Unit Conversions and Dimensional Analysis for the FE Exam
Unit conversion errors are among the most common and preventable mistakes on the FE exam. In the pressure of a timed exam, it is easy to mix up feet and meters, confuse kPa with Pa, or forget to convert minutes to seconds. Mastering dimensional analysis and common conversions can prevent these costly errors.
Why Dimensional Analysis Matters
Dimensional analysis is a method of checking equations and calculations by tracking units. It serves two critical purposes on the FE exam:
- Verification: If the units of your answer do not match what the question asks for, you know something went wrong.
- Problem setup: When you are unsure how to combine given information, dimensional analysis can guide you to the correct equation.
Fundamental Dimensions
All engineering quantities can be expressed in terms of fundamental dimensions:
- Mass (M): kilograms (kg), pounds-mass (lbm), slugs
- Length (L): meters (m), feet (ft), inches (in)
- Time (T): seconds (s), minutes (min), hours (hr)
- Temperature (Theta): Kelvin (K), Celsius (C), Fahrenheit (F), Rankine (R)
- Electric current (I): amperes (A)
SI (Metric) Units
The SI system is used in most FE exam problems. Key derived units:
- Force: Newton (N) = kg * m/s^2
- Pressure/Stress: Pascal (Pa) = N/m^2 = kg / (m * s^2)
- Energy/Work: Joule (J) = N * m = kg * m^2/s^2
- Power: Watt (W) = J/s = kg * m^2/s^3
Common SI prefixes:
- kilo (k) = 10^3
- mega (M) = 10^6
- giga (G) = 10^9
- milli (m) = 10^(-3)
- micro (mu) = 10^(-6)
US Customary Units
Some FE exam problems use US customary units. The key complication is the distinction between pounds-force (lbf) and pounds-mass (lbm):
- 1 lbf = 1 lbm * 32.174 ft/s^2 (at sea level, 1 lbm weighs 1 lbf)
- 1 slug = 1 lbf * s^2/ft (the mass unit consistent with lbf, ft, and s)
The distinction between lbf and lbm is a frequent source of errors. Always be clear about which one you are using.
Essential Conversion Factors
These conversions appear frequently on the FE exam:
Length
- 1 ft = 0.3048 m
- 1 in = 25.4 mm
- 1 mile = 5,280 ft = 1,609 m
Mass
- 1 kg = 2.205 lbm
- 1 slug = 14.59 kg
Force
- 1 kN = 224.8 lbf
- 1 lbf = 4.448 N
Pressure
- 1 atm = 101.325 kPa = 14.696 psi = 760 mmHg
- 1 MPa = 145.0 psi
- 1 bar = 100 kPa
Temperature
- T(K) = T(C) + 273.15
- T(R) = T(F) + 459.67
- T(F) = 1.8 * T(C) + 32
Energy
- 1 BTU = 1,055 J
- 1 kWh = 3,600,000 J = 3,412 BTU
- 1 hp = 746 W = 550 ft*lbf/s
Dimensional Analysis Technique
The key technique is to multiply by conversion factors expressed as fractions equal to 1. For example, to convert 50 miles per hour to meters per second:
50 mi/hr * (5,280 ft/mi) * (0.3048 m/ft) * (1 hr/3,600 s) = 22.35 m/s
Each conversion factor has a numerator and denominator with the same physical quantity but different units. The units you want to cancel go in the opposite position (numerator or denominator) from where they currently appear.
Common Unit Pitfalls on the FE Exam
- kPa vs Pa: Many formulas require pressure in Pa, but problems often give values in kPa. Remember to multiply by 1,000.
- mm vs m: Cross-sectional areas given in mm^2 must be converted to m^2 for use with SI stress formulas. Note: 1 mm^2 = 10^(-6) m^2.
- Degrees vs radians: Trigonometric functions in calculations typically require radians, but angles may be given in degrees.
- Absolute vs gauge pressure: P_abs = P_gauge + P_atm. Some formulas require absolute pressure.
- Per-second vs per-minute: Flow rates, rotational speeds, and other rates may be given in per-minute units but required in per-second units.
Strategy for the Exam
- Always write units next to every number in your calculation
- Cancel units as you go to verify the calculation is set up correctly
- Check that your final answer has the units the question asks for
- If the answer choices have specific units, convert your answer to those units before selecting
- Use the conversion tables in the NCEES Reference Handbook for less common conversions
Developing the habit of tracking units throughout every calculation is one of the simplest ways to improve your FE exam score. It catches errors that would otherwise go unnoticed and builds confidence in your answers.