📌 Mechanical: Statics

Equilibrium of Forces:

\( \sum F_x = 0, \sum F_y = 0 \)

Equilibrium of Moments:

\( \sum M = 0 \)

Force Magnitude:

\( F = \sqrt{F_x^2 + F_y^2} \)

Angle of Force:

\( \theta = \tan^{-1} \left( \frac{F_y}{F_x} \right) \)

Moment (Torque):

\( M = F d \sin \theta \)

Centroid (x-coordinate):

\( \bar{x} = \frac{\sum (x_i A_i)}{\sum A_i} \)

Centroid (y-coordinate):

\( \bar{y} = \frac{\sum (y_i A_i)}{\sum A_i} \)

Friction Force:

\( F_f = \mu N \)

Resultant Force (Vector Sum):

\( F_R = \sqrt{(\sum F_x)^2 + (\sum F_y)^2} \)

Truss Reaction (Method of Joints):

\( \sum F = 0 \) at each joint

Beam Shear Force:

\( V = \sum F_{\text{vertical}} \)

📌 Mechanical: Dynamics

Velocity:

\( v = \frac{\Delta x}{\Delta t} \)

Acceleration:

\( a = \frac{\Delta v}{\Delta t} \)

Newton’s Second Law:

\( F = m a \)

Kinematic Equation (Constant \( a \)):

\( v = v_0 + a t \)

Displacement (Constant \( a \)):

\( x = x_0 + v_0 t + \frac{1}{2} a t^2 \)

Velocity Squared:

\( v^2 = v_0^2 + 2 a (x - x_0) \)

Work:

\( W = F d \cos \theta \)

Kinetic Energy:

\( KE = \frac{1}{2} m v^2 \)

Potential Energy (Gravity):

\( PE = m g h \)

Power:

\( P = \frac{W}{t} = F v \cos \theta \)

Angular Velocity:

\( \omega = \frac{\Delta \theta}{\Delta t} \)

Centripetal Acceleration:

\( a_c = \frac{v^2}{r} = \omega^2 r \)

📌 Mechanical: Thermodynamics

Ideal Gas Law:

\( PV = nRT \)

First Law of Thermodynamics:

\( \Delta U = Q - W \)

Work (Constant Pressure):

\( W = P \Delta V \)

Enthalpy:

\( H = U + PV \)

Specific Heat (Constant Volume):

\( Q = m c_v \Delta T \)

Specific Heat (Constant Pressure):

\( Q = m c_p \Delta T \)

Efficiency (Carnot):

\( \eta = 1 - \frac{T_C}{T_H} \)

Heat Transfer (Conduction):

\( Q = k A \frac{\Delta T}{L} t \)

Entropy Change:

\( \Delta S = \frac{Q_{\text{rev}}}{T} \)

Thermal Expansion:

\( \Delta L = L_0 \alpha \Delta T \)

Gas Work (Isothermal):

\( W = nRT \ln \left( \frac{V_2}{V_1} \right) \)

📌 Mechanical: Materials

Stress:

\( \sigma = \frac{F}{A} \)

Strain:

\( \epsilon = \frac{\Delta L}{L_0} \)

Young’s Modulus:

\( E = \frac{\sigma}{\epsilon} \)

Shear Stress:

\( \tau = \frac{F}{A} \)

Shear Strain:

\( \gamma = \tan \theta \approx \theta \) (small angles)

Shear Modulus:

\( G = \frac{\tau}{\gamma} \)

Poisson’s Ratio:

\( \nu = -\frac{\epsilon_{\text{lateral}}}{\epsilon_{\text{longitudinal}}} \)

Bulk Modulus:

\( K = -\frac{\Delta P}{\Delta V / V} \)

Torsional Stress:

\( \tau = \frac{T r}{J} \) (where \( J \) is polar moment)

Bending Stress:

\( \sigma = \frac{M c}{I} \) (where \( I \) is moment of inertia)

Moment of Inertia (Rectangle):

\( I = \frac{b h^3}{12} \) (about centroidal axis)

📌 Electrical: Circuits

Ohm’s Law:

\( V = I R \)

Power (Electrical):

\( P = V I = I^2 R = \frac{V^2}{R} \)

Resistors in Series:

\( R_{\text{eq}} = R_1 + R_2 + \cdots + R_n \)

Resistors in Parallel:

\( \frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \)

Kirchhoff’s Current Law (KCL):

\( \sum I_{\text{in}} = \sum I_{\text{out}} \)

Kirchhoff’s Voltage Law (KVL):

\( \sum V = 0 \) (around a loop)

Capacitance:

\( C = \frac{Q}{V} \)

Capacitors in Parallel:

\( C_{\text{eq}} = C_1 + C_2 + \cdots + C_n \)

Capacitors in Series:

\( \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \cdots + \frac{1}{C_n} \)

RC Time Constant:

\( \tau = R C \)

Current in RC Circuit (Charging):

\( I(t) = \frac{V}{R} e^{-t / \tau} \)

📌 Electrical: Power

AC Power (Average):

\( P = V_{\text{rms}} I_{\text{rms}} \cos \phi \)

RMS Voltage:

\( V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}} \)

RMS Current:

\( I_{\text{rms}} = \frac{I_{\text{peak}}}{\sqrt{2}} \)

Power Factor:

\( \cos \phi = \frac{R}{Z} \) (where \( Z \) is impedance)

Impedance (RLC Circuit):

\( Z = \sqrt{R^2 + (X_L - X_C)^2} \)

Inductive Reactance:

\( X_L = \omega L = 2\pi f L \)

Capacitive Reactance:

\( X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C} \)

Resonant Frequency:

\( f_0 = \frac{1}{2\pi \sqrt{L C}} \)

Transformer Voltage Ratio:

\( \frac{V_{\text{secondary}}}{V_{\text{primary}}} = \frac{N_{\text{secondary}}}{N_{\text{primary}}} \)

Three-Phase Power:

\( P = \sqrt{3} V_L I_L \cos \phi \) (line values)

Efficiency:

\( \eta = \frac{P_{\text{out}}}{P_{\text{in}}} \times 100\% \)

📌 Civil: Structural Analysis

Beam Deflection (Cantilever, Point Load):

\( \delta = \frac{F L^3}{3 E I} \)

Beam Deflection (Simply Supported, Point Load):

\( \delta = \frac{F L^3}{48 E I} \) (at center)

Moment of Inertia (Circle):

\( I = \frac{\pi r^4}{4} \)

Shear Stress in Beam:

\( \tau = \frac{V Q}{I b} \) (where \( Q \) is first moment)

Buckling Load (Euler):

\( P_{cr} = \frac{\pi^2 E I}{L_e^2} \) (where \( L_e \) is effective length)

Slope of Beam (Point Load):

\( \theta = \frac{F L^2}{2 E I} \) (cantilever)

Reaction Force (Statically Determinate):

\( R = \sum F / \text{supports} \)

Truss Member Force (Method of Sections):

\( F = \frac{M}{d} \) (moment about cut point)

Column Slenderness Ratio:

\( \lambda = \frac{L_e}{r} \) (where \( r = \sqrt{I/A} \))

Stress Concentration Factor:

\( K_t = \frac{\sigma_{\text{max}}}{\sigma_{\text{nominal}}} \)

📌 Civil: Fluid Mechanics

Continuity Equation:

\( A_1 v_1 = A_2 v_2 \)

Bernoulli’s Equation:

\( P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2 \)

Hydrostatic Pressure:

\( P = \rho g h \)

Buoyant Force:

\( F_b = \rho g V_{\text{submerged}} \)

Reynolds Number:

\( Re = \frac{\rho v D}{\mu} \) (pipe flow)

Head Loss (Darcy-Weisbach):

\( h_L = f \frac{L}{D} \frac{v^2}{2g} \)

Friction Factor (Laminar):

\( f = \frac{64}{Re} \) (\( Re < 2000 \))

Pipe Flow Rate:

\( Q = A v \)

Drag Force:

\( F_D = \frac{1}{2} \rho v^2 C_D A \)

Lift Force:

\( F_L = \frac{1}{2} \rho v^2 C_L A \)

Manning’s Equation (Open Channel):

\( v = \frac{1}{n} R_h^{2/3} S^{1/2} \)

📌 Control Systems

Transfer Function:

\( G(s) = \frac{Y(s)}{U(s)} \)

First-Order System Response:

\( y(t) = y_{\text{ss}} (1 - e^{-t/\tau}) \) (step response)

Time Constant:

\( \tau = \frac{1}{a} \) (for \( \dot{y} + a y = b u \))

Second-Order System:

\( G(s) = \frac{\omega_n^2}{s^2 + 2 \zeta \omega_n s + \omega_n^2} \)

Natural Frequency:

\( \omega_n = \sqrt{\frac{k}{m}} \) (mass-spring system)

Damping Ratio:

\( \zeta = \frac{c}{2 \sqrt{k m}} \) (where \( c \) is damping coefficient)

Overshoot:

\( MP = e^{-\frac{\zeta \pi}{\sqrt{1 - \zeta^2}}} \times 100\% \)

Settling Time (2%):

\( t_s = \frac{4}{\zeta \omega_n} \)

Peak Time:

\( t_p = \frac{\pi}{\omega_n \sqrt{1 - \zeta^2}} \)

Steady-State Error (Step Input):

\( e_{ss} = \frac{1}{1 + K_p} \) (where \( K_p \) is position constant)

Laplace Transform (Derivative):

\( \mathcal{L}\{ \dot{y}(t) \} = s Y(s) - y(0) \)