7.1 Physics for Engineers

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.

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1. Classical Mechanics & Rotational Dynamics

Kinematics & Linear Kinetics

Linear motion under constant acceleration $a$ is described by the fundamental kinematic equations:

$$v = v_0 + at$$ $$s = v_0 t + \frac{1}{2} a t^2$$ $$v^2 = v_0^2 + 2as$$

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:

$$F = \frac{dp}{dt} = m a$$

• Work ($W$): $W = \int \mathbf{F} \cdot d\mathbf{s} = F s \cos\theta$
• Kinetic Energy ($K$): $K = \frac{1}{2} m v^2$
• Work-Energy Theorem: $W_{\text{net}} = \Delta K = K_f - K_i$
• Impulse-Momentum Principle: $J = \int F dt = \Delta p = m(v_f - v_i)$
• Mechanical Power ($P$): Rate of doing work: $P = \frac{dW}{dt} = \mathbf{F} \cdot \mathbf{v}$

Rotational Dynamics

For rigid bodies rotating about a fixed axis with angular velocity $\omega$ and angular acceleration $\alpha$:

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2. Thermodynamics & Heat Transfer

Laws of Thermodynamics

1. First Law (Energy Conservation): The change in internal energy $\Delta U$ of a closed system is equal to net heat added $Q$ minus net work done by the system $W$: $$\Delta U = Q - W$$
2. 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 $T_H$ and a cold reservoir at $T_C$ is strictly bounded by the Carnot Efficiency: $$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} = \frac{T_H - T_C}{T_H}$$ (Note: Temperatures $T_H$ and $T_C$ must always be converted to absolute units in Kelvin: $K = ^\circ\text{C} + 273.15$)

Heat Transfer Modes

• Conduction (Fourier's Law): Heat transfer through a solid medium via molecular vibration:

  $$q = -k A \frac{dT}{dx} \implies Q = \frac{k A (T_1 - T_2) t}{L}$$

  where $k$ is thermal conductivity ($\text{W}/(\text{m}\cdot\text{K})$), $A$ is cross-sectional area, $L$ is thickness, and $t$ is time.

• Convection (Newton's Law of Cooling): Heat transfer between a surface and a flowing fluid:

  $$q = h A (T_s - T_f)$$

  where $h$ is the convection heat transfer coefficient ($\text{W}/(\text{m}^2\cdot\text{K})$).

• Radiation (Stefan-Boltzmann Law): Thermal radiation emitted by a body at absolute temperature $T$:

  $$q = \varepsilon \sigma A T^4$$

  where $\sigma = 5.670 \times 10^{-8}\ \text{W}/(\text{m}^2\cdot\text{K}^4)$ is the Stefan-Boltzmann constant, and $\varepsilon$ is surface emissivity ($0 \le \varepsilon \le 1$).

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3. Electromagnetism & Field Fundamentals

Electrostatics & Coulomb's Law

The electrostatic force between two point charges $q_1$ and $q_2$ separated by distance $r$ in vacuum is:

$$F = \frac{1}{4\pi \varepsilon_0} \frac{q_1 q_2}{r^2}$$

where $\varepsilon_0 \approx 8.854 \times 10^{-12}\ \text{F/m}$ is the permittivity of free space.

Gauss's Law

The net electric flux through any closed surface equals the enclosed charge divided by $\varepsilon_0$:

$$\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$$

Electromagnetic Induction & Faraday's Law

An induced electromotive force (emf) $e$ is produced in a closed circuit whenever the magnetic flux $\Phi = \int \mathbf{B} \cdot d\mathbf{A}$ linking the circuit changes over time:

$$\ne = -N \frac{d\Phi}{dt}$$

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 $q$ moving with velocity $\mathbf{v}$ through electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$ experiences force:

$$\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})$$

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4. Waves, Sound & Optics

Simple Harmonic Motion (SHM) & Wave Motion

For a mass-spring system in SHM, the period of oscillation $T$ and frequency $f$ are:

$$T = 2\pi \sqrt{\frac{m}{k}}, \quad f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$$

For wave propagation with velocity $v$, frequency $f$, and wavelength $\lambda$:

$$v = f \lambda$$

Geometrical Optics & Refraction

• Snell's Law of Refraction: $$n_1 \sin\theta_1 = n_2 \sin\theta_2$$ where $n = c/v$ is the refractive index of the medium.
• Critical Angle & Total Internal Reflection: Occurs when light travels from a dense medium ($n_1$) into a less dense medium ($n_2 < n_1$) at an angle exceeding $\theta_c$: $$\sin\theta_c = \frac{n_2}{n_1}$$
• Thin Lens Formula: $$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$ where $f$ is focal length, $d_o$ is object distance, and $d_i$ is image distance.

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Solved Practice Examples

Example 1: Rotational Power & Torque Calculation

Problem: An electric motor generates a constant shaft torque of $180\ \text{N}\cdot\text{m}$ while rotating at $1,750\ \text{rpm}$. Calculate the mechanical output power in kilowatts (kW) and horsepower (hp).

Solution:

1. Convert rotational speed $\text{rpm}$ to angular velocity $\omega$ in $\text{rad/s}$: $$\omega = 1,750 \times \frac{2\pi}{60} = \frac{3,500 \pi}{60} \approx 183.26\ \text{rad/s}$$
2. Calculate mechanical power $P = \tau \omega$: $$P = 180 \times 183.26 = 32,986.7\ \text{W} \approx 32.99\ \text{kW}$$
3. Convert kW to mechanical horsepower ($1\ \text{hp} = 746\ \text{W}$): $$P_{\text{hp}} = \frac{32,986.7}{746} \approx 44.22\ \text{hp}$$

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Example 2: Carnot Engine Thermal Efficiency

Problem: A thermal power plant operates between a steam turbine inlet temperature of $520^\circ\text{C}$ and a condenser cooling water outlet temperature of $35^\circ\text{C}$. Determine the maximum theoretical Carnot thermal efficiency of this power plant.

Solution:

1. Convert inlet ($T_H$) and outlet ($T_C$) temperatures to absolute Kelvin: $$T_H = 520 + 273.15 = 793.15\ \text{K}$$ $$T_C = 35 + 273.15 = 308.15\ \text{K}$$
2. Compute Carnot efficiency $\eta_{\text{Carnot}}$: $$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} = 1 - \frac{308.15}{793.15} = 1 - 0.3885 = 0.6115 \quad (61.15\%)$$