon) = ________________
Convert the following measurements as indicated. Write answers in the blank space provided. Round to one decimal place.
185°F = ________________ °C
17.6 liters = ________________ gal
132 kg = ________________ lb
127 ft3 = ________________ yd3
129 mA = ________________ A
0.589 m = ________________ mm
58 cm = ________________ m
127mV = ________________ V
98 km/hour = ________________ mph
10.0°C = ________________ °F
2.56 sq ft = ________________ cm2
2.49 gal = ________________ mL
179cm3 = ________________ in3
Round to the nearest 32nd of an inch:
0.165 in = ________________
Round to the nearest 64th of an inch:
0.645 in = ________________
Read the following measuring devices.
English Vernier micrometer
Common metric micrometer
What is the largest size bolt, to the nearest 64th of an inch,
that will fit a hole 12 mm in
diameter?
size = _______________
A box has dimensions of 5 ft 6 in × 3 ft 9 in × 3 ft 4 in. How many cubic
meters is this?
volume = _______________
A car has a gas tank with a capacity of 50 L. If the car gets 33.5 miles per
gallon, how many km can the car travel on a full tank?
distance = _______________
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© 2002 by Al Lehnen; HTML-ized by Kevin Mirus (kmirus@madison.tec.wi.us or kjmirus@execpc.com). |
| This document was last modified Wednesday, August 21, 2002, 4:10 PM. |