Key Concepts

Why This Matters

Physics isn't just a collection of formulas to memorize—it's a framework for understanding how everything in the universe behaves, from a ball rolling across a table to light bending around a black hole. On your exam, you're being tested on your ability to connect these foundational principles: conservation laws, force interactions, field theory, and the quantum-relativistic revolution that reshaped our understanding of reality in the 20th century.

The concepts here build on each other in predictable ways. Classical mechanics gives you the tools for everyday motion, conservation laws reveal what stays constant even when systems change, and modern physics (quantum mechanics and relativity) shows where classical ideas break down. Don't just memorize equations—know why each principle matters and when it applies. That's what separates a 3 from a 5.

---

Classical Mechanics: Forces and Motion

These foundational principles describe how objects move and interact in everyday situations. Classical mechanics works beautifully for objects much larger than atoms moving much slower than light.

Newton's Laws of Motion

• First Law (Inertia)—objects resist changes to their motion; without a net external force, velocity remains constant
• Second Law gives you the quantitative relationship: $\vec{F} = m\vec{a}$, meaning acceleration is proportional to force and inversely proportional to mass
• Third Law (Action-Reaction) explains force pairs: when you push on a wall, the wall pushes back with equal magnitude in the opposite direction

Law of Universal Gravitation

• Every mass attracts every other mass with a force described by $F = G\frac{m_1 m_2}{r^2}$, where $G$ is the gravitational constant
• Inverse-square relationship—doubling the distance between objects reduces gravitational force to one-quarter its original value
• Explains orbital motion of planets, moons, and satellites; this is the "why" behind Kepler's empirical laws

---

Conservation Laws: What Stays Constant

Conservation laws are among the most powerful tools in physics. When a quantity is conserved, you can relate "before" and "after" states without knowing the messy details in between.

Conservation of Energy

• Energy transforms but never disappears—kinetic, potential, thermal, and chemical energy are all interconvertible
• Total energy in an isolated system remains constant, making this principle essential for solving collision and motion problems
• Work-energy theorem connects force and displacement to changes in kinetic energy: $W = \Delta KE$

Conservation of Momentum

• Total momentum stays constant in a closed system with no external forces; momentum is defined as $\vec{p} = m\vec{v}$
• Elastic collisions conserve both momentum and kinetic energy; inelastic collisions conserve only momentum
• Vector quantity—you must account for direction, which is why momentum problems often require component analysis

---

Thermodynamics: Energy, Entropy, and Limits

Thermodynamics governs energy transfer and the direction of natural processes. These laws explain why engines have efficiency limits and why time seems to flow in one direction.

Laws of Thermodynamics

• First Law is conservation of energy applied to heat and work: $\Delta U = Q - W$, where $U$ is internal energy, $Q$ is heat added, and $W$ is work done by the system
• Second Law introduces entropy—the total entropy of an isolated system never decreases, explaining why heat flows from hot to cold spontaneously
• Third Law sets an absolute limit: as temperature approaches absolute zero ($0 \text{ K}$), entropy of a perfect crystal approaches zero, making absolute zero unattainable

---

Electromagnetism: Fields and Waves

Electromagnetic theory unifies electricity and magnetism into a single framework. Changing electric fields create magnetic fields, and vice versa—this interplay produces light itself.

Electromagnetic Theory

• Maxwell's equations describe how electric and magnetic fields behave and interact with charged matter
• Electromagnetic waves arise from oscillating fields; they travel at the speed of light $c \approx 3 \times 10^8 \text{ m/s}$ and require no medium
• Light is an electromagnetic wave—this insight unified optics with electricity and magnetism, one of physics' great unifications

---

Quantum Mechanics: The Subatomic World

At atomic and subatomic scales, classical physics fails. Quantum mechanics introduces probability, wave functions, and fundamental limits on what we can know.

Wave-Particle Duality

• Particles exhibit wave behavior—electrons and photons create interference patterns in the double-slit experiment, a hallmark of wave phenomena
• Waves exhibit particle behavior—light delivers energy in discrete packets (photons), explaining the photoelectric effect
• De Broglie wavelength $\lambda = \frac{h}{p}$ relates a particle's momentum to its associated wavelength, where $h$ is Planck's constant

Quantum Mechanics Principles

• Quantization—energy, angular momentum, and other properties come in discrete values, not continuous ranges
• Heisenberg's Uncertainty Principle sets fundamental limits: $\Delta x \cdot \Delta p \geq \frac{\hbar}{2}$, meaning position and momentum cannot both be precisely known
• Quantum entanglement creates correlations between particles that persist regardless of distance—"spooky action at a distance" that Einstein famously distrusted

---

Relativity: Space, Time, and Gravity Reimagined

Einstein's theories revolutionized our understanding of space, time, and gravity. Special relativity handles high speeds; general relativity handles gravity and acceleration.

Special Relativity

• Principle of relativity—the laws of physics are identical for all observers in uniform (non-accelerating) motion
• Time dilation and length contraction occur at speeds approaching $c$; moving clocks run slow, and moving objects contract along the direction of motion
• Mass-energy equivalence is captured in $E = mc^2$, showing that mass itself is a form of energy

General Relativity

• Gravity is geometry—mass curves spacetime, and objects follow geodesics (shortest paths) through that curved geometry
• Predictions confirmed include gravitational lensing (light bending around massive objects), gravitational time dilation, and gravitational waves
• Black holes are regions where spacetime curvature becomes so extreme that nothing, not even light, can escape

---

Quick Reference Table

---

Self-Check Questions

1. Which two conservation laws apply to collisions, and how do elastic and inelastic collisions differ in what they conserve?

2. Both Newton's Law of Universal Gravitation and Coulomb's Law (for electric charges) follow an inverse-square relationship. What does this mathematical similarity suggest about how these forces behave with distance?

3. Compare and contrast the First and Second Laws of Thermodynamics: what does each tell you about energy, and what does only the Second Law explain?

4. How does the Heisenberg Uncertainty Principle differ from simple measurement error? What fundamental limit does it describe?

5. An FRQ asks you to explain why GPS satellites require relativistic corrections. Which effects from special relativity and general relativity would you need to address, and do they work in the same direction or opposite directions?