GATE GG flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Trace and determinant eigenvalues

For a square matrix A, the sum of all eigenvalues equals the trace and their product equals the determinant, allowing quick verification of computed roots.

Integrating factor solution

For dy/dx + P(x)y = Q(x), multiply both sides by exp(∫P dx); the left side becomes an exact derivative, and integration gives y.

Repeated characteristic root solution

When the auxiliary equation of a second-order constant-coefficient ODE has a repeated root r, the general solution is (C1 + C2 x)e^(rx).

Circular motion acceleration

A particle moving on a circle of radius r at speed v has centripetal acceleration v²/r directed toward the centre, plus tangential acceleration dv/dt.

Coriolis and centrifugal terms

In a frame rotating at angular velocity ω, apparent forces per unit mass are −2ω × v (Coriolis) and −ω × (ω × r) (centrifugal), both frame-dependent.

Principal stresses as eigenvalues

The three principal stresses are eigenvalues of the symmetric stress tensor; maximum shear stress equals half the difference between the largest and smallest principal stresses.

Simply supported beam central load

For span L and central point load W, maximum bending moment is WL/4 at midspan and maximum deflection is WL³/(48EI), with E the modulus and I the second moment.

Cantilever end-load deflection

A cantilever of length L with a point load W at its free end deflects WL³/(3EI) at the tip, and the fixed-end bending moment is WL.

Torsion of circular shaft

Shear stress is τ = Tr/J and twist angle is θ = TL/(GJ); for a solid shaft J = πd⁴/32, so stress peaks at the outer radius.

Ideal gas internal energy

For an ideal gas, internal energy depends only on temperature: ΔU = nC_vΔT for any process, while ΔH = nC_pΔT and C_p − C_v = R.

Ideal gas entropy change

Entropy change is ΔS = nC_v ln(T2/T1) + nR ln(V2/V1), or nC_p ln(T2/T1) − nR ln(P2/P1); both forms agree for an ideal gas.

Carnot and refrigerator COP

A reversible engine between T_H and T_C has efficiency 1 − T_C/T_H; a reversible refrigerator has COP = T_C/(T_H − T_C), using absolute temperatures.

Plane wall conduction resistance

Steady conduction through a plane wall gives q = kAΔT/L, so thermal resistance is L/(kA); series resistances add, and parallel paths combine as conductance sums.

Thevenin equivalent procedure

Compute open-circuit voltage for V_th, then deactivate independent sources (short voltage, open current) and find resistance seen at the terminals for R_th.

Ideal transformer relations

Voltage ratio equals turns ratio N1/N2, current ratio is its inverse, and an impedance Z2 on the secondary appears as (N1/N2)²Z2 on the primary.

HHV versus LHV

Higher heating value counts latent heat of water vapour in combustion products; lower heating value excludes it, differing by the water mass times latent heat.

Stoichiometric air requirement

Theoretical air mass per kg fuel equals oxygen mass required divided by 0.232, the oxygen mass fraction of air; excess air ratio scales actual to theoretical air.

Solar declination formula

Declination δ ≈ 23.45° sin[360(284 + n)/365] for day number n; it varies between about +23.45° at summer solstice and −23.45° at winter solstice.

Noon solar altitude angle

At solar noon, altitude α = 90° − |φ − δ|, where φ is latitude and δ is declination; the sun is overhead when φ equals δ.

Tilt factor for beam radiation

The tilt factor R_b is the ratio cos θ / cos θ_z, comparing beam incidence on a tilted surface with that on a horizontal surface at the same instant.

Flat-plate collector useful gain

Useful heat is Q_u = A_c[F_R(τα)G − F_R U_L(T_i − T_a)]; efficiency is Q_u divided by incident solar power A_c G.

PV fill factor

Fill factor is P_max/(V_oc·I_sc), a measure of cell quality; higher fill factor means the current-voltage curve is squarer and conversion is better.

PV temperature coefficient effect

Rising cell temperature raises short-circuit current slightly but lowers open-circuit voltage more, so module power typically drops about 0.3–0.5 percent per degree Celsius.

Wind power and Betz limit

Available wind power is ½ρAV³; the Betz limit states no turbine can extract more than 16/27, about 59.3 percent, of that power.

Tip-speed ratio

Tip-speed ratio λ = ωR/V compares blade tip speed to wind speed; power coefficient peaks near λ of 7 to 8 for three-blade horizontal-axis turbines.

Hydro power output

Electrical output is P = ρgQH times overall efficiency, where Q is volumetric flow and H is net head; Pelton wheels run best with bucket speed near half jet speed.

Battery C-rate definition

C-rate is charge or discharge current divided by rated capacity in ampere-hours; a 2C rate on a 50 Ah cell means 100 A, discharging nominally in half an hour.

Reversible fuel cell voltage

Reversible cell voltage is E = −ΔG/(nF); actual voltage is lower due to activation, ohmic, and concentration overpotentials, so efficiency falls with current density.

Biogas versus producer gas

Anaerobic digestion gives biogas with roughly 50–70 percent methane; gasification gives producer gas containing CO and H2 with lower heating value near 4–6 MJ per normal cubic metre.

Nuclear multiplication factor

The effective multiplication factor k compares neutrons in successive generations: k = 1 is critical, k > 1 supercritical, and k < 1 subcritical; each U-235 fission yields about 200 MeV.

Levelised cost of energy

LCOE divides total discounted lifetime costs by total discounted lifetime energy generated; it allows comparison of technologies with different capital costs and lifetimes.

Storage capacity sizing

Required storage equals the integral of load minus generation over the deficit period, divided by round-trip efficiency and allowable depth of discharge.