SSC JE Electrical cheat sheet

Bending equation: M/I = σ/y = E/R, where M is bending moment, I is second moment of area, σ is bending stress, y is distance from the neutral axis, E is Young's modulus and R is the radius of curvature.

For a rectangular section of width b and depth d, I = bd³/12 and the section modulus Z = bd²/6, so the maximum bending stress is σ = 6M/(bd²).

Shear stress in a beam is τ = VQ/(Ib), maximum at the neutral axis for a rectangular section (τmax = 3V/(2bd)) and zero at the extreme fibres.

Slope-deflection and moment-distribution methods assume the structure is linearly elastic and displacements are small; the fixed-end moment for a central point load W on a fixed beam is WL/8.

For a simply supported beam with a central point load W and span L, maximum deflection is δ = WL³/(48EI); for a uniformly distributed load w per unit length it is 5wL⁴/(384EI).

Euler buckling load for a pin-ended column is Pcr = π²EI/L²; effective length factors are 1.0 (both ends pinned), 0.5 (both fixed) and 2.0 (one fixed, one free).

Standard consistency of cement is the water content (as a percentage of cement mass) at which the Vicat plunger penetrates 5–7 mm from the bottom; initial setting time must be at least 30 minutes for ordinary Portland cement.

Workability of concrete is commonly specified by slump (in mm), and the target mean strength for mix design is fck plus 1.65 times the standard deviation for the usual 5% failure criterion.

Dumpy level: the line of collimation must be horizontal when the bubble is central; the rise or fall between two stations equals backsight minus foresight, and the reduced level of the next point equals previous reduced level plus rise or minus fall.

For an ideal gas, PV = mRT; for a polytropic process PVⁿ = C, work done is (P₁V₁ − P₂V₂)/(n − 1) for n ≠ 1, and isothermal work is P₁V₁ ln(V₂/V₁).

Carnot efficiency of a heat engine is η = 1 − T₂/T₁ with absolute temperatures; the coefficient of performance of a refrigerator is COP = T₂/(T₁ − T₂) for the reversible cycle.

Otto cycle efficiency is η = 1 − 1/r^(γ−1) where r is the compression ratio; Diesel cycle efficiency also depends on the cut-off ratio rc via η = 1 − [1/r^(γ−1)]·[(rc^γ − 1)/(γ(rc − 1))].

For steady flow, the continuity equation is A₁V₁ = A₂V₂ for incompressible fluid, and Bernoulli's equation P/ρg + V²/2g + z = constant applies along a streamline for ideal flow.

Reynolds number Re = ρVD/μ distinguishes laminar (Re < 2000 in pipes) from turbulent flow; Darcy–Weisbach head loss is hf = fLV²/(2gD).

For a simply supported beam with a central load W, maximum bending moment is WL/4; for a uniformly distributed load w over span L it is wL²/8.

The first law for a closed system is Q − W = ΔU; for a steady-flow open system it is Q − W = Δh + ΔKE + ΔPE per unit mass.

In a four-stroke engine, the firing order repeats once per two crankshaft revolutions; theoretical air required for complete combustion of a fuel is calculated from stoichiometry of carbon, hydrogen and sulphur.

Tool signature in ASA system gives back rake, side rake, end relief, side relief, end cutting edge and side cutting edge angles plus nose radius; cutting speed V = πDN/1000 in m/min when D is in mm and N in rpm.

In a series circuit the current is common and total resistance is the sum of resistances; in a parallel circuit the voltage is common and total conductance is the sum of conductances.

For a series RLC circuit, resonant frequency is fr = 1/(2π√(LC)); at resonance the impedance is purely resistive, is minimum, and current is maximum.

Power in an AC circuit is P = VI cos φ, where cos φ is the power factor; reactive power Q = VI sin φ and apparent power S = VI.

Three-phase power delivered to a balanced load is P = √3 V_L I_L cos φ, where V_L and I_L are line voltage and line current.

For a transformer, V₁/V₂ = N₁/N₂ = I₂/I₁; the emf equation is E = 4.44 f N Φm, where Φm is the maximum flux in the core.

For a DC shunt motor, back emf Eb = V − IaRa and torque T = 0.159 Φ Ia Z / A (or T ∝ Φ Ia); speed N is proportional to Eb/Φ.

The synchronous speed of an induction motor is Ns = 120f/P, where P is the number of poles; slip s = (Ns − N)/Ns and rotor frequency = sf.

Kirchhoff's laws: current law states that the algebraic sum of currents at a node is zero; voltage law states that the algebraic sum of emfs and potential drops around a closed loop is zero.

Thevenin's theorem replaces a linear network by an equivalent voltage source Vth in series with Rth, where Vth is the open-circuit voltage and Rth the resistance seen with sources replaced by their internal resistances.