ESE ME Prelims cheat sheet

SI base units: length m, mass kg, time s, current A, temperature K, amount mol, luminous intensity cd; derived units are products/quotients of these (N = kg·m/s², Pa = N/m², J = N·m, W = J/s).

Dimensional homogeneity rule: every additive term in a valid physical equation has identical dimensions; use it to reject wrong options and to derive exponent relations in dimensionless groups.

Standard prefixes: k = 10³, M = 10⁶, G = 10⁹, m = 10⁻³, µ = 10⁻⁶, n = 10⁻⁹; convert before computing to avoid order-of-magnitude errors.

Significant figures: multiplication/division keeps the least number of significant figures in the inputs; addition/subtraction keeps the least number of decimal places.

Error propagation: for y = f(x₁,x₂), relative uncertainty adds in quadrature for independent random errors: (σ_y/y)² = Σ(∂ln y/∂xᵢ)²σ_xᵢ².

Linear least squares: slope a = [nΣxy − ΣxΣy]/[nΣx² − (Σx)²], intercept b = (Σy − aΣx)/n; correlation coefficient r measures linear fit quality only.

Basic statistics: mean = Σx/n; sample standard deviation s = √[Σ(x−x̄)²/(n−1)]; coefficient of variation = s/mean (dimensionless).

Vector operations: dot product A·B = |A||B|cosθ (scalar, work); cross product |A×B| = |A||B|sinθ (vector, moment, direction by right-hand rule).

Numerical methods: Newton–Raphson x_{n+1} = x_n − f(x_n)/f′(x_n) converges quadratically near a simple root; trapezoidal rule error is O(h²), Simpson's 1/3 rule error O(h⁴).

Continuity for steady incompressible flow: A₁V₁ = A₂V₂; for compressible flow ṁ = ρAV = constant.

Bernoulli (steady, incompressible, inviscid, along a streamline): p/ρ + V²/2 + gz = constant; add head loss h_L and pump head for real systems.

Reynolds number Re = ρVD/µ = VD/ν; pipe flow is laminar below Re ≈ 2300 and turbulent above ≈ 4000; laminar f = 64/Re.

Darcy–Weisbach head loss h_f = f(L/D)(V²/2g); minor losses h_m = K(V²/2g) with K from fitting geometry.

Hydrostatic pressure p = p₀ + ρgh; centre of pressure on a submerged plane lies below the centroid at depth h_cp = h_c + I_c/(A·h_c).

Buoyancy: Archimedes' principle, F_b = ρ_fluid·g·V_displaced; a floating body displaces its own weight of fluid.

Continuity + momentum (control volume): ΣF = ṁ(V_out − V_in); used for thrust, pipe bends, and nozzle reactions.

Boundary-layer thickness grows as δ ∝ x/√Re_x (laminar); drag coefficient of a smooth sphere drops sharply near Re ≈ 3×10⁵ (drag crisis).

Pump and turbine similarity: Q ∝ ND³, H ∝ N²D², P ∝ N³D⁵; specific speed N_s classifies impeller type, and NPSH must exceed vapour-pressure margin to avoid cavitation.

First law for a closed system: Q − W = ΔU; for a steady-flow open system: q − w = Δh + ΔKE + ΔPE per unit mass.

Ideal gas: pV = mRT, R = R̄/M; cp − cv = R, γ = cp/cv; for air R ≈ 0.287 kJ/kg·K, γ ≈ 1.4.

Isentropic relations: T₂/T₁ = (p₂/p₁)^((γ−1)/γ), pV^γ = constant; valid for reversible adiabatic ideal-gas processes.

Carnot efficiency η = 1 − T_L/T_H (absolute temperatures); no engine between the same reservoirs can exceed it.

Second law: entropy of an isolated system never decreases; Δs = ∫δQ_rev/T, and for an ideal gas Δs = cp ln(T₂/T₁) − R ln(p₂/p₁).

Entropy generation S_gen = ΔS_system + ΔS_surroundings ≥ 0; irreversibility I = T₀S_gen is the lost work.

Rankine cycle efficiency is raised by higher boiler pressure, lower condenser pressure, and feedwater regeneration; reheating raises quality at turbine exit.

Air-standard Otto cycle η = 1 − 1/r^(γ−1); Diesel cycle η depends on compression ratio r and cut-off ratio r_c.

Psychrometry: relative humidity φ = p_v/p_sat; dew point is the temperature at which p_v equals saturation pressure; wet-bulb equals dry-bulb at saturation.

Fourier's law: q = −kA(dT/dx); plane wall resistance R = L/(kA), and resistances in series add like electrical resistances.

Newton's law of cooling: q = hA(T_s − T_∞); Biot number Bi = hL_c/k compares internal conduction resistance to external convection resistance.

Nusselt number Nu = hL/k; forced convection correlations: Nu = 0.023Re^0.8Pr^0.4 (turbulent, heating) and Nu = 3.66 (laminar fully developed, constant wall temperature).

Rayleigh number Ra = gβΔT L³/(να); free convection onset requires Ra above a critical value (≈10⁹ for vertical plates).

Stefan–Boltzmann law: q = εσA(T_s⁴ − T_surr⁴) with σ = 5.67×10⁻⁸ W/m²·K⁴; grey-body emissivity ε ≤ 1.

Lumped capacitance: T(t) − T_∞ = (T_i − T_∞)e^(−t/τ) with τ = ρVc/(hA), valid when Bi < 0.1.

Fins: efficiency η_f = tanh(mL)/(mL) for a rectangular fin, m = √(hP/(kA_c)); long-fin effectiveness ε_f = √(kP/(hA_c)).

Heat exchanger LMTD ΔT_lm = (ΔT₁ − ΔT₂)/ln(ΔT₁/ΔT₂), corrected by F for multipass; NTU = UA/C_min, ε = f(NTU, C_r).

Boiling curve regions in order of increasing ΔT_sat: natural convection, nucleate (highest h), critical heat flux, transition, film boiling; film condensation h ∝ (k³ρ²h_fg/µLΔT)^0.25.

Degrees of freedom (Kutzbach): F = 3(n − 1) − 2j − h for planar mechanisms, with n links, j lower pairs, h higher pairs.

Grashof condition: s + l ≤ p + q gives a crank-rocker (shortest link fixed); equality gives a change-point mechanism.

Velocity ratio of a simple gear train = N_driven/N_driver; for compound trains multiply stage ratios; planetary train uses the tabular or relative-velocity method.

Four-bar coupler-point motion is analysed with instantaneous centres; the number of instantaneous centres is n(n − 1)/2.

Cam follower: simple harmonic motion has maximum acceleration at the ends of stroke; cycloidal motion has zero acceleration at both ends and minimum jerk.

Flywheel: ΔE = Iω²(C_s) where C_s = (ω_max − ω_min)/ω_mean is the coefficient of fluctuation of speed; energy stored E = ½Iω².

Balancing: primary unbalanced force of a reciprocating mass varies with crank angle; secondary force oscillates at twice crank speed and is not removed by a single counterweight.

Governor: sensitiveness = (N_max − N_min)/N_mean; a governor is isochronous when equilibrium speed is constant over the sleeve-lift range.

Vibration: natural frequency ω_n = √(k/m), damping ratio ζ = c/(2√(km)); resonance occurs at ω/ω_n = 1, and transmissibility peaks near that ratio.

Factor of safety n = S_yt/σ or S_ut/σ depending on whether yielding or fracture governs; use distortion-energy (von Mises) theory for ductile materials.

Von Mises stress σ′ = √(σ₁² − σ₁σ₂ + σ₂²); Tresca criterion uses σ₁ − σ₃ = S_y and is more conservative.

Soderberg: 1/n = σ_m/S_y + σ_a/S_e; Goodman: 1/n = σ_m/S_ut + σ_a/S_e; Gerber is a parabolic fit between them.

Endurance limit estimate for steel: S_e′ = 0.5S_ut, then multiply by surface, size, load, temperature, and reliability factors; S_e′ ≤ 100 kpsi (700 MPa).

Shaft design under combined loading uses equivalent torque T_e = √(M² + T²) and equivalent moment M_e = ½(M + √(M² + T²)) with maximum-shear theory.

Bolted joints: preload F_i = 0.75F_p for reusable connections; joint stiffness C = k_b/(k_b + k_m), and bolt load F_b = F_i + C·P.

Welded joints: throat thickness = 0.707 × leg size for a 45° fillet; shear stress τ = P/(0.707 h L) for parallel fillet welds.

Rolling bearings: L₁₀ = (C/P)^p million revolutions, p = 3 for ball and 10/3 for roller bearings; rated life is 10⁶ revolutions at rated load C.

Keys and couplings: key shear τ = 2T/(d·b·L) and bearing stress σ = 4T/(d·h·L); a square key has b = h = d/4 typically.

Axial stress σ = P/A; elastic deformation δ = PL/(AE); thermal stress σ = αEΔT when expansion is fully constrained.

Bending stress σ = My/I with maximum at the extreme fibre; for a rectangle I = bh³/12, for a circle I = πd⁴/64.

Transverse shear stress τ = VQ/(Ib); maximum for a rectangle is 1.5V/A at the neutral axis, and 4V/(3A) for a solid circle.

Torsion: τ = Tr/J with J = πd⁴/32 for a solid shaft; angle of twist θ = TL/(GJ).

Euler buckling: P_cr = π²EI/(KL)²; effective length factor K = 1 pinned-pinned, 0.5 fixed-fixed, 0.7 fixed-pinned, 2 fixed-free.

Slenderness ratio λ = KL/r; long columns fail by buckling (Euler), short columns by yielding (Rankine–Gordon or Johnson parabola).

Principal stresses σ₁,₂ = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ_xy²]; maximum shear τ_max = √[((σ_x − σ_y)/2)² + τ_xy²].

Mohr's circle: centre at (σ_x + σ_y)/2 on the σ-axis, radius equal to maximum shear stress; principal planes are 90° apart on the circle.

Strain energy U = P²L/(2AE) for axial loading; resilience and toughness are measured by the area under the stress–strain curve up to yield and to fracture respectively.

Cutting: V = πDN/1000 (m/min) with D in mm and N in rpm; feed rate = f·N for turning and f_z·z·N for milling.

Merchant's shear angle relation: φ = 45° + α/2 − β/2; cutting force ratio and shear-plane area follow from φ.

Taylor's tool life: VT^n = C; n ≈ 0.1–0.15 for HSS, 0.2–0.3 for carbide, 0.4–0.5 for ceramic tools.

Machining time for turning = L/(f·N); for milling = (L + approach)/table feed; grinding uses wheel depth of cut and table speed.

Casting: Chvorinov's rule t = B(V/A)²; riser must solidify after the casting, so it needs a larger V/A than the casting.

Welding: arc voltage and current set heat input H = VI/v (J/mm); SMAW uses a consumable coated electrode, TIG uses a non-consumable tungsten electrode.

Metal forming: rolling draft Δh = h₀ − h_f; forging force rises with flow stress and contact area; extrusion pressure depends on reduction ratio ln(A₀/A_f).

Limits and fits: hole basis H has lower deviation zero; clearance fit means the minimum hole size exceeds the maximum shaft size.

Metrology: least count of a vernier calliper = 1 MSD − 1 VSD; a micrometer least count is pitch/number of circular-scale divisions.

Break-even quantity Q = F/(p − v) where F is fixed cost, p unit price, and v variable cost per unit; contribution margin is p − v.

Economic order quantity EOQ = √(2DS/H) with D annual demand, S ordering cost, H holding cost per unit per year.

Inventory: reorder point = demand rate × lead time + safety stock; safety stock scales with √(lead time) for independent demand.

Learning curve: time for the nth unit T_n = T₁·n^b with b = log(learning rate)/log 2; an 80% curve means the cumulative average falls to 80% at doubled output.

PERT: expected time t_e = (a + 4m + b)/6, variance σ² = ((b − a)/6)²; critical path has zero total float.

Work study: standard time = observed time × rating factor × (1 + allowance); the rating factor normalises the operator to a standard pace.

Line balancing: theoretical minimum number of workstations = ⌈Σ task times / cycle time⌉; cycle time = available time / required output.

Quality control: control limits on an x̄ chart are x̄̄ ± A₂R̄; process capability Cp = (USL − LSL)/(6σ) and Cpk accounts for centring.

Queuing and reliability: for M/M/1, utilisation ρ = λ/µ requires ρ < 1; series system reliability R_s = ΠR_i, and parallel redundancy raises availability.