Key Formulas & Equations
Strain: ε = ΔL/L₀
Young's Modulus: E = σ/ε
Hooke's Law: F = k·Δx
Power: P = V·I = I²·R = V²/R
Capacitance: C = Q/V
Inductance: V = L·(dI/dt)
Efficiency: η = Wout/Qin × 100%
Heat Transfer: Q = mcΔT
Fluid Pressure: P = ρ·g·h
Torsion: τ = Tr/J
Beam Deflection: δ = FL³/(3EI)
Buckling Load (Euler): Pcr = π²EI/L²
Mass Flow Rate: ṁ = ρ·A·v
Bernoulli's Equation: P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂
Reynolds Number: Re = ρvD/μ
Core Concepts
Statics & Dynamics
Study of forces and their effects on rigid bodies at rest (statics) or in motion (dynamics). Fundamental to structural and mechanical design.
Thermodynamics
The study of energy, heat, work, and the laws governing their conversion. Essential for engines, power plants, refrigeration, and HVAC systems.
Fluid Mechanics
Study of fluids (liquids and gases) at rest and in motion. Governs pipelines, aerodynamics, blood flow, and weather systems.
Strength of Materials
Analysis of how solid objects deform and fail under various loads. Critical for safe structural and mechanical design.
Circuit Theory
Study of electrical circuits, including voltage, current, resistance, and power. Foundation for all electrical and electronic engineering.
Control Systems
Design of systems that regulate the behavior of other systems using feedback loops. Used in robotics, aerospace, and automation.
Engineering Economics
Application of economic principles to engineering decisions, including cost analysis, depreciation, and return on investment.
Materials Science
Study of the properties and applications of materials including metals, ceramics, polymers, and composites.
Worked Examples
Example 1: Stress and Strain in a Steel Rod
Problem: A steel rod (E = 200 GPa) of length 2 m and cross-sectional area 0.0001 m² is subjected to a tensile force of 10,000 N. Calculate stress, strain, and elongation.
Solution:
Stress: σ = F/A = 10,000/0.0001 = 100 MPa
Strain: ε = σ/E = 100×10⁶/200×10⁹ = 0.0005
Elongation: ΔL = ε·L₀ = 0.0005 × 2 = 1 mm
Example 2: RC Circuit Charging
Problem: A 10 μF capacitor is charged through a 1 kΩ resistor from a 12V DC source. Calculate the time constant and voltage after 10 ms.
Solution:
Time constant: τ = RC = 1000 × 10×10⁻⁶ = 10 ms
Voltage: V(t) = V₀(1 - e^(-t/τ)) = 12(1 - e^(-1)) = 12 × 0.632 = 7.59V
Example 3: Heat Engine Efficiency
Problem: A Carnot engine operates between 400K and 300K. Calculate maximum efficiency and work output if 1000 J of heat is absorbed.
Solution:
Efficiency: η = 1 - Tc/Th = 1 - 300/400 = 0.25 = 25%
Work output: W = η × Qh = 0.25 × 1000 = 250 J
Practice Problems
Problem 1
A cantilever beam of length 1.5 m carries a point load of 500 N at the free end. If EI = 2000 N·m², find the maximum deflection.
Answer: δ = FL³/(3EI) = 500 × 3.375/(3 × 2000) = 0.281 m
Problem 2
A copper wire (resistivity = 1.68×10⁻⁸ Ω·m) has a diameter of 2 mm and length 50 m. Calculate its resistance.
Answer: R = ρL/A = 1.68×10⁻⁸ × 50/(π × 1×10⁻⁶) = 0.267 Ω
Problem 3
A gas at 300 K and 100 kPa is compressed isothermally to half its original volume. Find the final pressure.
Answer: P₂ = P₁V₁/V₂ = 100 × 2 = 200 kPa
Problem 4
A 3-phase motor draws 20 A at 415 V with power factor 0.85. Calculate the real power consumed.
Answer: P = √3 × V × I × pf = √3 × 415 × 20 × 0.85 = 12.2 kW
Problem 5
A pipe of diameter 0.1 m carries water at 2 m/s. Calculate the Reynolds number (ρ = 1000 kg/m³, μ = 0.001 Pa·s).
Answer: Re = ρvD/μ = 1000 × 2 × 0.1/0.001 = 200,000 (turbulent flow)
Problem 6
A spring stretches 5 cm under a 50 N load. What is its spring constant? How much will it stretch under 200 N?
Answer: k = F/x = 50/0.05 = 1000 N/m; x = 200/1000 = 0.2 m = 20 cm
Problem 7
A power plant produces 500 MW of electricity with 35% efficiency. Calculate the rate of heat input required.
Answer: Qin = W/η = 500/0.35 = 1429 MW