7.1 Physics for Engineers
Key Takeaways
- Newtonian mechanics governs physical kinetics via $F = ma$, impulse-momentum ($\\Delta p = F \\Delta t$), work-energy relations, and rotational torque $\\tau = I \\alpha$.
- The First Law of Thermodynamics specifies energy conservation ($\\Delta U = Q - W$), while the Second Law establishes entropy bounds and the Carnot efficiency limit $\\eta_{\\text{Carnot}} = 1 - \\frac{T_C}{T_H}$.
- Electromagnetism relies on Coulomb's law, Gauss's law ($\\oint \\mathbf{E} \\cdot d\\mathbf{A} = \\frac{Q_{\\text{enc}}}{\\varepsilon_0}$), Ampere's law, and Faraday's law of induction ($e = -N \\frac{d\\Phi}{dt}$) with Lenz's law establishing opposing polarity.
- Heat transfer occurs through conduction ($q = -k A \\frac{dT}{dx}$), convection ($q = h A \\Delta T$), and radiation ($q = \\varepsilon \\sigma A T^4$), which dictate equipment thermal rating and cooling.
- Waves and optics govern simple harmonic motion ($T = 2\\pi\\sqrt{\\frac{m}{k}}$), wave propagation ($v = f \\lambda$), Snell's law of refraction ($n_1 \\sin\\theta_1 = n_2 \\sin\\theta_2$), and total internal reflection.
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 $a$ is described by the fundamental kinematic equations:
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:
- 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 $\alpha$:
| Linear Quantity | Rotational Analog | Governing Relationship |
|---|---|---|
| Position $s$ | Angle $\theta$ | $s = r \theta$ |
| Velocity $v$ | Angular Velocity $\omega$ | $v = r \omega$ |
| Acceleration $a$ | Angular Acceleration $\alpha$ | $a_t = r \alpha$ |
| Mass $m$ | Moment of Inertia $I$ | $I = \int r^2 dm$ |
| Force $F$ | Torque $\tau$ | $\tau = I \alpha = r F \sin\theta$ |
| Momentum $p = mv$ | Angular Momentum $L$ | $L = I \omega$ |
| Kinetic Energy $K_L = \frac{1}{2}mv^2$ | Rotational KE $K_R$ | $K_R = \frac{1}{2} I \omega^2$ |
| Power $P = Fv$ | Rotational Power $P$ | $P = \tau \omega$ |
2. Thermodynamics & Heat Transfer
Laws of Thermodynamics
- 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 $W$:
- 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: (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: 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: 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 $T$: 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 ($0 \le \varepsilon \le 1$).
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:
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 $\varepsilon_0$:
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:
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:
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:
For wave propagation with velocity $v$, frequency $f$, and wavelength $\lambda$:
Geometrical Optics & Refraction
- Snell's Law of Refraction: 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 $\theta_c$:
- Thin Lens Formula: where $f$ is focal length, $d_o$ is object distance, and $d_i$ is image distance.
Solved Practice Examples
Example 1: Rotational Power & Torque Calculation
Problem: An electric motor generates a constant shaft torque of $180\ \text{N}\cdot\text{m}$ while rotating at $1,750\ \text{rpm}$. Calculate the mechanical output power in kilowatts (kW) and horsepower (hp).
Solution:
- Convert rotational speed $\text{rpm}$ to angular velocity $\omega$ in $\text{rad/s}$:
- Calculate mechanical power $P = \tau \omega$:
- Convert kW to mechanical horsepower ($1\ \text{hp} = 746\ \text{W}$):
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:
- Convert inlet ($T_H$) and outlet ($T_C$) temperatures to absolute Kelvin:
- Compute Carnot efficiency $\eta_{\text{Carnot}}$:
A rotating generator shaft delivers a torque of $250\ \text{N}\cdot\text{m}$ at a constant speed of $1,200\ \text{rpm}$. What is the mechanical power output of the shaft?
A heat engine operates between a heat source at $450^\circ\text{C}$ and a heat sink at $50^\circ\text{C}$. What is the maximum possible thermal efficiency of this engine?
A coil with 200 turns is placed in a magnetic field. If the magnetic flux through the coil decreases uniformly from $0.05\ \text{Wb}$ to $0.01\ \text{Wb}$ in a time interval of $0.1\ \text{seconds}$, what is the magnitude of the induced electromotive force (emf)?