Quick Reference Guide
Essential formulas, conversions, and data for mechanical engineering.
Fundamental Formulas
Statics
Equilibrium:
ΣF = 0 (sum of forces)
ΣM = 0 (sum of moments)
Moment:
M = F × d (force × perpendicular distance)
Center of Gravity (composite):
x̄ = Σ(A_i × x_i) / Σ A_i
ȳ = Σ(A_i × y_i) / Σ A_i
Dynamics
Newton's Second Law:
F = ma
Kinematic Equations (constant acceleration):
v = v₀ + at
x = x₀ + v₀t + ½at²
v² = v₀² + 2a(x - x₀)
Work and Energy:
W = F × d × cos(θ)
KE = ½mv²
PE = mgh
W_net = ΔKE
Power:
P = W/t = F × v
Momentum:
p = mv
J = FΔt = Δp
Rotational:
ω = v/r
α = a/r
τ = r × F
τ = Iα
KE_rot = ½Iω²
Mechanics of Materials
Stress and Strain:
σ = F/A
ε = ΔL/L₀
σ = Eε (Hooke's Law)
Axial Deformation:
ΔL = FL/(AE)
Bending Stress:
σ = My/I
Torsional Shear Stress:
τ = Tr/J
Shear Stress:
τ = F/A
Factor of Safety:
FOS = σ_failure / σ_actual
Thermodynamics
First Law:
ΔE = Q - W
Q - W = ΔU (closed system)
Ideal Gas Law:
PV = nRT
PV = mRT/M
Efficiency:
η = W_out / Q_in = 1 - Q_out/Q_in
Carnot Efficiency:
η_Carnot = 1 - T_cold/T_hot (T in Kelvin)
COP (Refrigeration):
COP = Q_cold / W_in
Heat Transfer:
q_cond = kA(T₁-T₂)/L
q_conv = hA(T_s-T_∞)
q_rad = εσA(T₁⁴-T₂⁴)
Fluid Mechanics
Pressure:
P = P₀ + ρgh
Continuity (incompressible):
A₁V₁ = A₂V₂ = Q
Bernoulli's Equation:
P₁ + ½ρV₁² + ρgh₁ = P₂ + ½ρV₂² + ρgh₂
Reynolds Number:
Re = ρVD/μ = VD/ν
Drag Force:
F_D = ½ρV²C_D A
Lift Force:
F_L = ½ρV²C_L A
Head Loss (Darcy-Weisbach):
h_L = f(L/D)(V²/2g)
Machine Design
Power-Torque-Speed:
P = Tω = T(2πN/60)
P [kW] = T[N·m] × N[rpm] / 9549
Shaft Torsion:
τ = 16T/(πd³) (solid shaft)
Gear Ratio:
GR = N_in/N_out = D_out/D_in = T_out/T_in
Spring Rate:
k = F/δ
k = Gd⁴/(8D³N_a)
Bearing Life:
L₁₀ = (C/P)³ (ball bearings)
L₁₀ = (C/P)^(10/3) (roller bearings)
Unit Conversions
Length
1 inch (in) = 25.4 mm
1 foot (ft) = 0.3048 m = 304.8 mm
1 mile = 1.609 km
1 mil = 0.001 in = 0.0254 mm
Mass
1 pound-mass (lbm) = 0.4536 kg
1 slug = 14.59 kg
1 metric ton = 1000 kg
1 ton (US) = 2000 lbm = 907.2 kg
Force
1 pound-force (lbf) = 4.448 N
1 kilogram-force (kgf) = 9.807 N
1 kip = 1000 lbf = 4448 N
Pressure
1 psi (lbf/in²) = 6,895 Pa = 6.895 kPa
1 bar = 100,000 Pa = 100 kPa
1 atm = 101,325 Pa = 101.3 kPa = 14.7 psi
1 ksi = 1000 psi = 6.895 MPa
1 mmHg = 133.3 Pa
1 inH₂O = 249.1 Pa
Energy/Work
1 Joule (J) = 1 N·m = 1 W·s
1 kWh = 3.6 MJ
1 BTU = 1,055 J
1 calorie = 4.184 J
1 ft·lbf = 1.356 J
Power
1 horsepower (hp) = 746 W = 0.746 kW
1 hp (metric) = 735.5 W
1 BTU/hr = 0.293 W
1 ton (refrigeration) = 3.517 kW
Temperature
K = °C + 273.15
°F = (9/5)°C + 32
°R = °F + 459.67 (Rankine)
ΔT(°C) = ΔT(K)
ΔT(°F) = ΔT(°R)
Velocity
1 mph = 0.447 m/s = 1.609 km/h
1 km/h = 0.278 m/s
1 knot = 0.514 m/s = 1.852 km/h
1 ft/s = 0.3048 m/s
Volume
1 gallon (US) = 3.785 L
1 cubic foot (ft³) = 28.32 L = 0.02832 m³
1 cubic inch (in³) = 16.39 cm³
1 barrel (oil) = 159 L = 42 gal
Torque
1 lbf·ft = 1.356 N·m
1 lbf·in = 0.113 N·m
Material Properties
Metals (at Room Temperature)
| Material | Density (kg/m³) | E (GPa) | σ_y (MPa) | σ_u (MPa) |
|---|---|---|---|---|
| Steel (mild) | 7850 | 200 | 250 | 400 |
| Steel (4340) | 7850 | 200 | 470 | 745 |
| Stainless 304 | 8000 | 193 | 215 | 505 |
| Aluminum 6061-T6 | 2700 | 69 | 240 | 290 |
| Aluminum 7075-T6 | 2800 | 72 | 505 | 570 |
| Titanium Ti-6Al-4V | 4430 | 114 | 880 | 950 |
| Copper | 8960 | 120 | 70 | 220 |
| Brass | 8500 | 100 | 200 | 350 |
| Cast Iron (gray) | 7200 | 100 | - | 200 |
E = Young's Modulus
σ_y = Yield Strength
σ_u = Ultimate Tensile Strength
Non-Metals
| Material | Density (kg/m³) | E (GPa) | σ_u (MPa) |
|---|---|---|---|
| Concrete | 2400 | 30 | 40 (compression) |
| Wood (pine) | 500 | 10 | 40 |
| ABS Plastic | 1050 | 2.3 | 40 |
| Nylon | 1150 | 3 | 75 |
| PEEK | 1300 | 3.6 | 90 |
| Polycarbonate | 1200 | 2.4 | 60 |
| Glass (soda-lime) | 2500 | 70 | 50 |
| Carbon Fiber (CFRP) | 1600 | 150 | 600 |
Thermal Properties
| Material | k (W/m·K) | c_p (J/kg·K) | α (µm/m·K) |
|---|---|---|---|
| Copper | 400 | 385 | 17 |
| Aluminum | 205 | 900 | 23 |
| Steel | 50 | 490 | 12 |
| Stainless Steel | 16 | 500 | 17 |
| Concrete | 1.4 | 880 | 12 |
| Glass | 1.0 | 840 | 9 |
| Air (20°C) | 0.026 | 1005 | - |
| Water | 0.6 | 4180 | - |
k = Thermal conductivity
c_p = Specific heat
α = Thermal expansion coefficient
Second Moments of Area
| Cross-Section | Axis | I |
|---|---|---|
| Rectangle (b × h) | Through centroid, parallel to b | bh³/12 |
| Rectangle (b × h) | Through base | bh³/3 |
| Circle (diameter d) | Through center | πd⁴/64 |
| Circle (radius r) | Through center | πr⁴/4 |
| Hollow circle (D, d) | Through center | π(D⁴-d⁴)/64 |
| Triangle (b × h) | Through base | bh³/12 |
For composite shapes: I_total = Σ(I_i + A_i × d_i²) using parallel axis theorem
Polar Moments of Inertia
| Shape | J |
|---|---|
| Solid circle (radius R) | πR⁴/2 = πD⁴/32 |
| Hollow circle (R_o, R_i) | π(R_o⁴ - R_i⁴)/2 |
Mass Moments of Inertia
| Shape | Axis | I |
|---|---|---|
| Point mass | Distance r | mr² |
| Thin rod (length L) | Through center, ⊥ | mL²/12 |
| Thin rod (length L) | Through end | mL²/3 |
| Solid cylinder (radius R) | Longitudinal axis | mR²/2 |
| Solid cylinder (radius R) | Through center of mass, ⊥ | m(R²/4 + L²/12) |
| Solid sphere (radius R) | Through center | 2mR²/5 |
| Hollow sphere (radius R) | Through center | 2mR²/3 |
| Disk (radius R) | Through center, ⊥ | mR²/2 |
Physical Constants
Gravitational acceleration (g):
Standard: 9.80665 m/s²
Approximate: 9.81 m/s²
Universal gas constant (R):
8.314 J/(mol·K)
Specific gas constant for air:
287 J/(kg·K)
Stefan-Boltzmann constant (σ):
5.67 × 10⁻⁸ W/(m²·K⁴)
Atmospheric pressure (standard):
101,325 Pa = 101.3 kPa = 1.013 bar = 14.7 psi
Speed of sound in air (20°C):
343 m/s
Density of water (4°C):
1000 kg/m³
Density of air (20°C, 1 atm):
1.2 kg/m³
Dynamic viscosity of water (20°C):
0.001 Pa·s = 1 cP (centipoise)
Standard Thread Sizes
ISO Metric (Coarse)
| Size | Pitch (mm) | Drill (mm) | Tensile Area (mm²) |
|---|---|---|---|
| M3 | 0.5 | 2.5 | 5.03 |
| M4 | 0.7 | 3.3 | 8.78 |
| M5 | 0.8 | 4.2 | 14.2 |
| M6 | 1.0 | 5.0 | 20.1 |
| M8 | 1.25 | 6.8 | 36.6 |
| M10 | 1.5 | 8.5 | 58.0 |
| M12 | 1.75 | 10.2 | 84.3 |
| M16 | 2.0 | 14.0 | 157 |
| M20 | 2.5 | 17.5 | 245 |
UNC (Unified National Coarse)
| Size | TPI | Drill | Tensile Area (in²) |
|---|---|---|---|
| #10-24 | 24 | #25 (0.1495") | 0.0175 |
| 1/4"-20 | 20 | #7 (0.201") | 0.0318 |
| 5/16"-18 | 18 | F (0.257") | 0.0524 |
| 3/8"-16 | 16 | 5/16" | 0.0775 |
| 1/2"-13 | 13 | 27/64" | 0.1419 |
| 5/8"-11 | 11 | 17/32" | 0.226 |
TPI = Threads Per Inch
Standard Tolerances (ISO 2768 Medium)
| Dimension Range | Linear ±(mm) | Angular ±(degrees) |
|---|---|---|
| 0.5 - 3 mm | 0.1 | 1° |
| 3 - 6 mm | 0.1 | 0.5° |
| 6 - 30 mm | 0.2 | 0.5° |
| 30 - 120 mm | 0.3 | 0.25° |
| 120 - 400 mm | 0.5 | 0.2° |
| 400 - 1000 mm | 0.8 | 0.1° |
Useful Relationships
Circumference: C = πd = 2πr
Area of circle: A = πr² = πd²/4
Volume of cylinder: V = πr²h = πd²h/4
Surface area of cylinder: A = 2πrh + 2πr²
Volume of sphere: V = (4/3)πr³
Surface area of sphere: A = 4πr²
Pythagorean theorem: c² = a² + b²
Arc length: s = rθ (θ in radians)
Radians to degrees: deg = rad × (180/π)
Degrees to radians: rad = deg × (π/180)
Quick Checks
Is my calculation reasonable?
Stress checks:
- Steel yield: ~250 MPa
- Aluminum yield: ~200-500 MPa
- If you calculate 2000 MPa in a simple part, check math!
Velocity checks:
- Walking: ~1.5 m/s
- Running: ~5 m/s
- Highway car: ~30 m/s (70 mph)
- Airplane: ~250 m/s (Mach 0.7)
Power checks:
- Human sustained: ~100 W
- Cyclist: ~200-400 W
- Car engine: 50-200 kW
- Large truck: 300-500 kW
End of Reference Guide
You've completed the mechanical engineering foundation course! Continue practicing, work on projects, and keep this reference handy.
Next: Return to README or review any chapter for deeper understanding.