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7.1 Physics for Engineers
PRC Registered Electrical Engineer REE
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1 Chapter 1: Overview, RA 7920 & Electrical Engineering Ethics
1.1 PRC Board of Electrical Engineering & REE Examination Overview 1.2 Republic Act No. 7920 (New Electrical Engineering Law) & License Classes 1.3 Code of Ethics & Professional Practice for Electrical Engineers
2 Chapter 2: Math — Algebra, Trig & Geometry
2.1 Algebraic Equations, Polynomials & Complex Numbers 2.2 Trigonometry — Identities, Triangles & Spherical Trig 2.3 Analytic & Solid Geometry
3 Chapter 3: Engineering Mathematics — Calculus & Differential Equations
3.1 Differential Calculus, Limits, Derivatives & Applications 3.2 Integral Calculus, Integration Techniques & Area/Volume Applications 3.3 Ordinary Differential Equations & Initial-Value Problems
4 Chapter 4: Engineering Mathematics — Advanced Math & Transform Methods
4.1 Laplace Transforms, Inverse Transforms & Circuit Applications 4.2 Fourier Series, Fourier Transforms & Frequency-Domain Analysis 4.3 Vector Analysis, Linear Algebra, Matrices & Complex Analysis
5 Chapter 5: Engineering Mathematics — Probability, Statistics & Engineering Data Analysis
5.1 Probability Theory, Counting Techniques & Bayes' Theorem 5.2 Engineering Data Analysis I — Descriptive Statistics & Probability Distributions 5.3 Engineering Data Analysis II — Confidence Intervals, Hypothesis Testing & Regression
6 Chapter 6: Engineering Mathematics — Numerical Methods & Analysis
6.1 Numerical Root-Finding & Solution of Linear Systems 6.2 Interpolation, Numerical Integration & Numerical Solution of ODEs
7 Chapter 7: ESAS — Physics, Chemistry & Computer Fundamentals
7.1 Physics for Engineers 7.2 General Chemistry & Materials Science 7.3 Computer Fundamentals, Programming & Data Logic 7.4 Logic Circuits, Switching Theory & Microprocessor Systems
8 Chapter 8: ESAS — Mechanics & Thermal-Fluid Engineering
8.1 Engineering Mechanics — Statics & Dynamics 8.2 Strength of Materials & Mechanics of Deformable Bodies 8.3 Thermal-Fluid Engineering — Thermodynamics & Fluid Mechanics
9 Chapter 9: ESAS — Engineering Economics, Management & Contracts
9.1 Engineering Economics & Depreciation 9.2 Engineering Management & Ethics 9.3 Philippine Electrical Engineering Law, Code & Contracts 9.4 Technopreneurship 101 & Management of Engineering Projects
10 Chapter 10: PEC 1 — Safety, Sizing & Protection
10.1 PEC Fundamentals, Safety Standards & Grounding/Bonding 10.2 Conductor Sizing, Ampacity Adjustment & Voltage Drop 10.3 Overcurrent Protection, Branch Circuit & Feeder Sizing 10.4 Basic Occupational Safety & Health (BOSH) and Electrical Safety Standards
11 Chapter 11: PEC 1 & 2 — Equipment & Utility Systems
11.1 PEC Equipment Installation & Wiring Methods 11.2 Utility Systems, Transformers & Switchgear 11.3 Special Occupancies & Hazardous Locations
12 Chapter 12: Electrical Professional — DC & AC Circuit Fundamentals
12.1 DC Circuit Analysis & Network Theorems 12.2 Single-Phase AC Circuits & Phasor Analysis 12.3 Balanced & Unbalanced Three-Phase AC Circuits
13 Chapter 13: Electrical Professional — EM Fields & Transients
13.1 Electrostatics, Magnetostatics & Maxwell's Equations 13.2 Electromagnetic Waves & Transmission Line Fields 13.3 First-Order & Second-Order Circuit Transients
14 Chapter 14: Electrical Professional — Electronics, Industrial Electronics & Communications
14.1 Electronics 1 & 2 — Semiconductor Devices & Analog Circuits 14.2 Electrical Apparatus & Devices — Industrial Electronics & Power Converters 14.3 Fundamentals of Electronic Communications
15 Chapter 15: Electrical Professional — Transformers & DC Machines
15.1 Single-Phase & Three-Phase Transformers 15.2 Transformer Connections & Special Transformers 15.3 DC Generators & DC Motors
16 Chapter 16: Electrical Professional — AC Synchronous & Induction Machinery
16.1 Synchronous Generators (Alternators) 16.2 Synchronous Motors & Special AC Machines 16.3 Three-Phase & Single-Phase Induction Motors
17 Chapter 17: Electrical Professional — Power Plants & Energy Conversion
17.1 Hydroelectric, Thermal & Combined Cycle Power Plants 17.2 Renewable Energy Systems & Geothermal Power Plants 17.3 Power Plant Economics, Load Curves & Energy Conversion
18 Chapter 18: Electrical Professional — Transmission & Distribution
18.1 Transmission Line Parameters & Performance Models 18.2 Mechanical Overhead Line Design, Underground Cables & Insulators 18.3 Distribution Systems, Substation Layouts & Power Flow Control
19 Chapter 19: Electrical Professional — Power System Analysis
19.1 The Per-Unit System, Base Conversion & Impedance Diagrams 19.2 Symmetrical Components & Unbalanced Fault Analysis 19.3 Load Flow, Economic Dispatch & Power System Stability
20 Chapter 20: Electrical Professional — Protection, Control, Instrumentation & Illumination
20.1 Power System Protection & Protective Relaying 20.2 Industrial Motor Control, Logic Circuits & PLC Automation 20.3 Electrical Instrumentation, Transducers & Measurement Systems 20.4 Illumination Engineering, Lighting Design & Photometrics 20.5 Feedback Control Systems & Research Methods
7.1 Physics for Engineers
Key Takeaways Explain This Topic AI
Newtonian mechanics governs physical kinetics via F = m a F = ma F = ma , impulse-momentum ( Δ p = F Δ t \Delta p = F \Delta t Δ p = F Δ t ), work-energy relations, and rotational torque τ = I α \tau = I \alpha τ = I α .
The First Law of Thermodynamics specifies energy conservation ( Δ U = Q − W \Delta U = Q - W Δ U = Q − W ), while the Second Law establishes entropy bounds and the Carnot efficiency limit η Carnot = 1 − T C T H \eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} η Carnot = 1 − T H T C .
Electromagnetism relies on Coulomb's law, Gauss's law ( ∮ E ⋅ d A = Q enc ε 0 \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} ∮ E ⋅ d A = ε 0 Q enc ), Ampere's law, and Faraday's law of induction ( e = − N d Φ d t e = -N \frac{d\Phi}{dt} e = − N d t d Φ ) with Lenz's law establishing opposing polarity.
Heat transfer occurs through conduction ( q = − k A d T d x q = -k A \frac{dT}{dx} q = − k A d x d T ), convection ( q = h A Δ T q = h A \Delta T q = h A Δ T ), and radiation ( q = ε σ A T 4 q = \varepsilon \sigma A T^4 q = ε σ A T 4 ), which dictate equipment thermal rating and cooling.
Waves and optics govern simple harmonic motion ( T = 2 π m k T = 2\pi\sqrt{\frac{m}{k}} T = 2 π k m ), wave propagation ( v = f λ v = f \lambda v = f λ ), Snell's law of refraction ( n 1 sin θ 1 = n 2 sin θ 2 n_1 \sin\theta_1 = n_2 \sin\theta_2 n 1 sin θ 1 = n 2 sin θ 2 ), and total internal reflection.
Last updated: August 2026
7.1 Physics for Engineers
Engineering Physics serves as a core foundation for the Engineering Sciences and Allied Subjects (ESAS) portion of the PRC Registered Electrical Engineer (REE) Licensure Examination. A rigorous understanding of mechanics, thermodynamics, heat transfer, electromagnetism, and wave optics is vital for solving board exam problems and analyzing physical systems.
- 1. Classical Mechanics & Rotational Dynamics
- Kinematics & Linear Kinetics
Linear motion under constant acceleration a a a is described by the fundamental kinematic equations: v = v 0 + a t v = v_0 + at v = v 0 + a t s = v 0 t + 1 2 a t 2 s = v_0 t + \frac{1}{2} a t^2 s = v 0 t + 2 1 a t 2 v 2 = v 0 2 + 2 a s v^2 = v_0^2 + 2as v 2 = v 0 2 + 2 a s
According to Newton's Second Law of Motion , the net external force acting on a body equals the time rate of change of its linear momentum: F = d p d t = m a F = \frac{dp}{dt} = m a F = d t d p = ma
Work ( W W W ): W = ∫ F ⋅ d s = F s cos θ W = \int \mathbf{F} \cdot d\mathbf{s} = F s \cos\theta W = ∫ F ⋅ d s = F s cos θ
Kinetic Energy ( K K K ): K = 1 2 m v 2 K = \frac{1}{2} m v^2 K = 2 1 m v 2
Work-Energy Theorem: W net = Δ K = K f − K i W_{\text{net}} = \Delta K = K_f - K_i W net = Δ K = K f − K i
Impulse-Momentum Principle: J = ∫ F d t = Δ p = m ( v f − v i ) J = \int F dt = \Delta p = m(v_f - v_i) J = ∫ F d t = Δ p = m ( v f − v i )
Mechanical Power ( P P P ): Rate of doing work: P = d W d t = F ⋅ v P = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v} P = d t d W = F ⋅ v
Rotational Dynamics
For rigid bodies rotating about a fixed axis with angular velocity ω \omega ω and angular acceleration α \alpha α :
- Linear Quantity Rotational Analog Governing Relationship
- Position s s s Angle θ \theta θ s = r θ s = r \theta s = r θ
- Velocity v v v Angular Velocity ω \omega ω v = r ω v = r \omega v = r ω
- Acceleration a a a Angular Acceleration α \alpha α a t = r α a_t = r \alpha a t = r α
- Mass m m m Moment of Inertia I I I I = ∫ r 2 d m I = \int r^2 dm I = ∫ r 2 d m
- Force F F F Torque τ \tau τ τ = I α = r F sin θ \tau = I \alpha = r F \sin\theta τ = I α = r F sin θ
- Momentum p = m v p = mv p = m v Angular Momentum L L L L = I ω L = I \omega L = I ω
Kinetic Energy K L = 1 2 m v 2 K_L = \frac{1}{2}mv^2 K L = 2 1 m v 2 Rotational KE K R K_R K R K R = 1 2 I ω 2 K_R = \frac{1}{2} I \omega^2 K R = 2 1 I ω 2
- Power P = F v P = Fv P = F v Rotational Power P P P P = τ ω P = \tau \omega P = τ ω
- 2. Thermodynamics & Heat Transfer
- Laws of Thermodynamics
First Law (Energy Conservation): The change in internal energy Δ U \Delta U Δ U of a closed system is equal to net heat added Q Q Q minus net work done by the system W W W : Δ U = Q − W \Delta U = Q - W Δ U = Q − W
Second Law (Entropy & Efficiency): Heat cannot spontaneously flow from a colder body to a hotter body. The thermal efficiency of any heat engine operating between a hot reservoir at absolute temperature T H T_H T H and a cold reservoir at T C T_C T C is strictly bounded by the Carnot Efficiency : η Carnot = 1 − T C T H = T H − T C T H \eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} = \frac{T_H - T_C}{T_H} η Carnot = 1 − T H T C = T H T H − T C (Note: Temperatures T H T_H T H and T C T_C T C must always be converted to absolute units in Kelvin: K = ∘ C + 273.15 K = ^\circ\text{C} + 273.15 K = ∘ C + 273.15 )
Heat Transfer Modes
Conduction (Fourier's Law): Heat transfer through a solid medium via molecular vibration: q = − k A d T d x ⟹ Q = k A ( T 1 − T 2 ) t L q = -k A \frac{dT}{dx} \implies Q = \frac{k A (T_1 - T_2) t}{L} q = − k A d x d T ⟹ Q = L k A ( T 1 − T 2 ) t
where k k k is thermal conductivity ( W / ( m ⋅ K ) \text{W}/(\text{m}\cdot\text{K}) W / ( m ⋅ K ) ), A A A is cross-sectional area, L L L is thickness, and t t t is time.
Convection (Newton's Law of Cooling): Heat transfer between a surface and a flowing fluid: q = h A ( T s − T f ) q = h A (T_s - T_f) q = h A ( T s − T f )
where h h h is the convection heat transfer coefficient ( W / ( m 2 ⋅ K ) \text{W}/(\text{m}^2\cdot\text{K}) W / ( m 2 ⋅ K ) ).
Radiation (Stefan-Boltzmann Law): Thermal radiation emitted by a body at absolute temperature T T T : q = ε σ A T 4 q = \varepsilon \sigma A T^4 q = ε σ A T 4
where σ = 5.670 × 10 − 8 W / ( m 2 ⋅ K 4 ) \sigma = 5.670 \times 10^{-8}\ \text{W}/(\text{m}^2\cdot\text{K}^4) σ = 5.670 × 1 0 − 8 W / ( m 2 ⋅ K 4 ) is the Stefan-Boltzmann constant, and ε \varepsilon ε is surface emissivity ( 0 ≤ ε ≤ 1 0 \le \varepsilon \le 1 0 ≤ ε ≤ 1 ).
- 3. Electromagnetism & Field Fundamentals
- Electrostatics & Coulomb's Law
The electrostatic force between two point charges q 1 q_1 q 1 and q 2 q_2 q 2 separated by distance r r r in vacuum is: F = 1 4 π ε 0 q 1 q 2 r 2 F = \frac{1}{4\pi \varepsilon_0} \frac{q_1 q_2}{r^2} F = 4 π ε 0 1 r 2 q 1 q 2
where ε 0 ≈ 8.854 × 10 − 12 F/m \varepsilon_0 \approx 8.854 \times 10^{-12}\ \text{F/m} ε 0 ≈ 8.854 × 1 0 − 12 F/m is the permittivity of free space.
Gauss's Law
The net electric flux through any closed surface equals the enclosed charge divided by ε 0 \varepsilon_0 ε 0 : ∮ E ⋅ d A = Q enc ε 0 \oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} ∮ E ⋅ d A = ε 0 Q enc
Electromagnetic Induction & Faraday's Law
An induced electromotive force (emf) e e e is produced in a closed circuit whenever the magnetic flux Φ = ∫ B ⋅ d A \Phi = \int \mathbf{B} \cdot d\mathbf{A} Φ = ∫ B ⋅ d A linking the circuit changes over time: ≠ = − N d Φ d t \ne = -N \frac{d\Phi}{dt} = = − N d t d Φ
According to Lenz's Law (represented by the negative sign), the direction of the induced current creates a magnetic field that opposes the change in magnetic flux that produced it.
Lorentz Force Law
A charge q q q moving with velocity v \mathbf{v} v through electric field E \mathbf{E} E and magnetic field B \mathbf{B} B experiences force: F = q ( E + v × B ) \mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B}) F = q ( E + v × B )
- 4. Waves, Sound & Optics
- Simple Harmonic Motion (SHM) & Wave Motion
For a mass-spring system in SHM, the period of oscillation T T T and frequency f f f are: T = 2 π m k , f = 1 2 π k m T = 2\pi \sqrt{\frac{m}{k}}, \quad f = \frac{1}{2\pi} \sqrt{\frac{k}{m}} T = 2 π k m , f = 2 π 1 m k
For wave propagation with velocity v v v , frequency f f f , and wavelength λ \lambda λ : v = f λ v = f \lambda v = f λ
Geometrical Optics & Refraction
Snell's Law of Refraction: n 1 sin θ 1 = n 2 sin θ 2 n_1 \sin\theta_1 = n_2 \sin\theta_2 n 1 sin θ 1 = n 2 sin θ 2 where n = c / v n = c/v n = c / v is the refractive index of the medium.
Critical Angle & Total Internal Reflection: Occurs when light travels from a dense medium ( n 1 n_1 n 1 ) into a less dense medium ( n 2 < n 1 n_2 < n_1 n 2 < n 1 ) at an angle exceeding θ c \theta_c θ c : sin θ c = n 2 n 1 \sin\theta_c = \frac{n_2}{n_1} sin θ c = n 1 n 2
Thin Lens Formula: 1 f = 1 d o + 1 d i \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} f 1 = d o 1 + d i 1 where f f f is focal length, d o d_o d o is object distance, and d i d_i d i is image distance.
- Solved Practice Examples
- Example 1: Rotational Power & Torque Calculation
Problem: An electric motor generates a constant shaft torque of 180 N ⋅ m 180\ \text{N}\cdot\text{m} 180 N ⋅ m while rotating at 1 , 750 rpm 1,750\ \text{rpm} 1 , 750 rpm . Calculate the mechanical output power in kilowatts (kW) and horsepower (hp).
Solution:
Convert rotational speed rpm \text{rpm} rpm to angular velocity ω \omega ω in rad/s \text{rad/s} rad/s : ω = 1 , 750 × 2 π 60 = 3 , 500 π 60 ≈ 183.26 rad/s \omega = 1,750 \times \frac{2\pi}{60} = \frac{3,500 \pi}{60} \approx 183.26\ \text{rad/s} ω = 1 , 750 × 60 2 π = 60 3 , 500 π ≈ 183.26 rad/s
Calculate mechanical power P = τ ω P = \tau \omega P = τ ω : P = 180 × 183.26 = 32 , 986.7 W ≈ 32.99 kW P = 180 \times 183.26 = 32,986.7\ \text{W} \approx 32.99\ \text{kW} P = 180 × 183.26 = 32 , 986.7 W ≈ 32.99 kW
Convert kW to mechanical horsepower ( 1 hp = 746 W 1\ \text{hp} = 746\ \text{W} 1 hp = 746 W ): P hp = 32 , 986.7 746 ≈ 44.22 hp P_{\text{hp}} = \frac{32,986.7}{746} \approx 44.22\ \text{hp} P hp = 746 32 , 986.7 ≈ 44.22 hp
Example 2: Carnot Engine Thermal Efficiency
Problem: A thermal power plant operates between a steam turbine inlet temperature of 520 ∘ C 520^\circ\text{C} 52 0 ∘ C and a condenser cooling water outlet temperature of 35 ∘ C 35^\circ\text{C} 3 5 ∘ C . Determine the maximum theoretical Carnot thermal efficiency of this power plant.
Solution:
Convert inlet ( T H T_H T H ) and outlet ( T C T_C T C ) temperatures to absolute Kelvin: T H = 520 + 273.15 = 793.15 K T_H = 520 + 273.15 = 793.15\ \text{K} T H = 520 + 273.15 = 793.15 K T C = 35 + 273.15 = 308.15 K T_C = 35 + 273.15 = 308.15\ \text{K} T C = 35 + 273.15 = 308.15 K
Compute Carnot efficiency η Carnot \eta_{\text{Carnot}} η Carnot : η Carnot = 1 − T C T H = 1 − 308.15 793.15 = 1 − 0.3885 = 0.6115 ( 61.15 % ) \eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} = 1 - \frac{308.15}{793.15} = 1 - 0.3885 = 0.6115 \quad (61.15\%) η Carnot = 1 − T H T C = 1 − 793.15 308.15 = 1 − 0.3885 = 0.6115 ( 61.15% )