-
First Law of Thermodynamics (Energy Conservation): [ \Delta U = Q - W ]
- Inputs: ( Q ) (Heat added to the system in J), ( W ) (Work done by the system in J)
- Output: ( \Delta U ) (Change in internal energy in J)
- Units: Joules (J)
- Tags: Energy, Heat transfer
-
Second Law of Thermodynamics (Entropy): [ \Delta S \geq \frac{Q}{T} ]
- Inputs: ( Q ) (Heat in J), ( T ) (Temperature in K)
- Output: ( \Delta S ) (Change in entropy in J/K)
- Units: Joules per Kelvin (J/K)
- Tags: Entropy, Heat transfer
-
Bernoulli’s Equation: [ P + \frac{1}{2} \rho v^2 + \rho gh = \text{constant} ]
- Inputs: ( P ) (Pressure in Pa), ( \rho ) (Density in kg/m³), ( v ) (Velocity in m/s), ( g ) (Gravity in m/s²), ( h ) (Height in m)
- Output: Energy per unit volume
- Units: Pascals (Pa), meters per second (m/s), meters (m)
- Tags: Fluid dynamics
-
Continuity Equation: [ A_1 v_1 = A_2 v_2 ]
- Inputs: ( A_1 ) (Area 1 in m²), ( v_1 ) (Velocity 1 in m/s), ( A_2 ) (Area 2 in m²), ( v_2 ) (Velocity 2 in m/s)
- Output: Relationship between velocities and areas
- Units: Square meters (m²), meters per second (m/s)
- Tags: Fluid dynamics
-
Fourier’s Law of Conduction: [ q = -k \frac{dT}{dx} ]
- Inputs: ( k ) (Thermal conductivity in W/(m·K)), ( \frac{dT}{dx} ) (Temperature gradient in K/m)
- Output: ( q ) (Heat flux in W/m²)
- Units: Watts per meter squared (W/m²)
- Tags: Heat conduction
-
Newton’s Law of Cooling (Convection): [ Q = h A (T_s - T_\infty) ]
- Inputs: ( h ) (Convective heat transfer coefficient in W/(m²·K)), ( A ) (Surface area in m²), ( T_s ) (Surface temperature in K), ( T_\infty ) (Fluid temperature in K)
- Output: ( Q ) (Heat transfer rate in W)
- Units: Watts (W)
- Tags: Heat convection
-
Stress and Strain: [ \sigma = \frac{F}{A}, \quad \epsilon = \frac{\Delta L}{L} ]
- Inputs: ( F ) (Force in N), ( A ) (Area in m²), ( \Delta L ) (Change in length in m), ( L ) (Original length in m)
- Output: ( \sigma ) (Stress in Pa), ( \epsilon ) (Strain, dimensionless)
- Units: Pascals (Pa), dimensionless
- Tags: Material strength, Deformation
-
Young’s Modulus: [ E = \frac{\sigma}{\epsilon} ]
- Inputs: ( \sigma ) (Stress in Pa), ( \epsilon ) (Strain, dimensionless)
- Output: ( E ) (Modulus of elasticity in Pa)
- Units: Pascals (Pa)
- Tags: Elasticity
-
Newton’s Second Law of Motion: [ F = m a ]
- Inputs: ( m ) (Mass in kg), ( a ) (Acceleration in m/s²)
- Output: ( F ) (Force in N)
- Units: Newtons (N)
- Tags: Mechanics, Dynamics
-
Transfer Function (Laplace Transform): [ G(s) = \frac{Y(s)}{U(s)} ]
- Inputs: ( Y(s) ) (Output in Laplace domain), ( U(s) ) (Input in Laplace domain)
- Output: ( G(s) ) (Transfer function, dimensionless)
- Units: Dimensionless
- Tags: Control systems
- Machining Time Calculation:
[
T = \frac{L}{f \times N}
]
- Inputs: ( L ) (Length of cut in mm), ( f ) (Feed rate in mm/rev), ( N ) (Spindle speed in rev/min)
- Output: ( T ) (Machining time in min)
- Units: Minutes (min)
- Tags: Manufacturing
- Hooke’s Law:
[
\sigma = E \epsilon
]
- Inputs: ( E ) (Modulus of elasticity in Pa), ( \epsilon ) (Strain, dimensionless)
- Output: ( \sigma ) (Stress in Pa)
- Units: Pascals (Pa)
- Tags: Material deformation
- Factor of Safety:
[
\text{FoS} = \frac{\text{Ultimate Strength}}{\text{Allowable Stress}}
]
- Inputs: Ultimate strength (Pa), Allowable stress (Pa)
- Output: Factor of Safety (dimensionless)
- Units: Dimensionless
- Tags: Safety, Design
- PID Control Algorithm:
[
u(t) = K_p e(t) + K_i \int e(t) , dt + K_d \frac{de(t)}{dt}
]
- Inputs: ( K_p ) (Proportional gain, dimensionless), ( K_i ) (Integral gain, dimensionless), ( K_d ) (Derivative gain, dimensionless), ( e(t) ) (Error signal)
- Output: ( u(t) ) (Control output)
- Units: Dimensionless
- Tags: Control systems
- Engine Efficiency:
[
\eta = \frac{W_{\text{out}}}{Q_{\text{in}}}
]
- Inputs: ( W_{\text{out}} ) (Work output in J), ( Q_{\text{in}} ) (Heat input in J)
- Output: ( \eta ) (Efficiency, dimensionless)
- Units: Dimensionless
- Tags: Engine performance
- Lift Equation:
[
L = \frac{1}{2} \rho v^2 S C_L
]
- Inputs: ( \rho ) (Air density in kg/m³), ( v ) (Velocity in m/s), ( S ) (Wing area in m²), ( C_L ) (Lift coefficient, dimensionless)
- Output: ( L ) (Lift in N)
- Units: Newtons (N)
- Tags: Aerodynamics
- Stress-Strain Relationship in Biomaterials:
[
\sigma = E \epsilon
]
- Inputs: ( E ) (Modulus of elasticity in Pa), ( \epsilon ) (Strain, dimensionless)
- Output: ( \sigma ) (Stress in Pa)
- Units: Pascals (Pa)
- Tags: Biomaterials
- Carnot Efficiency:
[
\eta = 1 - \frac{T_C}{T_H}
]
- Inputs: ( T_C ) (Cold reservoir temperature in K), ( T_H ) (Hot reservoir temperature in K)
- Output: ( \eta ) (Efficiency, dimensionless)
- Units: Dimensionless
- Tags: Thermodynamics, Energy
- Cooling Load Calculation:
[
Q = m \cdot C_p \cdot \Delta T
]
- Inputs: ( m ) (Mass flow rate in kg/s), ( C_p ) (Specific heat at constant pressure in J/(kg·K)), ( \Delta T ) (Temperature difference in K)
- Output: ( Q ) (Cooling load in W)
- Units: Watts (W)
- Tags: HVAC, Thermal systems
- Bending Stress:
[
\sigma = \frac{M c}{I}
]
- Inputs: ( M ) (Moment in Nm), ( c ) (Distance from neutral axis in m), ( I ) (Moment of inertia in m⁴)
- Output: ( \sigma ) (Bending stress in Pa)
- Units: Pascals (Pa)
- Tags: Structural analysis, Material strength
- Arrhenius Equation:
[
k = A e^{-\frac{E_a}{RT}}
]
- Inputs: ( A ) (Pre-exponential factor in s⁻¹), ( E_a ) (Activation energy in J/mol), ( R ) (Gas constant in J/(mol·K)), ( T ) (Temperature in K)
- Output: ( k ) (Rate constant in s⁻¹)
- Units: s⁻¹
- Tags: Reaction rates
- Gibbs Free Energy:
[
\Delta G = \Delta H - T \Delta S
]
- Inputs: ( \Delta H ) (Enthalpy change in J), ( T ) (Temperature in K), ( \Delta S ) (Entropy change in J/K)
- Output: ( \Delta G ) (Gibbs free energy change in J)
- Units: Joules (J)
- Tags: Thermodynamics, Reaction spontaneity
- Fick's Second Law:
[
\frac{\partial C}{\partial t} = D \frac{\partial^2 C}{\partial x^2}
]
- Inputs: ( D ) (Diffusion coefficient in m²/s), ( C ) (Concentration in mol/m³), ( t ) (Time in s), ( x ) (Position in m)
- Output: Concentration over time
- Units: m²/s, mol/m³
- Tags: Diffusion
- Michaelis-Menten Equation:
[
v = \frac{V_{max} [S]}{K_m + [S]}
]
- Inputs: ( V_{max} ) (Maximum rate in mol/s), ( [S] ) (Substrate concentration in mol/L), ( K_m ) (Michaelis constant in mol/L)
- Output: ( v ) (Reaction rate in mol/s)
- Units: mol/s
- Tags: Enzyme activity
- Hill's Equation for Muscle Contraction:
[
F = \frac{F_0 b}{v + b}
]
- Inputs: ( F_0 ) (Maximum isometric force in N), ( b ) (Constant related to the properties of the muscle in s⁻¹), ( v ) (Shortening velocity in m/s)
- Output: ( F ) (Force in N)
- Units: Newtons (N)
- Tags: Muscle dynamics
- Euler's Buckling Formula:
[
P_{cr} = \frac{\pi^2 E I}{(K L)^2}
]
- Inputs: ( E ) (Modulus of elasticity in Pa), ( I ) (Moment of inertia in m⁴), ( K ) (Column effective length factor, dimensionless), ( L ) (Length of column in m)
- Output: ( P_{cr} ) (Critical load in N)
- Units: Newtons (N)
- Tags: Buckling analysis
- Manning's Equation:
[
v = \frac{1}{n} R^{2/3} S^{1/2}
]
- Inputs: ( n ) (Manning's roughness coefficient, dimensionless), ( R ) (Hydraulic radius in m), ( S ) (Slope of the energy grade line, dimensionless)
- Output: ( v ) (Flow velocity in m/s)
- Units: m/s
- Tags: Open channel flow
-
Ohm's Law: [ V = IR ]
- Inputs: ( I ) (Current in A), ( R ) (Resistance in Ω)
- Output: ( V ) (Voltage in V)
- Units: Volts (V)
- Tags: Electrical circuits
-
Power Calculation: [ P = VI ]
- Inputs: ( V ) (Voltage in V), ( I ) (Current in A)
- Output: ( P ) (Power in W)
- Units: Watts (W)
- Tags: Power, Circuits
- Faraday's Law of Induction:
[
\mathcal{E} = -N \frac{d\Phi_B}{dt}
]
- Inputs: ( N ) (Number of turns, dimensionless), ( \Phi_B ) (Magnetic flux in Wb), ( t ) (Time in s)
- Output: ( \mathcal{E} ) (Electromotive force in V)
- Units: Volts (V)
- Tags: Electromagnetic induction
- BOD (Biochemical Oxygen Demand):
[
BOD = \frac{D_1 - D_2}{P}
]
- Inputs: ( D_1 ) (Initial dissolved oxygen in mg/L), ( D_2 ) (Final dissolved oxygen in mg/L), ( P ) (Volume of sample in L)
- Output: ( BOD ) (Biochemical oxygen demand in mg/L)
- Units: mg/L
- Tags: Water quality
- Henry's Law:
[
C = k_H \cdot P
]
- Inputs: ( k_H ) (Henry's law constant in mol/(L·atm)), ( P ) (Partial pressure of gas in atm)
- Output: ( C ) (Concentration of gas in solution in mol/L)
- Units: mol/L
- Tags: Gas solubility
-
EOQ (Economic Order Quantity):
[
EOQ = \sqrt{\frac{2DS}{H}}
]
-
Inputs: ( D ) (Demand rate in units/year), ( S ) (Order cost in
$/order), ( H ) (Holding cost in $ /unit/year) - Output: ( EOQ ) (Economic order quantity in units)
- Units: Units
- Tags: Inventory management
-
Inputs: ( D ) (Demand rate in units/year), ( S ) (Order cost in
- Little's Law:
[
L = \lambda W
]
- Inputs: ( \lambda ) (Arrival rate in units/time), ( W ) (Average time a unit spends in the system in time)
- Output: ( L ) (Average number of units in the system)
- Units: Units
- Tags: Queuing analysis
- Big O Notation:
[
O(f(n))
]
- Inputs: ( f(n) ) (Function representing the complexity)
- Output: Order of complexity
- Units: Dimensionless
- Tags: Algorithm efficiency
- Binary Search:
[
\text{Index} = \text{BinarySearch}(array, target)
]
- Inputs: ( array ) (Sorted array), ( target ) (Element to search for)
- Output: Index of the target element
- Units: Index (integer)
- Tags: Search algorithms
- Fourier Transform:
[
X(f) = \int_{-\infty}^{\infty} x(t) e^{-j2\pi ft} , dt
]
- Inputs: ( x(t) ) (Time-domain signal), ( t ) (Time in s), ( f ) (Frequency in Hz)
- Output: ( X(f) ) (Frequency-domain representation)
- Units: Dimensionless (for the transform result, often represented in terms of magnitude and phase)
- Tags: Signal analysis
- Amplitude Modulation (AM):
[
s(t) = [A + m(t)] \cos(2\pi f_c t)
]
- Inputs: ( A ) (Carrier amplitude), ( m(t) ) (Message signal), ( f_c ) (Carrier frequency in Hz), ( t ) (Time in s)
- Output: ( s(t) ) (Modulated signal)
- Units: Varies depending on input signals
- Tags: Communication systems
- Boolean Algebra:
[
Y = A \cdot B + \overline{C}
]
- Inputs: ( A ) (Binary input), ( B ) (Binary input), ( C ) (Binary input)
- Output: ( Y ) (Binary output)
- Units: Binary (0 or 1)
- Tags: Digital circuits
- Address Translation in Paging:
[
\text{Physical Address} = \text{Page Number} \times \text{Page Size} + \text{Offset}
]
- Inputs: Page Number (integer), Page Size (integer), Offset (integer)
- Output: Physical Address (integer)
- Units: Address units (bytes)
- Tags: Operating systems