| name | formula-lookup |
| description | Look up any mechanical engineering formula with full derivation, term table, units, and worked numerical example. Covers all ME domains from Shigley, Hibbeler, Roark, AISC, ASME. |
| metadata | {"priority":7,"promptSignals":{"phrases":["formula for","equation for","what is the formula","how do you calculate","formula lookup","find the formula"],"minScore":3}} |
Formula Lookup — Complete Skill
Response Format (Always Use This)
For any formula request:
1. Formula Name & Domain
State: what it calculates, which domain (SOM, dynamics, fluids, heat transfer, etc.)
2. Formula (KaTeX)
Full equation in proper mathematical notation.
3. Term Table
| Symbol | Name | Meaning | Imperial units | SI units |
|---|
| σ | Normal stress | Stress at fiber distance c from NA | psi, ksi | Pa, MPa |
| ... | ... | ... | ... | ... |
4. Assumptions & Validity
When does this formula apply? When does it break down?
(e.g., "Valid for elastic behavior, σ < Sy")
5. Derivation (Key Steps)
Starting from first principles, show the logical path to the formula.
For students: derivation builds physical intuition.
6. Related Formulas
What other formulas connect to this one?
(e.g., "Flexure formula → shear formula VQ/Ib → transverse shear stress")
7. Numerical Example
Plug in real numbers: show substitution, check units, interpret result.
Quick Reference: Most Requested Formulas
Strength of Materials
σ = Mc/I (bending stress)
σ = P/A (axial stress)
τ = VQ/Ib (transverse shear stress)
τ = Tc/J (torsional shear)
σ_von = √(σ_x² - σ_xσ_y + σ_y² + 3τ_xy²) (von Mises)
δ = PL/AE (axial deformation)
δ = PL³/3EI (cantilever tip)
δ = 5wL⁴/384EI (SS midspan, UDL)
Dynamics
F = ma (Newton 2nd)
T = Iα (rotational Newton)
ω_n = √(k/m) (natural frequency)
X/X_st = 1/|1-r²| (DMF, undamped)
E_k = ½mv², E_p = ½kx² (energy)
Fluid Mechanics
Q = A×V (continuity)
P₁/γ + V₁²/2g + z₁ = P₂/γ + V₂²/2g + z₂ + h_L (Bernoulli)
h_f = f(L/D)(V²/2g) (Darcy-Weisbach)
Re = ρVD/μ (Reynolds)
τ_w = μ(du/dy) (Newton's viscosity)
Heat Transfer
q = kA ΔT/L (conduction)
q = hA ΔT (convection)
q = εσA T⁴ (radiation)
Nu = hL/k (Nusselt)
Bi = hL/k (Biot)
Fo = αt/L² (Fourier)
Machine Design
n = Sy/σ or n = Se/σ_a (factor of safety)
F_i = 0.75 A_t S_p (bolt preload)
T = K d F_i (bolt torque)
W_t = 63025 H/n (gear tangential force, lbf, in, rpm, hp)
L₁₀ = (C/P)³ × 10⁶/60n (bearing life, hours)
Thermodynamics
PV = mRT (ideal gas)
q - w = Δh (steady flow energy)
η_Carnot = 1 - T_L/T_H (max efficiency)
η_Rankine = w_net/q_in (steam cycle)
COP = q_L/w_c (refrigeration)
When Formula Is Unknown
If you cannot recall the exact formula:
- State the principle it comes from (Hooke's law, Newton's law, energy conservation...)
- Derive it from first principles
- Cite the reference: Shigley Ch.X, Hibbeler Ch.X, Roark Table X, AISC Eq. X
- Always verify dimensions before presenting
Units Matter — Always Check
Every formula answer must include unit verification:
[N][m]/[m⁴] = [N/m²] = [Pa] ✓
[kg][m/s²][m] = [N·m] = [J] ✓
Never present a dimensionally inconsistent result.
