GATE Mechanical Engineering (ME) Calculators
One calculator per formula in the GATE Mechanical Engineering syllabus, grouped by section. Enter your numbers to get the full answer with step-by-step working.
53 calculators · 10 converters · common conversions
Engineering Mathematics
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Eigenvalue Calculator
det(A − λI) = 0
Σλ = trace A, Πλ = det A
Example: The matrix [[4, 1], [2, 3]]: Eigenvalues: λ₁ = 5, λ₂ = 2.
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Simpson’s Rule Calculator
I = h/3 [y₀ + 4(y₁ + y₃ + …) + 2(y₂ + y₄ + …) + yₙ]
h step size, n even number of intervals
Example: y = 1/(1 + x²) at x = 0, 0.25, …, 1 (h = 0.25): Integral I ≈ 0.785392.
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- Binomial Distribution Calculator – Trials n, probability p and successes k → exact, cumulative and tail probabilities, mean and variance.
- Poisson Distribution Calculator – Mean rate λ and count k → exact, cumulative and tail Poisson probabilities.
- Standard Deviation Calculator – Paste a list of numbers → standard deviation, variance, mean, median, MAD and CV with steps.
- Area Calculator – Pick a shape, enter its sides in any unit, and get the area in m², ft², acres or hectares with steps.
Mechanics of Materials
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Beam Deflection Calculator
δ = PL³/3EI, wL⁴/8EI, PL³/48EI, 5wL⁴/384EI
P point load, w load per length, L span, E Young’s modulus, I second moment of area
Example: A 2 m steel cantilever (E = 200 GPa, I = 8 × 10⁶ mm⁴) with 10 kN at the free end: Maximum deflection δ (free end) = 16.6667 mm.
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Bending Stress Calculator
σ = M y / I = M / Z
M bending moment, y distance from the neutral axis, I second moment of area, Z = I/y
Example: A 100 × 200 mm rectangular beam carrying a 20 kN·m bending moment: Maximum bending stress σ = 30 MPa.
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- Mohr’s Circle Calculator – σx, σy and τxy → principal stresses, maximum shear, principal angle and stresses on a plane.
- Euler Buckling Calculator – End conditions, E, I, length and area → Euler critical load, critical stress and slenderness.
- Shaft Torsion Calculator – Torque, diameters, G and length → maximum shear stress, polar moment, angle of twist and power.
- Thin Cylinder Calculator – Pressure, diameter and thickness → hoop and longitudinal stress, strains and volumetric strain.
- Thermal Stress Calculator – E, α, temperature change, length and gap → thermal stress, free expansion and thermal force.
- Elastic Constants Calculator – Any two of E, G, K and ν → the other elastic constants and Lamé’s λ.
- Strain Energy Calculator – Load type, load, length, area and E → maximum stress, extension and strain energy stored.
- Tensile Stress Calculator – Load + round, rectangular or tube section → tensile or yield stress in MPa, N/mm², psi, ksi.
- Young's Modulus Calculator – Stress and strain → Young's modulus, or E with one of them → the other (Hooke's law).
- Thermal Expansion Calculator – Material or α + length + temperature change → expansion ΔL and new length.
Theory of Machines and Vibrations
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Natural Frequency Calculator
ωn = √(k/m), fn = ωn/2π, ζ = c/2√(km)
m mass, k stiffness, c damping coefficient
Example: A 10 kg mass on a 4000 N/m spring: Natural frequency fn = 3.1831 Hz.
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Transmissibility Calculator
TR = √[1 + (2ζr)²] ÷ √[(1 − r²)² + (2ζr)²]
r = ω/ωn frequency ratio, ζ damping ratio
Example: A machine running at twice the natural frequency with ζ = 0.1: Transmissibility TR = 0.355862.
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- Gear Train Calculator – Input speed and teeth on each meshing pair → speed ratio, output speed, direction and torque.
- Flywheel Calculator – Fluctuation of energy, mean speed and Cs → moment of inertia, mass and speed limits.
- Belt Tension Calculator – Maximum tension, belt mass, speed, μ and wrap angle → T₁, T₂, centrifugal tension and power.
- Critical Speed Calculator – Static deflection, or rotor mass and shaft stiffness → critical whirling speed in rpm.
Machine Design
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Helical Spring Calculator
τ = Kw 8PD/πd³, δ = 8PD³n/Gd⁴
Kw = (4C − 1)/(4C − 4) + 0.615/C, C = D/d
Example: A spring with d = 10 mm, D = 80 mm, 10 active coils and G = 80 GPa under 1000 N: Maximum shear stress τ (Wahl) = 241.206 MPa.
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Fatigue Factor of Safety Calculator
1/n = σa/Se + σm/Sut (Goodman)
Soderberg uses Syt; Gerber nσa/Se + (nσm/Sut)² = 1
Example: σm = 100 MPa, σa = 80 MPa, Se = 250 MPa, Sut = 600 MPa, Syt = 450 MPa: Factor of safety n (Goodman) = 2.05479.
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Bearing Life Calculator
L10 = (C/P)ᵏ × 10⁶ rev, P = XFr + YFa
k = 3 ball, 10/3 roller; L10h = L10 × 10⁶ / 60N
Example: A ball bearing with C = 30 kN under a 5 kN radial load at 1000 rpm: Rating life L10h = 3600 h.
Fluid Mechanics
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Bernoulli Equation Calculator
p₁/ρg + V₁²/2g + z₁ = p₂/ρg + V₂²/2g + z₂ + hL
p pressure, V velocity, z elevation, hL head loss
Example: Water at 200 kPa and 2 m/s accelerating to 6 m/s while rising 3 m: Pressure at 2, p₂ = 154.57 kPa.
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Hydrostatic Force Calculator
F = ρ g A h̄, h* = h̄ + IG sin²θ / (A h̄)
h̄ depth of the centroid, IG second moment about the centroid, θ angle to the free surface
Example: A vertical 2 m wide, 3 m high gate with its top edge 1 m below water: Total pressure force F = 147.15 kN.
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- Metacentric Height Calculator – Floating block dimensions and specific gravity, or I, V and BG → metacentric height, stability and buoyancy.
- Venturi Meter Calculator – Inlet and throat diameters, manometer reading or Δp and Cd → actual and theoretical discharge.
- Reynolds Number Calculator – Velocity, diameter, density and viscosity in any units → Re and flow regime.
- Pipe Pressure Drop Calculator – Flow + pipe size + material + fluid → pressure drop, velocity, Re and friction factor.
Heat Transfer
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LMTD Calculator
LMTD = (ΔT₁ − ΔT₂) ÷ ln(ΔT₁/ΔT₂)
counter-flow ΔT₁ = Th,i − Tc,o, ΔT₂ = Th,o − Tc,i
Example: Counter-flow: hot 150 → 100 °C, cold 30 → 70 °C, Q = 200 kW, U = 500 W/m²·K: LMTD = 74.8888 °C.
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NTU Effectiveness Calculator
ε = [1 − e^(−NTU(1 − Cr))] ÷ [1 − Cr e^(−NTU(1 − Cr))]
NTU = UA/Cmin, Cr = Cmin/Cmax (counter-flow)
Example: UA = 3000 W/K, Ch = 4000 W/K, Cc = 2000 W/K, hot in at 120 °C, cold in at 20 °C: Effectiveness ε = 0.690785.
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- Heat Conduction Calculator – Geometry, conductivities and temperatures → heat flow, thermal resistance and interface temperatures.
- Fin Efficiency Calculator – Fin size, h, k and temperatures → fin efficiency, heat transfer, effectiveness and mL.
- Radiation Heat Transfer Calculator – Temperatures, emissivities and area → net radiation, emissive power and Wien peak wavelength.
- Lumped Capacitance Calculator – Body size, h and properties → Biot number, time constant, and time to a temperature or temperature after a time.
Thermodynamics
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Carnot Efficiency Calculator
η = 1 − TL/TH
COPR = TL/(TH − TL), COPHP = TH/(TH − TL), absolute temperatures
Example: An engine between 600 K and 300 K: Carnot efficiency η = 50 %.
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Otto Cycle Calculator
η = 1 − 1/r^(γ−1)
r compression ratio, γ = cp/cv
Example: r = 8, γ = 1.4, intake 300 K and 100 kPa, peak 2000 K: Air-standard efficiency η = 56.4725 %.
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- Diesel Cycle Calculator – Compression ratio, cut-off ratio, γ and intake state → efficiency, temperatures, work and MEP.
- Brayton Cycle Calculator – Pressure ratio, temperatures and component efficiencies → thermal efficiency, work and back work ratio.
- Polytropic Process Calculator – Initial state, final pressure, n and γ → final volume and temperature, work, heat and ΔU.
- Air–Fuel Ratio Calculator – Fuel formula and excess air → stoichiometric and actual air–fuel ratio, φ and λ.
- Ideal Gas Law Calculator – Solve PV = nRT for P, V, n or T in any pressure, volume and temperature units.
Manufacturing and Industrial Engineering
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Taylor Tool Life Calculator
V Tⁿ = C
V cutting speed, T tool life, n exponent (≈ 0.1 HSS, 0.25 carbide), C constant
Example: V = 100 m/min with n = 0.25 and C = 300: Tool life T = 81 min.
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Machining Time Calculator
t = L / (f N), MRR = V f d
V = πDN/1000 m/min, f feed per rev, d depth of cut
Example: Turning a 50 mm bar at 500 rpm, 0.2 mm/rev, 2 mm depth over 200 mm: Machining time t = 2 min.
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- PERT Calculator – Optimistic, most likely and pessimistic times on the critical path → expected duration, σ and completion probability.
- M/M/1 Queue Calculator – Arrival and service rates → utilisation, L, Lq, W, Wq and P₀.
- EOQ Calculator – Annual demand, ordering and holding cost → EOQ, total cost and reorder point.
- Moving Average Calculator – Time series and window → moving-average forecasts and accuracy.
- Exponential Smoothing Calculator – Time series and α → exponentially smoothed forecasts and accuracy.
Unit Converters for GATE Mechanical (ME)
- Pressure Converter – Pa, kPa, psi, bar, inH₂O, mmHg, torr, atm
- Force Converter – N, kN, lbf, kgf, kip, dyn, ozf, gf
- Torque Converter – N·m, lbf·ft, lbf·in, kgf·cm, kgf·m, ozf·in
- Power Converter – W, kW, MW, hp, PS, BTU/h, TR, kcal/h
- Energy Converter – J, kJ, kWh, kcal, cal, BTU, therm, eV
- Temperature Converter – °C, °F, K, °R
- Heat Transfer Coefficient Converter – W/(m²·K), BTU/(h·ft²·°F), kcal/(h·m²·°C), U-value
- Thermal Conductivity Converter – W/(m·K), mW/(m·K), BTU/(h·ft·°F), k-value, kcal/(h·m·°C)
- Dynamic Viscosity Converter – Pa·s, mPa·s, cP, P, lbf·s/ft²
- Kinematic Viscosity Converter – m²/s, mm²/s, cSt, St, ft²/s
Common GATE Mechanical (ME) Conversions
How to use these GATE ME calculators
Each calculator matches one formula in the GATE Mechanical Engineering syllabus and is grouped under the same section headings as the official syllabus. Inputs use the units GATE questions are set in (mm, MPa, GPa, kN, rpm), and g is taken as 9.81 m/s² as in GATE papers.
Use them to check your hand calculations while practising previous-year questions. Every result shows the formula with your numbers substituted and the intermediate values, so you can see where your working differs. The full answer is shown unrounded, followed by the same value cut to fewer decimals, which helps when matching a numerical answer type (NAT) range.
Calculators are not allowed in the exam beyond the on-screen virtual calculator, so the aim is to understand each relation, not to depend on the tool. The formula, variables and a worked example are given on every page.