GATE ME Formula Sheet for Quick Revision
GATE Mechanical Engineering important formulas on one page: SOM, TOM and vibrations, design, fluids, thermodynamics, heat transfer, manufacturing and IE.
The formulas that GATE Mechanical Engineering numerical questions use most, subject by subject. Use this page for revision after studying each lesson.
Strength of materials
| Formula | Meaning |
|---|---|
| δ = PL/(AE), σ = My/I, τ = Tr/J | Axial, bending and torsion |
| E = 2G(1 + ν) | Relation between elastic constants |
| σ₁,₂ = (σx + σy)/2 ± √(((σx − σy)/2)² + τ²) | Principal stresses |
| σ_h = pd/(2t), σ_l = pd/(4t) | Thin cylinder hoop and longitudinal stress |
| P_cr = π²EI/L_e² | Euler buckling load |
See Mechanics and strength of materials.
Theory of machines and vibrations
| Formula | Meaning |
|---|---|
| ω_n = √(k/m), f_n = ω_n/(2π) | Natural frequency of a spring–mass system |
| ζ = c/(2√(km)) | Damping ratio |
| Logarithmic decrement δ = 2πζ/√(1 − ζ²) | From successive amplitudes: δ = ln(x₁/x₂) |
| Gear ratio = N₁/N₂ = T₂/T₁ | Speed and teeth of meshing gears |
| Grübler: F = 3(n − 1) − 2j₁ − j₂ | Degrees of freedom of a planar mechanism |
See Theory of machines and vibrations.
Machine design
| Formula | Meaning |
|---|---|
| Factor of safety = failure stress / working stress | Design margin |
| Goodman: σa/Se + σm/Sut = 1/N | Fatigue under mean and alternating stress |
| L₁₀ = (C/P)^3 million revolutions | Ball bearing life (exponent 10/3 for roller bearings) |
| Max shear stress theory: τ_max = S_y/(2N) | Tresca criterion |
See Machine design.
Fluid mechanics
| Formula | Meaning |
|---|---|
| p/ρg + V²/2g + z = constant | Bernoulli along a streamline |
| h_f = fLV²/(2gD); laminar f = 64/Re | Darcy–Weisbach |
| F = ρQ(V₁ − V₂) | Momentum equation (jet force) |
| Re = ρVD/μ | Reynolds number |
See Fluid mechanics.
Thermodynamics and applications
| Formula | Meaning |
|---|---|
| Q − W = ΔU (closed), q − w = Δh + ΔKE + ΔPE (open) | First law |
| η_Carnot = 1 − T_L/T_H | Carnot efficiency (kelvin) |
| COP_R = T_L/(T_H − T_L), COP_HP = COP_R + 1 | Carnot refrigerator and heat pump |
| η_Otto = 1 − 1/r^(γ−1) | Otto cycle efficiency |
| pv = RT, pv^n = constant | Ideal gas and polytropic process |
See Thermodynamics and applications.
Heat transfer
| Formula | Meaning |
|---|---|
| Q = kAΔT/L | Conduction through a plane wall |
| R = L/(kA) (conduction), 1/(hA) (convection) | Thermal resistances add in series |
| Q = σεA(T₁⁴ − T₂⁴) | Radiation, σ = 5.67 × 10⁻⁸ W/m²K⁴ |
| LMTD = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂) | Heat exchanger |
| Bi = hL_c/k | Lumped analysis valid when Bi < 0.1 |
See Heat transfer.
Manufacturing and industrial engineering
| Formula | Meaning |
|---|---|
| VT^n = C | Taylor tool life |
| Merchant: 2φ + β − α = 90° | Shear angle relation |
| EOQ = √(2DS/H) | Economic order quantity |
| Break-even quantity = fixed cost / (price − variable cost) | Break-even analysis |
See Materials and manufacturing and Industrial engineering.
Check yourself
A spring of stiffness 400 N/m carries a 4 kg mass. What is the natural frequency in rad/s?
Show answer and explanation
Answer: 10ω_n = √(k/m) = √100 = 10 rad/s.
A Carnot engine works between 600 K and 300 K. What is its efficiency?
Show answer and explanation
Answer: 0.5η = 1 − 300/600 = 0.5.
Annual demand is 1000 units, ordering cost ₹50 per order and holding cost ₹4 per unit per year. What is the EOQ?
Show answer and explanation
Answer: 158.11EOQ = √(2 × 1000 × 50/4) = √25000 ≈ 158.11 units.
A wall 0.2 m thick with k = 1 W/mK and area 10 m² has a temperature difference of 40 K. What is the heat flow in W?
Show answer and explanation
Answer: 2000Q = kAΔT/L = 1 × 10 × 40/0.2 = 2000 W.
A planar mechanism has 4 links and 4 lower pairs (revolute joints). How many degrees of freedom does it have?
Show answer and explanation
Answer: 1F = 3(4 − 1) − 2 × 4 = 9 − 8 = 1 (the four-bar linkage).