Fundamental Engineering Equations

Why This Matters

Engineering equations aren't just formulas to memorize—they're the universal language that describes how the physical world behaves. When you're tested on these relationships, you're being asked to demonstrate that you understand conservation principles, proportional relationships, and system behavior across mechanical, electrical, thermal, and fluid systems. The same underlying concepts (energy conservation, equilibrium, material response) appear in different forms depending on the domain.

Here's the key insight: these equations connect to each other through fundamental physical principles. Bernoulli's equation and the Continuity equation both stem from conservation laws. Hooke's Law and the stress-strain relationship both describe how materials respond to forces. Don't just memorize $F = ma$—know why it matters and when to apply it. That conceptual understanding is what separates strong exam performance from rote recall.

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Conservation Principles: Energy and Mass

The most powerful equations in engineering stem from one elegant idea: certain quantities in closed systems remain constant. Energy transforms but doesn't disappear. Mass flowing into a pipe must flow out. Master these, and you've unlocked the foundation of system analysis.

Conservation of Energy

• Energy transforms but is never created or destroyed—this principle governs everything from roller coasters to power plants
• Efficiency calculations depend on tracking energy through a system, identifying where useful work is lost to heat or friction
• Applies across all engineering domains: mechanical work, electrical power, thermal energy, and chemical reactions all follow this rule

Bernoulli's Equation

• Relates pressure, velocity, and elevation in flowing fluids: $P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}$
• Energy conservation for fluids—pressure energy, kinetic energy, and potential energy trade off along a streamline
• Critical for aerodynamics and hydraulics: explains why airplane wings generate lift and why water speeds up through a narrow pipe

Continuity Equation

• Mass flow rate stays constant in a closed system: $A_1 v_1 = A_2 v_2$ for incompressible flow
• Explains velocity changes when fluid moves through pipes of varying diameter—smaller area means faster flow
• Foundation for pipe system design, from plumbing to industrial fluid transport

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Force and Motion Relationships

These equations describe how objects respond to applied forces. The core principle: forces cause acceleration, and that relationship is always proportional. Whether you're analyzing a car crash or designing a bridge, these relationships predict behavior.

Newton's Second Law of Motion

• $F = ma$—net force equals mass times acceleration, the foundation of all dynamics problems
• Acceleration is proportional to force and inversely proportional to mass: double the force, double the acceleration; double the mass, halve it
• Essential for free-body diagrams and analyzing motion in mechanical systems, robotics, and vehicle dynamics

Ideal Gas Law

• $PV = nRT$ relates pressure, volume, temperature, and amount of gas through the universal gas constant $R$
• Predicts gas behavior under changing conditions—compress a gas and pressure rises; heat it and it expands
• Assumes ideal conditions: works well at low pressures and high temperatures, breaks down near phase changes

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Material Response and Deformation

When forces act on solid materials, the material responds by deforming. These equations describe elastic behavior—the predictable, reversible response that occurs before permanent damage. Understanding these relationships is critical for structural safety.

Hooke's Law

• $F = kx$—spring force is proportional to displacement from equilibrium, with $k$ as the spring constant
• Defines elastic behavior: materials return to original shape when force is removed, as long as you stay within the elastic limit
• Used in springs, shock absorbers, and vibration analysis—anywhere you need controlled, predictable force response

Stress-Strain Relationship

• $\sigma = E\varepsilon$ relates stress (force per area) to strain (deformation per length) through Young's modulus $E$
• $E$ measures material stiffness: steel has high $E$ (stiff), rubber has low $E$ (flexible)
• Critical for structural design—ensures beams, columns, and frames won't deform excessively under load

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Electrical Circuit Analysis

Electrical systems follow their own set of proportional relationships. Voltage drives current through resistance, just as force drives acceleration through mass. The mathematical structure mirrors mechanical systems, making cross-domain thinking powerful.

Ohm's Law

• $V = IR$—voltage equals current times resistance, the fundamental relationship for circuit analysis
• Resistance opposes current flow: higher resistance means less current for a given voltage
• Foundation for all circuit design: from simple resistor networks to complex electronic systems

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Heat and Fluid Transport

These equations describe how energy and matter move through systems. The driving principle: flow occurs from high to low (temperature, pressure, concentration), and the rate depends on material properties and geometry.

Fourier's Law of Heat Conduction

• Heat flows from hot to cold at a rate proportional to the temperature gradient: $q = -k\frac{dT}{dx}$
• Thermal conductivity $k$ determines how easily heat moves through a material—metals conduct well, foams insulate
• Essential for heat exchanger and insulation design, directly impacting energy efficiency in buildings and industrial processes

Navier-Stokes Equations

• Govern all viscous fluid motion by balancing inertia, pressure, viscous forces, and external forces
• Computationally intensive—no general analytical solution exists, making these equations the backbone of CFD (computational fluid dynamics) simulations
• Critical for aerodynamics, weather modeling, and blood flow analysis—anywhere complex fluid behavior matters

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Quick Reference Table

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Self-Check Questions

1. Which two equations both express conservation principles for fluid systems, and how do they differ in what quantity they conserve?

2. Identify the equation you would use to determine how much a steel beam deflects under load. What material property does it require?

3. Compare Ohm's Law and Newton's Second Law: what analogous quantities do voltage, current, and resistance correspond to in mechanical systems?

4. If an FRQ asks you to explain why air speeds up over an airplane wing, which equations would you combine, and what does each contribute to the explanation?

5. Both Hooke's Law and the stress-strain relationship describe elastic behavior. When would you use one versus the other, and what's the key difference in their application?