ISRO Scientist Mechanical cheat sheet

Thermal efficiency of a Carnot engine: η = 1 − T_L/T_H with absolute temperatures; no real engine between the same reservoirs can exceed this value.

Otto cycle efficiency: η = 1 − 1/r^(γ−1), where r = V_1/V_2 is the compression ratio and γ = c_p/c_v; Diesel cycle efficiency additionally depends on the cut-off ratio ρ = V_3/V_2.

Reynolds number Re = ρVD/μ = VD/ν. For pipe flow, Re < 2300 is laminar, 2300–4000 transitional, and Re > 4000 turbulent; hydraulic diameter D_h = 4A/P for non-circular ducts.

Darcy–Weisbach head loss h_f = f(L/D)(V²/2g); for laminar pipe flow f = 64/Re exactly, independent of wall roughness.

Euler beam bending: M/EI = 1/R = d²y/dx²; maximum bending stress σ = My/I. For a simply supported beam under central point load W: δ_max = WL³/(48EI).

Torsion of a solid circular shaft: τ_max = Tr/J with J = πd⁴/32; the angle of twist θ = TL/(GJ). Use J = π(d_o⁴ − d_i⁴)/32 for a hollow shaft.

Mohr's circle: principal stresses σ_1,2 = (σ_x + σ_y)/2 ± √[((σ_x − σ_y)/2)² + τ_xy²]; maximum in-plane shear τ_max = √[((σ_x − σ_y)/2)² + τ_xy²].

Von Mises (distortion energy) criterion: σ_vm = √(σ_1² − σ_1σ_2 + σ_2²) for plane stress; yielding occurs when σ_vm equals the tensile yield strength S_y. Tresca uses σ_1 − σ_3 = S_y.

S-N fatigue approach: endurance limit S_e applies only below about 10⁶–10⁷ cycles; apply Marin modifying factors (surface, size, load, temperature, reliability) to the rotating-beam specimen value.

First Law for a steady-flow open system: q − w = Δh + ΔKE + ΔPE per unit mass; for a nozzle and diffuser w = 0, and for a throttling device h is constant (isenthalpic).

Entropy change of an ideal gas with constant specific heats: Δs = c_p ln(T_2/T_1) − R ln(P_2/P_1); entropy generation must be ≥ 0 for any real process.

Carnot COP of a refrigerator = T_L/(T_H − T_L); of a heat pump = T_H/(T_H − T_L). Actual COP = refrigeration effect / work input.

Bernoulli equation along a streamline for steady, incompressible, inviscid flow: P/ρg + V²/2g + z = constant; it is invalid across a pump, turbine, or shock.

Continuity: A_1V_1 = A_2V_2 for incompressible flow. For compressible flow, mass flow ṁ = ρAV is constant and choking occurs at Mach 1 at the minimum area.

For a normal shock: M_2 < 1 always, stagnation temperature is constant, stagnation pressure drops, and static pressure and entropy rise across the shock.

Metal casting: Chvorinov's rule solidification time t = B(V/A)², where V/A is the modulus; a riser must have a larger modulus than the casting it feeds.

Welding: heat input per unit length H = ηVI/v; in arc welding the voltage–arc length characteristic and the power source characteristic intersect at the operating point.

Metal cutting (Merchant): shear angle φ satisfies 2φ + β − α = 90° for minimum energy; chip thickness ratio r = t_1/t_2 = sin φ / cos(φ − α).

Grinding and machining economics: Taylor tool life VT^n = C; increasing cutting speed reduces tool life, and the exponent n is about 0.1–0.15 for HSS and 0.2–0.4 for carbide.