01 Units of Measurement Concept Overview
UNITS OF MEASUREMENT | CONCEPT OVERVIEW The TOPIC of UNITS can be referenced on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
CONCEPT INTRO: A
MEASUREMENT
is
the
ASSIGNMENT
of
a
NUMBER
and
UNIT
to
CHARACTERIZE a PHYSICAL PARAMETER, which can be compared to other PHYSICAL PARAMETERS. A PHYSICAL PARAMETER is a way of DEFINING a PHYSICAL SITUATION such as LENGTH, FORCE, TIME, such that the DIMENSIONS of a PHYSICAL PROPERTY can be MEASURED. A UNIT specifies the PHYSICAL DIMENSION and MAGNITUDE of the REFERENCE QUANTITIES being measured, and provides DEFINITION to a PHYSICAL PARAMETER. The NUMBER indicates the QUANTITY of REFERENCE UNITS contained in the QUANTITY being measured. ACCURACY describes how CLOSE a MEAUSREMENT is to the ACTUAL VALUE of the QUANTITY that is measured.
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PRECISION describes how CLOSE two or more MEASUREMENTS are relative to OTHER MEASUREMENTS, regardless of how close the MEASUREMENTS are to the ACTUAL VALUE that is physically measured.
SIGNIFICANT FIGURES: The TOPIC of SIGNIFICANT FIGURES is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. When working with EXPERIMENTAL VALUES in SCIENCE and ENGINEERING, we are typically concerned with the PRECISION of VALUES used in CALCULATIONS. SIGNIFICANT FIGURES are used to ACCOUNT for the UNAVOIDABLE UNCERTAINTY in the MEASUREMENT of VALUES. When performing CALCULATIONS, you will want to take NOTE of the number of SIGNIFICANT FIGURES of each value, to maintain a consistent level of PRECISION in your CALCULATIONS. There are FIVE RULES that govern the application of SIGNIFICANT FIGURES in calculations: 1. Non-zero digits are always SIGNIFICANT. For example, the number 4.8383 has 5 significant figures and the number 5.21 contains 3 significant figures. 2. Any zeros between two significant digits or nonzero digits are SIGNIFICANT. For example, the number 1.08 has 3 significant figures and the number 3.2808 has5 significant figures. Made with
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3. Zeros before the decimal point are placeholders and NOT SIGNIFICANT. For example, the number 0.000482 has three significant figures and the number 0.005011 has four significant figures 4. Zeros after the decimal and after figures are SIGNIFICANT. For example, the number 0.3210 has four significant figures, and the number 0.120 has three significant figures. 5. Exponential digits in scientific notation are NOT SIGNIFICANT. For example, the number 1.38 x 106 has three significant figures, and the number 0.1082 x 10-23 has four significant figures. SIGNIFICANT FIGURES IN ARITHMETIC OPERATIONS: The TOPIC of SIGNIFICANT FIGURES in ARITHMETIC OPERATIONS is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. As SIGNIFICANT FIGURES are used to reflect the AMOUNT of PRECISION in MEASUREMENTS, there are two rules that govern the ARITHMETIC OPERATIONS of SIGNIFICANT FIGURES in CALCULATIONS. Outside of these TWO RULES, the normal ORDER of OPERATIONS should be followed for all CALCULATIONS. Rule 1: Addition and Subtraction When ADDING or SUBTRACTING values, the FINAL VALUE must have only as many DECIMALS as the LEAST PRECISE MEASUREMENT with the LEAST number of DECIMAL PLACES. Made with
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In ADDITION, or SUBTRACTION, the answer cannot have MORE DIGITS to the RIGHT of the decimal point than either of the original numbers. For example, letβs CALCULATE the SUM of three MASS measurements given as 153 g, 1.8 g, and 9.16 g: 153 π + 1.8 π + 9.16 π = 163.96 π As the LEAST PRECISE MEASUREMENT is 153 g with 0 DECIMAL PLACES, the FINAL VALUE of the SUM is shown as: 164 π Rule 2: Multiplication and Division When MULTIPLYING or DIVIDING, the FINAL VALUE can only have as many SIGNIFICANT FIGURES as the LEAST PRECISE MEASUREMENT with the LEAST number of SIGNIFICANT FIGURES. In MULTIPLICATION, and DIVISION, the number of SIGNIFICANT FIGURES in the PRODUCT or QUOTIENT, is determined by the LEAST PRECISE MEASUREMENT that has the FEWEST NUMBER of SIGNIFICANT FIGURES. For example, letβs CALCULATE the VALUE requiring the MULTIPLICATION and DIVISION of three values as shown as below: 34.78 Γ 11.7 0.17 Made with
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As the LEAST PRECISE MEASUREMENT is 0.17 with 2 SIGNIFICANT FIGURES, the final ANSWER must have TWO SIGNIFICANT FIGURES. You take the 2393.682353 value from your CALCULATOR and round it to 2,400 and EXPRESS it in SCIENTIFIC NOTATION with 2 SIGNIFICANT FIGURES as: 2.4 Γ 102
SCIENTIFIC NOTATION: The TOPIC of SCIENTIFIC NOTATION is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. SCIENTIFIC NOTATION is a standardized method of representing EXTREMELY LARGE or EXTREMELY SMALL NUMBERS in simple EXPONENTIAL TERMS that are written as the PRODUCT of a REAL NUMBER and a POWER of 10. The COEFFICIENT is defined as a non-zero INTEGER that represents a NUMERICAL VALUE, typically with respect to the NUMBER of SIGNIFICANT FIGURES in the problem. POSTIVE VALUES in the EXPONENT place represent how many ZEROES there are in the number being represented.
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NEGATIVE VALUES in the EXPONENT place represent how many ZEROES there are between the NUMBER and the DECIMAL PLACE.
Note that the STANDARD POSITION for the DECIMAL PLACE is always just to the RIGHT of the first NON-ZERO digit in the number. Also, it is the first NON-ZERO digit counting from the LEFT of the NUMBER. The TABLE below lists out some EXAMPLES of VALUES in SCIENTIFIC NOTATION: Power
Scientific Notation
Value
-3
1 x 10-3
0.001
-2
1 x 10-2
0.01
-1
1 x 10-1
0.1
1
1 x 101
10
2
1 x 102
100
3
1 x 103
1,000
4
1 x 104
10,000
5
1 x 105
100,000
6
1 x 106
1,000,000
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When MULTIPLYING numbers written in SCIENTIFIC NOTATION, multiply the COEFFICIENTS, and ADD the EXPONENTS. For example, letβs CALCULATE the PRODUCT of two VALUES given in SCIENTIFIC NOTATION as: (3 Γ 106 ) Γ (2 Γ 105 ) MULTIPLYING the COEFFICIENTS of each VALUE, we calculate the COEFFICIENT of the PRODUCT of the two values as: 3Γ2= 6 ADDING the EXPONENTS of each VALUE, we calculate the EXPONENT of the PRODUCT of two values as: 6 + 5 = 11 Therefore, we can CALCULATE the PRODUCT of the TWO VALUES in SCIENTIFIC NOTATION as: (3 Γ 106 ) Γ (2 Γ 105 ) = 6 Γ 1011
When DIVIDING numbers written in SCIENTIFIC NOTATION, divide the COEFFICIENTS, and SUBTRACT the EXPONENT of the DENOMINATOR from the EXPONENT of the NUMERATOR. Made with
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For example, letβs CALCULATE the QUOTIENT of two VALUES given in SCIENTIFIC NOTATION as: (8 Γ 1010 ) (4 Γ 103 ) DIVIDING the COEFFICIENTS of each VALUE, we calculate the COEFFICIENT of the QUOTIENT of the two values as: 8 =2 4 SUBTRACTING the EXPONENT of the VALUE in the NUMERATOR from the value in the DENOMINATOR, we calculate the EXPONENT of the PRODUCT of two values as: 10 β 3 = 7 Therefore, we can CALCULATE the QUOTIENT of the TWO VALUES in SCIENTIFIC NOTATION as: (8 Γ 1010 ) = 2 Γ 107 (4 Γ 103 )
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SYSTEMS OF UNITS: The TOPIC of SYSTEMS of UNITS can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. For the Fundamentals of Engineering Exam, you will be responsible for UNDERSTANDING both the SI (METRIC) SYSTEM and the U.S. CUSTOMARY SYSTEM (USCS) for UNITS. In the UNITED STATES, most measurement are made with the U.S. CUSTOMARY SYSTEM (USCS) which uses FRACTIONS to MODIFY units. The METRIC SYSTEM, also known as the INTERNATIONAL SYSTEM OF UNITS (SI), is a DECIMAL based SYSTEM OF MEASUREMENT based upon POWERS of TEN using PREFIXES to MODIFY the BASE UNITS. In all SYSTEMS of MEASUREMENT, the BASE UNITS are those from which all the others can be DERIVED. The TABLE of METRIC PREFIXES can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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There are 16 METRIC PREFIXES that are you responsible for working with on the exam. There is no need to memorize anything as all of the PREFIXES, SYMBOLS, and CORRESPONDING VALUES for the POWER of TEN are provided in a table to reference as the one below. METRIC PREFIXES Multiple
Prefix
Symbol
10-18
Atto
a
10-15
Femto
f
10-12
Pico
p
10-9
Nano
n
10-6
Micro
10-3
Milli
m
10β2
Centi
c
10-1
Deci
d
101
Deka
da
102
Hecto
h
103
Kilo
k
106
Mega
M
109
Giga
G
1012
Tera
T
1015
Peta
P
1018
Exa
E
ΞΌ
For example, using a METRIC PREFIX, we can rewrite 1,000 meters (m) as one 1 kilometer (km) using the METRIC PREFIX kilo- (k). Made with
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The TABLE of UNITS for COMMON PHYSICAL PARAMETERS is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this reference and understand its application independent of the NCEES Supplied Reference Handbook. For each PHYSICAL QUANTITY, SYSTEMS of UNITS have DEFINED a DESIGNATED base UNIT as shown in the table below. Physical Quantity
U.S. Customary Unit
SI Base Unit
Length
Inch (in)
Meter (m)
Mass
Pound-Mass (lb-m)
Kilogram (kg)
Temperature
Degree Fahrenheit (Β°πΉ)
Kelvin (K)
Volume
Gallon (gal)
Liter (L)
Time
Second (s)
Second (s)
Force
Pound-force (lb-f)
Newton (N)
MASS: The TOPIC of MASS can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. MASS (π) is a MEASURE of the amount of MATTER in a SUBSTANCE or OBJECT. MASS (m) represents the NUMBER OF ATOMS in an OBJECT in conjunction with the DENSITY of each of those ATOMS; the mass of an object is a FUNDAMENTAL.
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The most commonly used unit for MASS is the KILGORAM. The KILOGRAM is an SI unit that is denoted using the symbol βkgβ. In the USCS system of units, both force and mass are called pounds. Therefore, one must distinguish the pound-force (lb-f) from the pound-mass (lb-m). LENGTH: The TOPIC of LENGTH can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. LENGTH (L) is DEFINED as the measured CHANGE in DISPLACEMENT of a DIMENSION between a defined STARTING POINT and defined ENDING POINT. The most commonly used unit for LENGTH is the METER. The METER is an SI unit that is denoted using the symbol βmβ. In the USCS system of units, length is measured using BASE UNITS of INCHES given as "ππ". As the USCS system of units is based on FRACTIONS, length is also measured to a CONVERSION FACTOR of 12 inches in a foot, where UNITS of FEET are represented as βftβ.
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Depending on the application in problems using the USCS system of units, HEIGHT and LENGTH are commonly given in UNITS of FEET, and smaller dimensions are given in UNITS of INCHES. FORCE: The TOPIC of FORCE can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. A FORCE is a PUSH or PULL upon an OJBECT or RIGID body resulting from the objectβs INTERACTION with another OBJECT, and represents the CAPACITY to do WORK or cause PHYSICAL CHANGE. The most commonly used unit for FORCE is the NEWTON. The NEWTON is an SI unit that is denoted using the symbol βNβ. In the USCS system of units, both force and mass are called pounds. Therefore, one must distinguish the pound-force (lb-f) from the pound-mass (lb-m). This unit is derived from NEWTONβS SECOND LAW OF MOTION which states that a FORCE of 1 N causes a mass of 1 kg to accelerate 1 π/π 2 . With this general definition, a more conventional statement of a NEWTON can be defined as 1 ππ β π/π 2 . Remember this fact, for it can be a simple conceptual problem given to you come exam day, and further, a simple point in your direction.
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Though we are probably more familiar and comfortable using SI units, the exam specifies that it is fair game to see either SI or USCS units. For this reason, we should be prepared and comfortable working problems in both worlds. The standard US unit for a FORCE is the POUND denoted with a symbol πππ or πππ . Similar to the NEWTON, a FORCE of 1 lbf causes a mass of 1 slug to accelerate at 1 ft/s 2 . USCS units can be a confusing way of representing mass because universally the mass of an object is reported as a weight in πππ or πππ . However, the weight of an object in pounds is not a mass, but actually the force of gravity acting on the mass. Consequently, an extra step to determine an objectβs mass in SLUGS, given its WEIGHT, must be completed using the following conversion:
ππ ππ’ππ =
πππ π
The FUNDAMENTAL CONSTANT for the ACCELERATION of GRAVITY can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Where:
π = 32.174
ππ‘ π 2
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VOLUME: The TOPIC of VOLUME can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. VOLUME is the AMOUNT of SPACE occupied by an OBJECT or SUBSTANCE such as a SOLID, LIQUID, or GAS. The most commonly used unit for VOLUME is the CUBIC METER (π3 ), which is a DERIVED unit from the METER (m), the SI UNIT for LENGTH. Another common UNIT of VOLUME in the SI system is the liter (L), which is commonly represented as MILLITERS (mL).
In the USCS system of units, VOLUME is given in UNITS of GALLONS, which are denoted as βgalβ.
The FORMULA for SPECIFIC VOLUME can be referenced under the SUBJECT of THERMODYNAMICS on page 87 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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The SPECIFIC VOLUME is the volume occupied by a UNIT MASS of substance. It is expressed in units of ft 3 /lbm or m3 /kg. It is calculated as the total volume divided by the mass:
π=
π π
Where: β’ π is the volume of the system in units of ft 3 or m3 β’ π is the mass of the system in units of lbm or kg β’ π is the specific volume of the system in units of ft 3 /lbm or m3 /kg PRESSURE: The FORMULA for PRESSURE can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. PRESSURE, (π), is defined as the FORCE per unit AREA that is PERPENDICULAR to the FORCE:
π=
πΉ π΄
The most commonly used unit for PRESSURE is the PASCAL which is equal to one NEWTON per SQUARE METER (π/π2 ). The PASCAL is an SI unit that is denoted using the symbol βPaβ. Made with
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In the USCS system of units, PRESSURE is represented as PSI, which is equal to one POUND-FORCE per SQUARE INCH (ππ/ππ2 ). Pressure is measured as GAUGE PRESSURE, the difference between ABSOLUTE and the ATMOSPHERIC. Gauge pressure is either positive or negative (vacuum).
The AMBIENT PRESSURE is pressure at which the vapor pressure is equal to the surrounding pressure. This is the pressure at which a liquid boils. The FORMULA for ABSOLUTE PRESSURE can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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We use the following formula to relate absolute pressure, atmospheric pressure, and gauge pressure. ππππ = πππ‘π + ππππ’ππ Where: ππππ = Absolute Pressure πππ‘π = Atmospheric Pressure πππππ = Gauge Pressure (positive or negative) DENSITY: The TOPIC of DENSITY can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. DENSITY (π) is RATIO of the MASS of a SUBSTANCE to the VOLUME occupied by that SUBSTANCE. The FORMULA for DENSITY can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The FORMULA for DENSITY calculates DENSITY as the QUOTIENT of the MASS (π) divided by the VOLUME (π) or the INVERSE of the SPECIFIC VOLUME (π£) and is given as: Made with
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π₯π π 1 1 = = ππ π = π₯πβ0 π₯π π π π
π = lim
Where: β’ π is the DENSITY β’ m is the MASS β’ V is the VOLUME β’ v is the SPECIFIC VOLUME DENSITY is expressed in DIFFERENT units depending on the PHASE of the SUBSANCE. The density of SOLIDS is usually expressed in UNITS of grams per cubic centimeter (π/ππ3 ), the density of LIQUIDS is commonly expressed in UNITS of grams per milliliter (π/ππΏ), and the density of GASES is commonly expressed in UNITS of grams per liter (π/πΏ).
ENERGY:
The
TOPIC
of
ENERGY
can
be
referenced
under
the
SUBJECT
of
THERMODYNAMICS on page 87 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. ENERGY is the ability to do work. When the work is actually being done, the energy is considered KINETIC, and when the work is waiting to be done and there is the potential do work, the energy is considered POTENTIAL. Made with
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The most commonly used unit for ENERGY is the JOULE. The JOULE is an SI unit that is denoted using the symbol βJβ. ENERGY is always CONSERVED, it is neither CREATED nor DESTROYED, but is transformed from one form to another. KINETIC ENERGY (πΎπΈ) is the ENERGY that results from an objectβs MOTION. The measurement of kinetic energy in an object is CALCULATED based on the objectβs MASS and VELOCITY. POTENTIAL ENERGY (ππΈ) is the STORED ENERGY that results from an objectβs position or arrangement of parts that have the POTENTIAL to become kinetic energy. The MEASUREMENT of POTENTIAL ENERGY in an object is calculated based on the objectβs mass, height, or distance. In CHEMISTRY, POTENIAL ENERGY (PE) is stored in CHEMICAL BONDS, which are the FORCES that hold ATOMS together in COMPOUNDS. INTERNAL ENERGY (π) accounts for all of the ENERGY of a SUBSTANCE, representing the SUM of the KINETIC and POTENTIAL energies of the mass comprising the system. WORK (π) is the PROCESS of transferring ENERGY across the boundaries of a system in the form of mechanical energy across the boundaries of a system.
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TEMPERATURE: The TOPIC of TEMPERATURE can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. TEMPERATURE, (π), is the thermodynamic property of a substance that depends on the ENERGY CONTENT. A VARIANCE, or difference in TEMPERATURE, (π₯π), is required for HEAT ENERGY to FLOW. When you measure the temperature of something, youβre measuring the average KINETIC ENERGY of the individual PARTICLES.
A TEMPERATURE SCALE is a unit measurement system for determining the ENERGY of the particles in a material. For the purposes of the FE Exam, the four commonly used temperature scales that we should be familiar with are: Fahrenheit, Celsius, Rankine, and Kelvin. The FAHRENHEIT TEMPERATURE SCALE is based on 32Β°πΉ for the FREEZING POINT of water and 212Β°πΉ for the BOILING POINT of water. A degree on the FAHRENHEIT scale represents 1/180 of the temperature range between the freezing and boiling points of water.
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The CELSIUS TEMPERATURE SCALE is based on 0Β°πΆ for the FREEZING POINT and 100Β°πΆ for the BOILING POINT of water. A degree on the CELSIUS scale represents 1/100 of the temperature range between the freezing and boiling points of water. The ABSOLUTE TEMPERATURE SCALE is a commonly used scale of temperature units in thermodynamic problems, and is given in units of RANKINE (Β°R) and KELVIN (K). As there are no NEGATIVE TEMPERATURES on the ABSOLUTE TEMPERATURE SCALE, the coldest theoretical temperature is ABSOLUTE ZERO, which is represented as 0 K on the Kelvin temperature scale and 0Β°π on the Rankine Scale. ABSOLUTE ZERO represents the temperature at which the THERMAL MOTION of atoms and molecules reaches its MINIMUM. The KELVIN TEMPERATURE SCALE is the absolute temperature scale used in the International Engineering Systems and SI System of units. The Kelvin scale of temperature uses degrees CELSIUS from ABSOLUTE ZERO, and is based on 0 K being defined as absolute zero. The RANKINE TEMPERATURE SCALE is the absolute temperature scale used in the American Engineering System and in the English System of Units. The Rankine scale of temperature uses degrees FAHRENEHTI from ABSOLUTE ZERO, and is based on 0Β°π being defined as absolute zero.
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The FORMULAS for the UNIT CONVERSIONS OF TEMPERATURE can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The formula for converting a CELSIUS temperature to degrees FAHRENHEIT is expressed using the following expression: ππΉ = 1.8 ππΆ + 32 Where: β’ ππΉ is the temperature given in degrees Fahrenheit(Β°πΉ) β’ ππΆ is the temperature given units of degrees Celsius (Β°πΆ)
The formula for converting a FAHRENHEIT temperature to degrees CELSIUS is expressed using the following expression: ππΆ = (ππΉ β 32)/1.8 Where: β’ ππΉ is the temperature given in degrees Fahrenheit(Β°πΉ) β’ ππΆ is the temperature given units of degrees Celsius (Β°πΆ)
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The formula for converting a FAHRENHEIT temperature to the Absolute Temperature Scale in units of degrees RANKINE (Β°π ) is expressed using the following expression: ππ = ππΉ + 459.67
π₯ππ = π₯ππΉ
Where: β’ ππ is the temperature given in degrees Rankine (Β°π ) β’ ππΉ is the temperature given in degrees Fahrenheit(Β°πΉ) The formula for converting a CELSIUS temperature to the Absolute Temperature Scale in units of KELVIN (πΎ) is expressed using the following expression: ππΎ = ππΆ + 273.15
π₯ππΎ = π₯ππΆ
Where: β’ ππ is the temperature given in units of Kelvin (πΎ) β’ ππΆ is the temperature given units of degrees Celsius (Β°πΆ)
HEAT: The TOPIC of HEAT is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook.
HEAT is a MEASURE of the TOTAL amount of ENERGY a substance possesses.
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The most commonly used unit for HEAT is the CALORIE. The CALORIE is an SI unit that is given as βcalβ and represents the AMOUNT of ENERGY it takes to raise the TEMPERATURE of 1 gram of water 1Β°πΆ.
HEAT and TEMPERATURE are both MEASURES of ENERGY.
HEAT, however, is not the same as TEMPERATURE, as HEAT measures the TOTAL ENERGY of a SUBSTANCE, and TEMPERATURE measures the AVERAGE ENERGY of a SUBSTANCE.
HEAT ENERGY (π) is the ENERGY transferred across the BOUNDARIES of a SYSTEM because of the VARIATION in TEMPERATURE. HEAT ENERGY is the RESULT of the MOVEMENT of atoms, molecules, or ions transferring from one solid, liquid, or gas to another. SUBSTANCES change TEMPERATURE when HEATED, but not all substances have their temperature RAISED to the same extent when EQUAL amounts of HEAT are added. SPECIFIC HEAT is the amount of HEAT required to raise the TEMPERATURE of one gram of a SUBSTANCE by one degrees CELSIUS, or other defined TEMPERATURE UNIT.
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SPECIFIC HEAT AT CONSTANT PRESSURE is represented by πΆπ and given in units of π΅π‘π’/πππ-π or ππ½/ππ-πΎ. SPECIFIC HEAT AT CONSTANT VOLUME is represented by βπΆπ β and given in units of π΅π‘π’.
CONVERSION FACTORS: The TABLE of CONVERSION FACTORS can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A CONVERSION FACTOR is an EQUALITY or NUMERICAL VALUE that can be used to CONVERT between UNITS from one UNIT to another. DIMENSIONAL ANALYSIS, also known as the UNIT FACTOR METHOD, is the process of using CONVERSION FACTORS to convert between UNITS and SYSTEMS of UNITS, without changing the PHYSICAL DEFINITION of the VALUE. The goal of DIMENSIONAL ANALYSIS is to go from a GIVEN UNIT to the DESIRED UNIT by MULTIPLYING the given unit by a CONVERSION FACTOR, in which the NUMERATOR and the DENOMINATOR must represent the same quantity.
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Three Steps of Dimensional Analysis: 1. Write down the GIVEN MEASUREMENT with the NUMERICAL VALUE and associated UNIT. Make sure to take not of the PHYSICAL DIMENSION being represented by the unit. For example, if you see kilograms, you should immediately realize this is a MEASURE of MASS, or if you saw mL, you should immediately be thinking VOLUME. 2. MULTIPLY the MEASUREMENT by one or more CONVERSION FACTORS. The UNIT in each DENOMINATOR must CANCEL OUT units of the original measurement, leaving the DESIRED UNITS in the NUMERATOR. For example, if you convert 72 hours to UNITS of DAYS, you would use the CONVERSION FACTOR of 1 day/24 hours, to cancel out the units of HOUR, and be left with an answer of 3 DAYS. 3. Perform the CALCULATION and report the answer to the proper SIGNIFICANT FIGURES based on the MEASUREMENTS given in the problem statement.
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CONCEPT EXAMPLE: A gold bar is measured to have a mass of 25 kilograms. Given the bar is made of pure gold, the volume of gold in the bar calculated in cubic inches (ππ3 ) is most close to: A. 70 B. 80 C. 90 D. 100
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SOLUTION: The TOPIC of DENSITY can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem we are looking to SOLVE for the VOLUME of the gold in the bar, given the MASS of BAR and ASSUMPTION that the bar is COMPRISED entirely of GOLD. DENSITY (π) is RATIO of the MASS of a SUBSTANCE to the VOLUME occupied by that SUBSTANCE. As we are told the bar is comprised of the element GOLD, we can look up the DENSITY for GOLD in the Reference Handbook, and relate that to the GIVEN MASS to SOLVE for the VOLUME of GOLD in the BAR. The TABLE for PROPERTIES of METALS can be referenced under the SUBJECT of MATERIALS SCIENCE on page 61 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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Referencing the TABLE for the PROPERTIES of METAL in the REFERENCE HANDBOOK, we find the DENSITY of GOLD is given as: π = 19,281 ππ/π3 The FORMULA for DENSITY can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The FORMULA for DENSITY calculates DENSITY as the QUOTIENT of the MASS (π) divided by the VOLUME (π) or the INVERSE of the SPECIFIC VOLUME (π£) and is given as: π₯π π 1 1 = = ππ π = π₯πβ0 π₯π π π π Made with by Prepineer | Prepineer.com
π = lim
Where: β’ π is the DENSITY β’ m is the MASS β’ V is the VOLUME β’ v is the SPECIFIC VOLUME Letβs re-write the FORMULA for DENSITY to ISOLATE the term for VOLUME (V):
π=
π π
Plugging in the GIVEN MASS and REFERENCE DENSITY for GOLD, we CALCULATE the VOLUME of the BAR as:
π=
25 ππ = 0.0013 π3 3 19,281 ππ/π
In the USCS system of units, LENGTH is measured using BASE UNITS of INCHES given as "ππ". Therefore, we will need to CONVERT the BASE UNITS of VOLUME from CUBIC METERS (π3 ) to CUBIC INCHES (ππ3 ) to CALCULATE the VOLUME of the GOLD in USCS UNITS. The TABLE of CONVERSION FACTORS can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Made with
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A CONVERSION FACTOR is an EQUALITY or NUMERICAL VALUE that can be used to CONVERT between UNITS from one UNIT to another. Looking at the TABLE of CONVERSION FACTORS in the REFERENCE HANDBOOK, we can use DIMENSIONAL ANALYSIS to CONVERT from CUBIC METERS (π3 ) to LITERS (πΏ) and then from LITERS (πΏ) to CUBIC INCHES (ππ3 ). Three Steps of Dimensional Analysis: 1. Write down the GIVEN MEASUREMENT with the NUMERICAL VALUE and associated UNIT. As we are working with UNITS of VOLUME, we need to take note that all UNITS are to THIRD POWER. ππππ’ππ = 0.0013 π3 2. MULTIPLY the MEASUREMENT by one or more CONVERSION FACTORS. The UNIT in each DENOMINATOR must CANCEL OUT units of the original measurement, leaving the DESIRED UNITS in the NUMERATOR. a. The FIRST CONVERSION will be CONVERTING from CUBIC METERS (π3 ) to LITERS (πΏ):
πΆπππ£πππ πππ πΉπππ‘ππ:
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b. The SECOND CONVERSION will be CONVERTING from LITERS (πΏ) to CUBIC INCHES (ππ3 ):
πΆπππ£πππ πππ πΉπππ‘ππ:
1πΏ 61.02 ππ3
3. Perform the CALCULATION and report the answer to the proper SIGNIFICANT FIGURES based on the MEASUREMENTS given in the problem statement: 1,000 πΏ 61.02 ππ3 0.0013 π Γ ( )Γ( ) = 79.326 ππ2 = 80 ππ2 3 1π 1πΏ 3
Therefore, the correct answer choice is B. ππ
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CONCEPT INTRO: A
MEASUREMENT
is
the
ASSIGNMENT
of
a
NUMBER
and
UNIT
to
CHARACTERIZE a PHYSICAL PARAMETER, which can be compared to other PHYSICAL PARAMETERS. A PHYSICAL PARAMETER is a way of DEFINING a PHYSICAL SITUATION such as LENGTH, FORCE, TIME, such that the DIMENSIONS of a PHYSICAL PROPERTY can be MEASURED. A UNIT specifies the PHYSICAL DIMENSION and MAGNITUDE of the REFERENCE QUANTITIES being measured, and provides DEFINITION to a PHYSICAL PARAMETER. The NUMBER indicates the QUANTITY of REFERENCE UNITS contained in the QUANTITY being measured. ACCURACY describes how CLOSE a MEAUSREMENT is to the ACTUAL VALUE of the QUANTITY that is measured.
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PRECISION describes how CLOSE two or more MEASUREMENTS are relative to OTHER MEASUREMENTS, regardless of how close the MEASUREMENTS are to the ACTUAL VALUE that is physically measured.
SIGNIFICANT FIGURES: The TOPIC of SIGNIFICANT FIGURES is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. When working with EXPERIMENTAL VALUES in SCIENCE and ENGINEERING, we are typically concerned with the PRECISION of VALUES used in CALCULATIONS. SIGNIFICANT FIGURES are used to ACCOUNT for the UNAVOIDABLE UNCERTAINTY in the MEASUREMENT of VALUES. When performing CALCULATIONS, you will want to take NOTE of the number of SIGNIFICANT FIGURES of each value, to maintain a consistent level of PRECISION in your CALCULATIONS. There are FIVE RULES that govern the application of SIGNIFICANT FIGURES in calculations: 1. Non-zero digits are always SIGNIFICANT. For example, the number 4.8383 has 5 significant figures and the number 5.21 contains 3 significant figures. 2. Any zeros between two significant digits or nonzero digits are SIGNIFICANT. For example, the number 1.08 has 3 significant figures and the number 3.2808 has5 significant figures. Made with
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3. Zeros before the decimal point are placeholders and NOT SIGNIFICANT. For example, the number 0.000482 has three significant figures and the number 0.005011 has four significant figures 4. Zeros after the decimal and after figures are SIGNIFICANT. For example, the number 0.3210 has four significant figures, and the number 0.120 has three significant figures. 5. Exponential digits in scientific notation are NOT SIGNIFICANT. For example, the number 1.38 x 106 has three significant figures, and the number 0.1082 x 10-23 has four significant figures. SIGNIFICANT FIGURES IN ARITHMETIC OPERATIONS: The TOPIC of SIGNIFICANT FIGURES in ARITHMETIC OPERATIONS is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. As SIGNIFICANT FIGURES are used to reflect the AMOUNT of PRECISION in MEASUREMENTS, there are two rules that govern the ARITHMETIC OPERATIONS of SIGNIFICANT FIGURES in CALCULATIONS. Outside of these TWO RULES, the normal ORDER of OPERATIONS should be followed for all CALCULATIONS. Rule 1: Addition and Subtraction When ADDING or SUBTRACTING values, the FINAL VALUE must have only as many DECIMALS as the LEAST PRECISE MEASUREMENT with the LEAST number of DECIMAL PLACES. Made with
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In ADDITION, or SUBTRACTION, the answer cannot have MORE DIGITS to the RIGHT of the decimal point than either of the original numbers. For example, letβs CALCULATE the SUM of three MASS measurements given as 153 g, 1.8 g, and 9.16 g: 153 π + 1.8 π + 9.16 π = 163.96 π As the LEAST PRECISE MEASUREMENT is 153 g with 0 DECIMAL PLACES, the FINAL VALUE of the SUM is shown as: 164 π Rule 2: Multiplication and Division When MULTIPLYING or DIVIDING, the FINAL VALUE can only have as many SIGNIFICANT FIGURES as the LEAST PRECISE MEASUREMENT with the LEAST number of SIGNIFICANT FIGURES. In MULTIPLICATION, and DIVISION, the number of SIGNIFICANT FIGURES in the PRODUCT or QUOTIENT, is determined by the LEAST PRECISE MEASUREMENT that has the FEWEST NUMBER of SIGNIFICANT FIGURES. For example, letβs CALCULATE the VALUE requiring the MULTIPLICATION and DIVISION of three values as shown as below: 34.78 Γ 11.7 0.17 Made with
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As the LEAST PRECISE MEASUREMENT is 0.17 with 2 SIGNIFICANT FIGURES, the final ANSWER must have TWO SIGNIFICANT FIGURES. You take the 2393.682353 value from your CALCULATOR and round it to 2,400 and EXPRESS it in SCIENTIFIC NOTATION with 2 SIGNIFICANT FIGURES as: 2.4 Γ 102
SCIENTIFIC NOTATION: The TOPIC of SCIENTIFIC NOTATION is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook. SCIENTIFIC NOTATION is a standardized method of representing EXTREMELY LARGE or EXTREMELY SMALL NUMBERS in simple EXPONENTIAL TERMS that are written as the PRODUCT of a REAL NUMBER and a POWER of 10. The COEFFICIENT is defined as a non-zero INTEGER that represents a NUMERICAL VALUE, typically with respect to the NUMBER of SIGNIFICANT FIGURES in the problem. POSTIVE VALUES in the EXPONENT place represent how many ZEROES there are in the number being represented.
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NEGATIVE VALUES in the EXPONENT place represent how many ZEROES there are between the NUMBER and the DECIMAL PLACE.
Note that the STANDARD POSITION for the DECIMAL PLACE is always just to the RIGHT of the first NON-ZERO digit in the number. Also, it is the first NON-ZERO digit counting from the LEFT of the NUMBER. The TABLE below lists out some EXAMPLES of VALUES in SCIENTIFIC NOTATION: Power
Scientific Notation
Value
-3
1 x 10-3
0.001
-2
1 x 10-2
0.01
-1
1 x 10-1
0.1
1
1 x 101
10
2
1 x 102
100
3
1 x 103
1,000
4
1 x 104
10,000
5
1 x 105
100,000
6
1 x 106
1,000,000
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When MULTIPLYING numbers written in SCIENTIFIC NOTATION, multiply the COEFFICIENTS, and ADD the EXPONENTS. For example, letβs CALCULATE the PRODUCT of two VALUES given in SCIENTIFIC NOTATION as: (3 Γ 106 ) Γ (2 Γ 105 ) MULTIPLYING the COEFFICIENTS of each VALUE, we calculate the COEFFICIENT of the PRODUCT of the two values as: 3Γ2= 6 ADDING the EXPONENTS of each VALUE, we calculate the EXPONENT of the PRODUCT of two values as: 6 + 5 = 11 Therefore, we can CALCULATE the PRODUCT of the TWO VALUES in SCIENTIFIC NOTATION as: (3 Γ 106 ) Γ (2 Γ 105 ) = 6 Γ 1011
When DIVIDING numbers written in SCIENTIFIC NOTATION, divide the COEFFICIENTS, and SUBTRACT the EXPONENT of the DENOMINATOR from the EXPONENT of the NUMERATOR. Made with
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For example, letβs CALCULATE the QUOTIENT of two VALUES given in SCIENTIFIC NOTATION as: (8 Γ 1010 ) (4 Γ 103 ) DIVIDING the COEFFICIENTS of each VALUE, we calculate the COEFFICIENT of the QUOTIENT of the two values as: 8 =2 4 SUBTRACTING the EXPONENT of the VALUE in the NUMERATOR from the value in the DENOMINATOR, we calculate the EXPONENT of the PRODUCT of two values as: 10 β 3 = 7 Therefore, we can CALCULATE the QUOTIENT of the TWO VALUES in SCIENTIFIC NOTATION as: (8 Γ 1010 ) = 2 Γ 107 (4 Γ 103 )
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SYSTEMS OF UNITS: The TOPIC of SYSTEMS of UNITS can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. For the Fundamentals of Engineering Exam, you will be responsible for UNDERSTANDING both the SI (METRIC) SYSTEM and the U.S. CUSTOMARY SYSTEM (USCS) for UNITS. In the UNITED STATES, most measurement are made with the U.S. CUSTOMARY SYSTEM (USCS) which uses FRACTIONS to MODIFY units. The METRIC SYSTEM, also known as the INTERNATIONAL SYSTEM OF UNITS (SI), is a DECIMAL based SYSTEM OF MEASUREMENT based upon POWERS of TEN using PREFIXES to MODIFY the BASE UNITS. In all SYSTEMS of MEASUREMENT, the BASE UNITS are those from which all the others can be DERIVED. The TABLE of METRIC PREFIXES can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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There are 16 METRIC PREFIXES that are you responsible for working with on the exam. There is no need to memorize anything as all of the PREFIXES, SYMBOLS, and CORRESPONDING VALUES for the POWER of TEN are provided in a table to reference as the one below. METRIC PREFIXES Multiple
Prefix
Symbol
10-18
Atto
a
10-15
Femto
f
10-12
Pico
p
10-9
Nano
n
10-6
Micro
10-3
Milli
m
10β2
Centi
c
10-1
Deci
d
101
Deka
da
102
Hecto
h
103
Kilo
k
106
Mega
M
109
Giga
G
1012
Tera
T
1015
Peta
P
1018
Exa
E
ΞΌ
For example, using a METRIC PREFIX, we can rewrite 1,000 meters (m) as one 1 kilometer (km) using the METRIC PREFIX kilo- (k). Made with
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The TABLE of UNITS for COMMON PHYSICAL PARAMETERS is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this reference and understand its application independent of the NCEES Supplied Reference Handbook. For each PHYSICAL QUANTITY, SYSTEMS of UNITS have DEFINED a DESIGNATED base UNIT as shown in the table below. Physical Quantity
U.S. Customary Unit
SI Base Unit
Length
Inch (in)
Meter (m)
Mass
Pound-Mass (lb-m)
Kilogram (kg)
Temperature
Degree Fahrenheit (Β°πΉ)
Kelvin (K)
Volume
Gallon (gal)
Liter (L)
Time
Second (s)
Second (s)
Force
Pound-force (lb-f)
Newton (N)
MASS: The TOPIC of MASS can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. MASS (π) is a MEASURE of the amount of MATTER in a SUBSTANCE or OBJECT. MASS (m) represents the NUMBER OF ATOMS in an OBJECT in conjunction with the DENSITY of each of those ATOMS; the mass of an object is a FUNDAMENTAL.
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The most commonly used unit for MASS is the KILGORAM. The KILOGRAM is an SI unit that is denoted using the symbol βkgβ. In the USCS system of units, both force and mass are called pounds. Therefore, one must distinguish the pound-force (lb-f) from the pound-mass (lb-m). LENGTH: The TOPIC of LENGTH can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. LENGTH (L) is DEFINED as the measured CHANGE in DISPLACEMENT of a DIMENSION between a defined STARTING POINT and defined ENDING POINT. The most commonly used unit for LENGTH is the METER. The METER is an SI unit that is denoted using the symbol βmβ. In the USCS system of units, length is measured using BASE UNITS of INCHES given as "ππ". As the USCS system of units is based on FRACTIONS, length is also measured to a CONVERSION FACTOR of 12 inches in a foot, where UNITS of FEET are represented as βftβ.
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Depending on the application in problems using the USCS system of units, HEIGHT and LENGTH are commonly given in UNITS of FEET, and smaller dimensions are given in UNITS of INCHES. FORCE: The TOPIC of FORCE can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. A FORCE is a PUSH or PULL upon an OJBECT or RIGID body resulting from the objectβs INTERACTION with another OBJECT, and represents the CAPACITY to do WORK or cause PHYSICAL CHANGE. The most commonly used unit for FORCE is the NEWTON. The NEWTON is an SI unit that is denoted using the symbol βNβ. In the USCS system of units, both force and mass are called pounds. Therefore, one must distinguish the pound-force (lb-f) from the pound-mass (lb-m). This unit is derived from NEWTONβS SECOND LAW OF MOTION which states that a FORCE of 1 N causes a mass of 1 kg to accelerate 1 π/π 2 . With this general definition, a more conventional statement of a NEWTON can be defined as 1 ππ β π/π 2 . Remember this fact, for it can be a simple conceptual problem given to you come exam day, and further, a simple point in your direction.
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Though we are probably more familiar and comfortable using SI units, the exam specifies that it is fair game to see either SI or USCS units. For this reason, we should be prepared and comfortable working problems in both worlds. The standard US unit for a FORCE is the POUND denoted with a symbol πππ or πππ . Similar to the NEWTON, a FORCE of 1 lbf causes a mass of 1 slug to accelerate at 1 ft/s 2 . USCS units can be a confusing way of representing mass because universally the mass of an object is reported as a weight in πππ or πππ . However, the weight of an object in pounds is not a mass, but actually the force of gravity acting on the mass. Consequently, an extra step to determine an objectβs mass in SLUGS, given its WEIGHT, must be completed using the following conversion:
ππ ππ’ππ =
πππ π
The FUNDAMENTAL CONSTANT for the ACCELERATION of GRAVITY can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Where:
π = 32.174
ππ‘ π 2
the acceleration due to gravity Made with
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VOLUME: The TOPIC of VOLUME can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. VOLUME is the AMOUNT of SPACE occupied by an OBJECT or SUBSTANCE such as a SOLID, LIQUID, or GAS. The most commonly used unit for VOLUME is the CUBIC METER (π3 ), which is a DERIVED unit from the METER (m), the SI UNIT for LENGTH. Another common UNIT of VOLUME in the SI system is the liter (L), which is commonly represented as MILLITERS (mL).
In the USCS system of units, VOLUME is given in UNITS of GALLONS, which are denoted as βgalβ.
The FORMULA for SPECIFIC VOLUME can be referenced under the SUBJECT of THERMODYNAMICS on page 87 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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The SPECIFIC VOLUME is the volume occupied by a UNIT MASS of substance. It is expressed in units of ft 3 /lbm or m3 /kg. It is calculated as the total volume divided by the mass:
π=
π π
Where: β’ π is the volume of the system in units of ft 3 or m3 β’ π is the mass of the system in units of lbm or kg β’ π is the specific volume of the system in units of ft 3 /lbm or m3 /kg PRESSURE: The FORMULA for PRESSURE can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. PRESSURE, (π), is defined as the FORCE per unit AREA that is PERPENDICULAR to the FORCE:
π=
πΉ π΄
The most commonly used unit for PRESSURE is the PASCAL which is equal to one NEWTON per SQUARE METER (π/π2 ). The PASCAL is an SI unit that is denoted using the symbol βPaβ. Made with
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In the USCS system of units, PRESSURE is represented as PSI, which is equal to one POUND-FORCE per SQUARE INCH (ππ/ππ2 ). Pressure is measured as GAUGE PRESSURE, the difference between ABSOLUTE and the ATMOSPHERIC. Gauge pressure is either positive or negative (vacuum).
The AMBIENT PRESSURE is pressure at which the vapor pressure is equal to the surrounding pressure. This is the pressure at which a liquid boils. The FORMULA for ABSOLUTE PRESSURE can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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We use the following formula to relate absolute pressure, atmospheric pressure, and gauge pressure. ππππ = πππ‘π + ππππ’ππ Where: ππππ = Absolute Pressure πππ‘π = Atmospheric Pressure πππππ = Gauge Pressure (positive or negative) DENSITY: The TOPIC of DENSITY can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. DENSITY (π) is RATIO of the MASS of a SUBSTANCE to the VOLUME occupied by that SUBSTANCE. The FORMULA for DENSITY can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The FORMULA for DENSITY calculates DENSITY as the QUOTIENT of the MASS (π) divided by the VOLUME (π) or the INVERSE of the SPECIFIC VOLUME (π£) and is given as: Made with
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π₯π π 1 1 = = ππ π = π₯πβ0 π₯π π π π
π = lim
Where: β’ π is the DENSITY β’ m is the MASS β’ V is the VOLUME β’ v is the SPECIFIC VOLUME DENSITY is expressed in DIFFERENT units depending on the PHASE of the SUBSANCE. The density of SOLIDS is usually expressed in UNITS of grams per cubic centimeter (π/ππ3 ), the density of LIQUIDS is commonly expressed in UNITS of grams per milliliter (π/ππΏ), and the density of GASES is commonly expressed in UNITS of grams per liter (π/πΏ).
ENERGY:
The
TOPIC
of
ENERGY
can
be
referenced
under
the
SUBJECT
of
THERMODYNAMICS on page 87 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. ENERGY is the ability to do work. When the work is actually being done, the energy is considered KINETIC, and when the work is waiting to be done and there is the potential do work, the energy is considered POTENTIAL. Made with
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The most commonly used unit for ENERGY is the JOULE. The JOULE is an SI unit that is denoted using the symbol βJβ. ENERGY is always CONSERVED, it is neither CREATED nor DESTROYED, but is transformed from one form to another. KINETIC ENERGY (πΎπΈ) is the ENERGY that results from an objectβs MOTION. The measurement of kinetic energy in an object is CALCULATED based on the objectβs MASS and VELOCITY. POTENTIAL ENERGY (ππΈ) is the STORED ENERGY that results from an objectβs position or arrangement of parts that have the POTENTIAL to become kinetic energy. The MEASUREMENT of POTENTIAL ENERGY in an object is calculated based on the objectβs mass, height, or distance. In CHEMISTRY, POTENIAL ENERGY (PE) is stored in CHEMICAL BONDS, which are the FORCES that hold ATOMS together in COMPOUNDS. INTERNAL ENERGY (π) accounts for all of the ENERGY of a SUBSTANCE, representing the SUM of the KINETIC and POTENTIAL energies of the mass comprising the system. WORK (π) is the PROCESS of transferring ENERGY across the boundaries of a system in the form of mechanical energy across the boundaries of a system.
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TEMPERATURE: The TOPIC of TEMPERATURE can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, 9.4 Version for Computer Based Testing. TEMPERATURE, (π), is the thermodynamic property of a substance that depends on the ENERGY CONTENT. A VARIANCE, or difference in TEMPERATURE, (π₯π), is required for HEAT ENERGY to FLOW. When you measure the temperature of something, youβre measuring the average KINETIC ENERGY of the individual PARTICLES.
A TEMPERATURE SCALE is a unit measurement system for determining the ENERGY of the particles in a material. For the purposes of the FE Exam, the four commonly used temperature scales that we should be familiar with are: Fahrenheit, Celsius, Rankine, and Kelvin. The FAHRENHEIT TEMPERATURE SCALE is based on 32Β°πΉ for the FREEZING POINT of water and 212Β°πΉ for the BOILING POINT of water. A degree on the FAHRENHEIT scale represents 1/180 of the temperature range between the freezing and boiling points of water.
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The CELSIUS TEMPERATURE SCALE is based on 0Β°πΆ for the FREEZING POINT and 100Β°πΆ for the BOILING POINT of water. A degree on the CELSIUS scale represents 1/100 of the temperature range between the freezing and boiling points of water. The ABSOLUTE TEMPERATURE SCALE is a commonly used scale of temperature units in thermodynamic problems, and is given in units of RANKINE (Β°R) and KELVIN (K). As there are no NEGATIVE TEMPERATURES on the ABSOLUTE TEMPERATURE SCALE, the coldest theoretical temperature is ABSOLUTE ZERO, which is represented as 0 K on the Kelvin temperature scale and 0Β°π on the Rankine Scale. ABSOLUTE ZERO represents the temperature at which the THERMAL MOTION of atoms and molecules reaches its MINIMUM. The KELVIN TEMPERATURE SCALE is the absolute temperature scale used in the International Engineering Systems and SI System of units. The Kelvin scale of temperature uses degrees CELSIUS from ABSOLUTE ZERO, and is based on 0 K being defined as absolute zero. The RANKINE TEMPERATURE SCALE is the absolute temperature scale used in the American Engineering System and in the English System of Units. The Rankine scale of temperature uses degrees FAHRENEHTI from ABSOLUTE ZERO, and is based on 0Β°π being defined as absolute zero.
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The FORMULAS for the UNIT CONVERSIONS OF TEMPERATURE can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The formula for converting a CELSIUS temperature to degrees FAHRENHEIT is expressed using the following expression: ππΉ = 1.8 ππΆ + 32 Where: β’ ππΉ is the temperature given in degrees Fahrenheit(Β°πΉ) β’ ππΆ is the temperature given units of degrees Celsius (Β°πΆ)
The formula for converting a FAHRENHEIT temperature to degrees CELSIUS is expressed using the following expression: ππΆ = (ππΉ β 32)/1.8 Where: β’ ππΉ is the temperature given in degrees Fahrenheit(Β°πΉ) β’ ππΆ is the temperature given units of degrees Celsius (Β°πΆ)
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The formula for converting a FAHRENHEIT temperature to the Absolute Temperature Scale in units of degrees RANKINE (Β°π ) is expressed using the following expression: ππ = ππΉ + 459.67
π₯ππ = π₯ππΉ
Where: β’ ππ is the temperature given in degrees Rankine (Β°π ) β’ ππΉ is the temperature given in degrees Fahrenheit(Β°πΉ) The formula for converting a CELSIUS temperature to the Absolute Temperature Scale in units of KELVIN (πΎ) is expressed using the following expression: ππΎ = ππΆ + 273.15
π₯ππΎ = π₯ππΆ
Where: β’ ππ is the temperature given in units of Kelvin (πΎ) β’ ππΆ is the temperature given units of degrees Celsius (Β°πΆ)
HEAT: The TOPIC of HEAT is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the NCEES Supplied Reference Handbook.
HEAT is a MEASURE of the TOTAL amount of ENERGY a substance possesses.
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The most commonly used unit for HEAT is the CALORIE. The CALORIE is an SI unit that is given as βcalβ and represents the AMOUNT of ENERGY it takes to raise the TEMPERATURE of 1 gram of water 1Β°πΆ.
HEAT and TEMPERATURE are both MEASURES of ENERGY.
HEAT, however, is not the same as TEMPERATURE, as HEAT measures the TOTAL ENERGY of a SUBSTANCE, and TEMPERATURE measures the AVERAGE ENERGY of a SUBSTANCE.
HEAT ENERGY (π) is the ENERGY transferred across the BOUNDARIES of a SYSTEM because of the VARIATION in TEMPERATURE. HEAT ENERGY is the RESULT of the MOVEMENT of atoms, molecules, or ions transferring from one solid, liquid, or gas to another. SUBSTANCES change TEMPERATURE when HEATED, but not all substances have their temperature RAISED to the same extent when EQUAL amounts of HEAT are added. SPECIFIC HEAT is the amount of HEAT required to raise the TEMPERATURE of one gram of a SUBSTANCE by one degrees CELSIUS, or other defined TEMPERATURE UNIT.
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SPECIFIC HEAT AT CONSTANT PRESSURE is represented by πΆπ and given in units of π΅π‘π’/πππ-π or ππ½/ππ-πΎ. SPECIFIC HEAT AT CONSTANT VOLUME is represented by βπΆπ β and given in units of π΅π‘π’.
CONVERSION FACTORS: The TABLE of CONVERSION FACTORS can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. A CONVERSION FACTOR is an EQUALITY or NUMERICAL VALUE that can be used to CONVERT between UNITS from one UNIT to another. DIMENSIONAL ANALYSIS, also known as the UNIT FACTOR METHOD, is the process of using CONVERSION FACTORS to convert between UNITS and SYSTEMS of UNITS, without changing the PHYSICAL DEFINITION of the VALUE. The goal of DIMENSIONAL ANALYSIS is to go from a GIVEN UNIT to the DESIRED UNIT by MULTIPLYING the given unit by a CONVERSION FACTOR, in which the NUMERATOR and the DENOMINATOR must represent the same quantity.
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Three Steps of Dimensional Analysis: 1. Write down the GIVEN MEASUREMENT with the NUMERICAL VALUE and associated UNIT. Make sure to take not of the PHYSICAL DIMENSION being represented by the unit. For example, if you see kilograms, you should immediately realize this is a MEASURE of MASS, or if you saw mL, you should immediately be thinking VOLUME. 2. MULTIPLY the MEASUREMENT by one or more CONVERSION FACTORS. The UNIT in each DENOMINATOR must CANCEL OUT units of the original measurement, leaving the DESIRED UNITS in the NUMERATOR. For example, if you convert 72 hours to UNITS of DAYS, you would use the CONVERSION FACTOR of 1 day/24 hours, to cancel out the units of HOUR, and be left with an answer of 3 DAYS. 3. Perform the CALCULATION and report the answer to the proper SIGNIFICANT FIGURES based on the MEASUREMENTS given in the problem statement.
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CONCEPT EXAMPLE: A gold bar is measured to have a mass of 25 kilograms. Given the bar is made of pure gold, the volume of gold in the bar calculated in cubic inches (ππ3 ) is most close to: A. 70 B. 80 C. 90 D. 100
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SOLUTION: The TOPIC of DENSITY can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem we are looking to SOLVE for the VOLUME of the gold in the bar, given the MASS of BAR and ASSUMPTION that the bar is COMPRISED entirely of GOLD. DENSITY (π) is RATIO of the MASS of a SUBSTANCE to the VOLUME occupied by that SUBSTANCE. As we are told the bar is comprised of the element GOLD, we can look up the DENSITY for GOLD in the Reference Handbook, and relate that to the GIVEN MASS to SOLVE for the VOLUME of GOLD in the BAR. The TABLE for PROPERTIES of METALS can be referenced under the SUBJECT of MATERIALS SCIENCE on page 61 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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Referencing the TABLE for the PROPERTIES of METAL in the REFERENCE HANDBOOK, we find the DENSITY of GOLD is given as: π = 19,281 ππ/π3 The FORMULA for DENSITY can be referenced under the SUBJECT of FLUID MECHANICS on page 103 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. The FORMULA for DENSITY calculates DENSITY as the QUOTIENT of the MASS (π) divided by the VOLUME (π) or the INVERSE of the SPECIFIC VOLUME (π£) and is given as: π₯π π 1 1 = = ππ π = π₯πβ0 π₯π π π π Made with by Prepineer | Prepineer.com
π = lim
Where: β’ π is the DENSITY β’ m is the MASS β’ V is the VOLUME β’ v is the SPECIFIC VOLUME Letβs re-write the FORMULA for DENSITY to ISOLATE the term for VOLUME (V):
π=
π π
Plugging in the GIVEN MASS and REFERENCE DENSITY for GOLD, we CALCULATE the VOLUME of the BAR as:
π=
25 ππ = 0.0013 π3 3 19,281 ππ/π
In the USCS system of units, LENGTH is measured using BASE UNITS of INCHES given as "ππ". Therefore, we will need to CONVERT the BASE UNITS of VOLUME from CUBIC METERS (π3 ) to CUBIC INCHES (ππ3 ) to CALCULATE the VOLUME of the GOLD in USCS UNITS. The TABLE of CONVERSION FACTORS can be referenced under the SUBJECT of UNITS on page 1 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Made with
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A CONVERSION FACTOR is an EQUALITY or NUMERICAL VALUE that can be used to CONVERT between UNITS from one UNIT to another. Looking at the TABLE of CONVERSION FACTORS in the REFERENCE HANDBOOK, we can use DIMENSIONAL ANALYSIS to CONVERT from CUBIC METERS (π3 ) to LITERS (πΏ) and then from LITERS (πΏ) to CUBIC INCHES (ππ3 ). Three Steps of Dimensional Analysis: 1. Write down the GIVEN MEASUREMENT with the NUMERICAL VALUE and associated UNIT. As we are working with UNITS of VOLUME, we need to take note that all UNITS are to THIRD POWER. ππππ’ππ = 0.0013 π3 2. MULTIPLY the MEASUREMENT by one or more CONVERSION FACTORS. The UNIT in each DENOMINATOR must CANCEL OUT units of the original measurement, leaving the DESIRED UNITS in the NUMERATOR. a. The FIRST CONVERSION will be CONVERTING from CUBIC METERS (π3 ) to LITERS (πΏ):
πΆπππ£πππ πππ πΉπππ‘ππ:
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1,000 πΏ 1 π3
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b. The SECOND CONVERSION will be CONVERTING from LITERS (πΏ) to CUBIC INCHES (ππ3 ):
πΆπππ£πππ πππ πΉπππ‘ππ:
1πΏ 61.02 ππ3
3. Perform the CALCULATION and report the answer to the proper SIGNIFICANT FIGURES based on the MEASUREMENTS given in the problem statement: 1,000 πΏ 61.02 ππ3 0.0013 π Γ ( )Γ( ) = 79.326 ππ2 = 80 ππ2 3 1π 1πΏ 3
Therefore, the correct answer choice is B. ππ
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