Common Physical Quantities and Units#
Common Units#
Physical Quantity |
Symbol |
SI Unit |
CGS Unit |
Imperial/US Unit |
BG (British Gravitational) Unit |
|---|---|---|---|---|---|
Length |
\(L\) |
meter (m) |
centimeter (cm) |
foot (ft), inch (in) |
foot (ft) |
Mass |
\(m\) |
kilogram (kg) |
gram (g) |
pound-mass (lbm) |
slug |
Time |
\(t\) |
second (s) |
second (s) |
second (s), minute (min) |
second (s) |
Speed |
\(v\) |
meter per second (m/s) |
centimeter per second (cm/s) |
foot per second (ft/s), miles per hour (mph) |
foot per second (ft/s) |
Acceleration |
\(a\) |
meter per second squared (m/s²) |
centimeter per second squared (cm/s²) |
foot per second squared (ft/s²) |
foot per second squared (ft/s²) |
Force |
\(F\) |
newton (N) = kg·m/s² |
dyne = g·cm/s² |
pound-force (lbf) |
pound-force (lbf) |
Energy |
\(E\) |
joule (J) = N·m |
erg = dyne·cm |
British thermal unit (BTU), foot-pound (ft·lb) |
foot-pound (ft·lb) |
Work |
\(W\) |
joule (J) |
erg |
foot-pound (ft·lb) |
foot-pound (ft·lb) |
Power |
\(P\) |
watt (W) = J/s |
erg per second (erg/s) |
horsepower (hp) |
horsepower (hp) |
Pressure |
\(p\) |
pascal (Pa) = N/m² |
barye (Ba) = dyne/cm² |
pound per square inch (psi) |
pound per square foot (psf) |
Density |
\(\rho\) |
kilogram per cubic meter (kg/m³) |
gram per cubic centimeter (g/cm³) |
pound per cubic foot (lb/ft³) |
slug per cubic foot (slug/ft³) |
Volume |
\(V\) |
cubic meter (m³) |
cubic centimeter (cm³) |
cubic foot (ft³), gallon (gal) |
cubic foot (ft³) |
Temperature |
\(T\) |
kelvin (K), Celsius (°C) |
Celsius (°C) |
Fahrenheit (°F) |
Fahrenheit (°F) |
Electric Current |
\(I\) |
ampere (A) |
statampere (statA) |
ampere (A) |
ampere (A) |
Electric Charge |
\(q\) |
coulomb (C) |
statcoulomb (statC) |
coulomb (C) |
coulomb (C) |
Voltage (Electric Potential) |
\(V\) |
volt (V) |
statvolt (statV) |
volt (V) |
volt (V) |
Resistance |
\(R\) |
ohm (Ω) |
statohm (statΩ) |
ohm (Ω) |
ohm (Ω) |
Capacitance |
\(C\) |
farad (F) |
statfarad (statF) |
farad (F) |
farad (F) |
Inductance |
\(L\) |
henry (H) |
stathenry (statH) |
henry (H) |
henry (H) |
Frequency |
\(f\) |
hertz (Hz) = 1/s |
cycles per second (cps) |
hertz (Hz) |
hertz (Hz) |
Angular Velocity |
\(\omega\) |
radian per second (rad/s) |
radian per second (rad/s) |
revolution per minute (rpm) |
revolution per minute (rpm) |
Momentum |
\(p\) |
kilogram meter per second (kg·m/s) |
gram centimeter per second (g·cm/s) |
slug·ft/s |
slug·ft/s |
Impulse |
\(J\) |
newton-second (N·s) |
dyne-second (dyn·s) |
pound-force second (lbf·s) |
pound-force second (lbf·s) |
Torque |
\(\tau\) |
newton-meter (N·m) |
dyne-centimeter (dyn·cm) |
pound-force foot (lbf·ft) |
pound-force foot (lbf·ft) |
Magnetic Flux |
\(\Phi\) |
weber (Wb) |
maxwell |
weber (Wb) |
weber (Wb) |
Magnetic Field Strength |
\(B\) |
tesla (T) |
gauss (G) |
gauss (G) |
gauss (G) |
Heat |
\(Q\) |
joule (J) |
calorie (cal) |
British thermal unit (BTU) |
British thermal unit (BTU) |
Luminous Intensity |
\(I_v\) |
candela (cd) |
stilb (sb) |
candela (cd) |
candela (cd) |
Radioactivity |
\(A\) |
becquerel (Bq) |
curie (Ci) |
curie (Ci) |
curie (Ci) |
Notes#
SI (International System of Units) is the standard unit system used in science and engineering.
CGS (Centimeter-Gram-Second) system is common in some theoretical physics fields.
Imperial/US Customary units are commonly used in the United States for daily measurements and some engineering fields.
Some physical quantities, such as angular velocity or frequency, are expressed in consistent units across systems, but they may be expressed differently in certain contexts (e.g., revolutions per minute in the Imperial system).
Unit Conversion Table#
Quantity |
SI Unit |
Imperial/US Unit |
CGS Unit |
BG Unit |
|---|---|---|---|---|
Length |
1 meter (m) |
3.28084 feet (ft), 39.3701 inches (in) |
100 centimeters (cm) |
3.28084 feet (ft) |
Length |
1 kilometer (km) |
0.621371 miles (mi) |
\(10^5\) centimeters (cm) |
3280.84 feet (ft) |
Length |
1 millimeter (mm) |
0.0393701 inches (in) |
0.1 centimeters (cm) |
0.00328084 feet (ft) |
Length |
1 micrometer (µm) |
3.93701e-5 inches (in) |
0.0001 centimeters (cm) |
3.28084e-6 feet (ft) |
Length |
1 nanometer (nm) |
3.93701e-8 inches (in) |
10 angstroms (Å) |
3.28084e-9 feet (ft) |
Mass |
1 kilogram (kg) |
2.20462 pounds (lb) |
1000 grams (g) |
0.0685218 slugs |
Mass |
1 gram (g) |
0.035274 ounces (oz) |
1 gram (g) |
6.85218e-5 slugs |
Mass |
1 slug |
32.174 pounds (lb) |
14,593 grams (g) |
1 slug |
Mass |
1 milligram (mg) |
0.0154324 grains (gr) |
0.001 grams (g) |
6.85218e-8 slugs |
Area |
1 square meter (m²) |
10.7639 square feet (ft²) |
\(10^4\) square centimeters (cm²) |
10.7639 square feet (ft²) |
Area |
1 hectare (ha) |
2.47105 acres |
\(10^8\) square centimeters (cm²) |
107,639 square feet (ft²) |
Volume |
1 cubic meter (m³) |
35.3147 cubic feet (ft³), 61023.7 cubic inches (in³) |
\(10^6\) cubic centimeters (cm³) |
35.3147 cubic feet (ft³) |
Volume |
1 liter (L) |
0.264172 gallons (gal), 33.814 fluid ounces (fl oz) |
1000 cubic centimeters (cm³) |
0.0353147 cubic feet (ft³) |
Volume |
1 milliliter (mL) |
0.033814 fluid ounces (fl oz) |
1 cubic centimeter (cm³) |
3.53147e-5 cubic feet (ft³) |
Speed |
1 meter/second (m/s) |
2.23694 miles/hour (mph), 3.28084 feet/second (ft/s) |
100 centimeters/second (cm/s) |
3.28084 feet/second (ft/s) |
Speed |
1 kilometer/hour (km/h) |
0.621371 miles/hour (mph) |
27.7778 centimeters/second (cm/s) |
0.911344 feet/second (ft/s) |
Acceleration |
1 meter/second² (m/s²) |
3.28084 feet/second² (ft/s²) |
100 centimeters/second² (cm/s²) |
3.28084 feet/second² (ft/s²) |
Force |
1 newton (N) |
0.224809 pound-force (lbf) |
\(10^5\) dynes (dyn) |
0.224809 pound-force (lbf) |
Energy |
1 joule (J) |
0.000947817 BTU, 0.737562 foot-pound (ft·lb) |
\(10^7\) ergs |
0.737562 foot-pound (ft·lb) |
Energy |
1 kilojoule (kJ) |
0.947817 BTU |
\(10^{10}\) ergs |
737.562 foot-pound (ft·lb) |
Work |
1 joule (J) |
0.737562 foot-pound (ft·lb) |
\(10^7\) ergs |
0.737562 foot-pound (ft·lb) |
Power |
1 watt (W) |
0.00134102 horsepower (hp) |
\(10^7\) erg/second (erg/s) |
0.00134102 horsepower (hp) |
Pressure |
1 pascal (Pa) |
0.000145038 psi (pound/inch²) |
10 baryes (Ba) |
0.0208854 pound per square foot (psf) |
Pressure |
1 atmosphere (atm) |
14.696 psi |
\(1.01325 \times 10^6\) dyn/cm² |
2116.22 pound per square foot (psf) |
Temperature |
0 Celsius (°C) |
32 Fahrenheit (°F) |
273.15 Kelvin (K) |
32 Fahrenheit (°F) |
Momentum |
1 kilogram meter/second (kg·m/s) |
0.737562 slug·ft/s |
\(10^5\) gram centimeter/second (g·cm/s) |
0.737562 slug·ft/s |
Torque |
1 newton meter (N·m) |
0.737562 pound-foot (lbf·ft) |
\(10^7\) dyne centimeter (dyn·cm) |
0.737562 pound-foot (lbf·ft) |
Angular Displacement |
1 radian (rad) |
57.2958 degrees (°) |
1 radian (rad) |
57.2958 degrees (°) |
Angular Velocity |
1 radian/second (rad/s) |
9.5493 revolutions per minute (rpm) |
1 radian/second (rad/s) |
9.5493 revolutions per minute (rpm) |
Heat |
1 joule (J) |
0.000238846 kilocalories (kcal), 0.000947817 BTU |
\(10^7\) ergs |
0.000947817 BTU |
Luminous Intensity |
1 candela (cd) |
0.092903 foot-candle |
1 stilb (sb) |
0.092903 foot-candle |
Electric Current |
1 ampere (A) |
1 ampere (A) |
3.33564 × \(10^{-10}\) statA |
1 ampere (A) |
Electric Charge |
1 coulomb (C) |
1 coulomb (C) |
\(3.33564 × 10^{-10}\) statC |
1 coulomb (C) |
Voltage |
1 volt (V) |
1 volt (V) |
\(299.79 \times 10^6\) statvolt |
1 volt (V) |
Resistance |
1 ohm (Ω) |
1 ohm (Ω) |
\(8.9876 \times 10^{11}\) statohm |
1 ohm (Ω) |
Capacitance |
1 farad (F) |
1 farad (F) |
\(1.11265 \times 10^{-12}\) statfarad |
1 farad (F) |
Inductance |
1 henry (H) |
1 henry (H) |
\(8.9876 \times 10^{11}\) statH |
1 henry (H) |
Notes:#
Length: Includes conversions between meters, feet, inches, and centimeters.
Mass: Includes conversions between kilograms, pounds, ounces, and grams.
Area: Includes conversions between square meters, square feet, hectares, acres, and square centimeters.
Volume: Includes conversions between liters, gallons, cubic feet, cubic inches, and cubic centimeters.
Speed: Includes conversions between meters per second, miles per hour, feet per second, and centimeters per second.
Energy: Includes conversions between joules, foot-pounds, BTU, and ergs.
Pressure: Includes conversions between pascals, psi, atmospheres, and dynes/cm².
Electricity & Magnetism: Includes units for electric current, electric charge, voltage, resistance, capacitance, inductance, and magnetic field strength.
The imperial system and the British Gravitational (BG) system are not exactly the same, though they are related and share some units.
Key Differences Between Imperial and BG Systems#
Measurement of Force:#
In the BG system, force is a primary unit and is measured in pounds-force (lbf). This system is structured around the gravitational force, so mass is often derived based on weight.
In the imperial system, measurements traditionally used pounds (lb) as a unit of mass. However, in modern contexts (especially engineering), “pound” usually refers to force, which aligns it more closely with the BG system.
Units of Mass and Weight:#
In the BG system, mass is measured in slugs. The slug is a derived unit, defined such that one slug accelerates at (1 , \text{ft/s}^2) under a force of (1 , \text{lbf}).
In contrast, the imperial system uses pound-mass (lbm), often with gravitational conversions. For example, (1 , \text{lbm}) weighs (1 , \text{lbf}) under standard gravity (32.174 ft/s²), which can create confusion without a consistent handling of gravity.
Consistent Unit Systems:#
The BG system is designed to be internally consistent with a clear differentiation between mass and force, which helps avoid conversion issues between mass (slugs) and force (pound-force).
The imperial system does not inherently provide this internal consistency, so conversions between force and mass may require gravitational constants depending on the context.
Example in Fluid Mechanics Context#
When calculating forces, pressures, or dynamic interactions in fluid mechanics:
The BG system is often used because it provides consistency for force-based calculations.
In the imperial system, conversions may be necessary to clarify if pounds are representing mass or force.
In summary, while both systems use similar units (e.g., feet, pounds), they apply them differently. The BG system offers a consistent approach for engineering calculations by defining mass in terms of slugs, aligning more directly with how the SI system separates kilograms (mass) and newtons (force).
Note
Slugs vs. Pounds in the British Gravitational (BG) System
In the British Gravitational (BG) system, there are two common ways to express mass-related quantities in fluid mechanics: slugs and pounds. These units represent different concepts and are often used in different contexts within fluid mechanics.
1. Mass Density vs. Weight Density
Slugs per cubic foot (\(\text{slug/ft}^3\)):
This is a measure of mass density in the BG system. It corresponds to the mass per unit volume of a fluid, similar to how kg/m³ is used in the SI system.
Pounds per cubic foot (\(\text{lb/ft}^3\)):
This is a measure of weight density (also called specific weight), which represents the weight of the fluid per unit volume. This unit incorporates the force of gravity because it measures the fluid’s weight in pounds.
2. Key Difference Between Slug and Pound Units
Slug (\(\text{slug}\)):
A slug is a unit of mass in the BG system.
The relationship between mass and weight in the BG system is defined by:
\[ W = mg \]Where:
\(W\) is the weight (in pounds, \(\text{lb}\)),
\(m\) is the mass (in slugs, \(\text{slug}\)),
\(g\) is the acceleration due to gravity (typically \(32.174 \, \text{ft/s}^2\)).
One slug is the amount of mass that, under standard Earth gravity, weighs 1 pound-force per unit of acceleration.
Pound (\(\text{lb}\)):
A pound is a unit of force (weight) in the BG system.
When we use \(\text{lb/ft}^3\), we are expressing weight density (the weight per unit volume). This value includes the force due to gravity.
3. Conversion Between Slug and Pound
To convert between weight density (\(\text{lb/ft}^3\)) and mass density (\(\text{slug/ft}^3\)), you need to account for the acceleration due to gravity (\(g\)):
For example, the weight density of water is approximately \(62.4 \, \text{lb/ft}^3\). To convert this to mass density in slugs per cubic foot:
4. When Slugs are Used
In fluid mechanics, slugs/ft³ is often used when dealing with mass density because it simplifies calculations involving dynamics. By using slugs, you can apply Newton’s second law (\(F = ma\)) directly without introducing a conversion factor for gravity, as slugs are already based on the relationship between mass and weight under standard gravity.
5. Common Mistake: Mixing Slugs and Pounds
It’s easy to confuse mass density and weight density in the BG system because both use units related to mass and force. Keep in mind:
\(\text{lb/ft}^3\) refers to weight density, which measures the weight of the fluid in pounds per cubic foot.
\(\text{slug/ft}^3\) is the correct unit for mass density in the BG system.
Always convert between the two by dividing or multiplying by the acceleration due to gravity (\(g = 32.174 \, \text{ft/s}^2\)).
6. Summary of Differences
Quantity |
Symbol |
Definition |
Units in BG System |
Units in SI System |
|---|---|---|---|---|
Mass Density |
\(\rho_{\text{mass}}\) |
Mass per unit volume |
\(\text{slug/ft}^3\) |
\(\text{kg/m}^3\) |
Weight Density |
\(\rho_{\text{weight}}\) |
Weight per unit volume (includes gravity) |
\(\text{lb/ft}^3\) |
\(\text{N/m}^3\) |
Mass Density: Refers to the mass contained in a given volume, and in the BG system, it is measured in slugs/ft³.
Weight Density: Refers to the weight of the fluid per unit volume, typically measured in lb/ft³, and incorporates the effect of gravity.
Practical Example: Calculating Density
Let’s calculate the mass density of air using its typical weight density at sea level (\(\approx 0.0765 \, \text{lb/ft}^3\)):
In this case, the mass density of air is about \(0.00238 \, \text{slug/ft}^3\), while its weight density remains \(0.0765 \, \text{lb/ft}^3\).
Units Cheat Sheet#
Core Unit Systems#
SI (MKS): meter (m), kilogram (kg), second (s), newton (N), joule (J), pascal (Pa)
CGS: centimeter (cm), gram (g), second (s), dyne, erg, barye
Imperial/US: foot (ft), pound-mass (lbm), second (s), pound-force (lbf), foot-pound (ft·lb), psi
British Gravitational (BG): foot (ft), slug (mass), second (s), pound-force (lbf)
Quick Conversions#
Length: 1 m = 3.281 ft = 39.37 in
Mass: 1 kg = 2.205 lbm = 0.0685 slugs
Force: 1 N = 0.225 lbf = \(10^5\) dyn
Energy: 1 J = 0.7376 ft·lb = \(10^7\) erg = 0.000948 BTU
Pressure: 1 Pa = 0.000145 psi = 10 Ba
Density (water):
SI: \(1000 \,\text{kg/m}^3\)
BG (mass density): \(1.94 \,\text{slug/ft}^3\)
BG (weight density): \(62.4 \,\text{lb/ft}^3\)
Slug vs. Pound (BG System)#
Slug = mass unit
Pound-force (lbf) = force (weight)
Relation:
\[ W = m g \quad\;\; (g = 32.174 \,\text{ft/s}^2) \]Mass density: slugs/ft³ (like kg/m³)
Weight density: lb/ft³ (like N/m³)
👉 Example:
Water: \(62.4 \,\text{lb/ft}^3 \;\Rightarrow\; 1.94 \,\text{slug/ft}^3\)
Air: \(0.0765 \,\text{lb/ft}^3 \;\Rightarrow\; 0.00238 \,\text{slug/ft}^3\)
Key Reminder#
SI is the universal standard
BG is preferred for fluid mechanics in Imperial contexts (avoids \(g_c\) confusion)
Always check if “lb” means mass (lbm) or force (lbf)