Comprehensive Physics Equations and Concepts: Chapters 1–30

Chapter 1: Representing Motion

Displacement, Velocity, and Time

Motion can be described using displacement, velocity, and time. Displacement (Δx) is the change in position, velocity (v) is the rate of change of displacement, and time (t) is the duration of motion.

• Displacement: $\Delta x = x_f - x_i$

• Average Velocity: $v = \frac{\Delta x}{\Delta t}$

Vectors and Their Representation

Vectors have both magnitude and direction. The magnitude and direction of a vector \( \vec{A} \) can be found using its components.

• Magnitude: $A = \sqrt{A_x^2 + A_y^2}$

• Direction: $\theta = \tan^{-1}(A_y/A_x)$

• Components: $A_x = A \cos \theta$, $A_y = A \sin \theta$

Chapter 2: Motion in One Dimension

Kinematic Equations

These equations describe the motion of objects moving in a straight line with constant acceleration.

• $v = v_i + a \Delta t$

• $x = x_i + v_i \Delta t + \frac{1}{2}a(\Delta t)^2$

• $v^2 = v_i^2 + 2a(x - x_i)$

Chapter 3: Vectors and Motion in Two Dimensions

Projectile Motion

Projectile motion involves two-dimensional motion under constant acceleration due to gravity.

• Horizontal motion: $x = v_{ix} t$

• Vertical motion: $y = v_{iy} t - \frac{1}{2}gt^2$

Chapter 4: Forces and Newton's Laws of Motion

Newton's Laws

Newton's laws describe the relationship between forces and motion.

• First Law (Inertia): An object remains at rest or in uniform motion unless acted upon by a net force.

• Second Law: $\vec{F}_{net} = m\vec{a}$

• Third Law: For every action, there is an equal and opposite reaction.

Chapter 5: Applying Newton's Laws

Friction and Tension

Friction opposes motion, and tension is the force transmitted through a string or rope.

• Kinetic friction: $f_k = \mu_k N$

• Static friction: $f_s \leq \mu_s N$

Chapter 6: Circular Motion, Orbits, and Gravity

Circular Motion

Objects moving in a circle experience centripetal acceleration directed toward the center.

• Centripetal acceleration: $a_c = \frac{v^2}{r}$

• Gravitational force: $F = G\frac{m_1 m_2}{r^2}$

Chapter 7: Rotational Motion

Rotational Kinematics and Dynamics

Rotational motion is described by angular displacement, velocity, and acceleration. The moment of inertia quantifies rotational inertia.

• Angular displacement: $\theta = \theta_i + \omega_i t + \frac{1}{2}\alpha t^2$

• Moment of inertia (I): Depends on mass distribution.

Rotational Kinetic Energy

• $K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2$

Chapter 8: Equilibrium and Elasticity

Conditions for Equilibrium

An object is in equilibrium if the net force and net torque on it are zero.

• $\sum F_x = 0$, $\sum F_y = 0$, $\sum \tau = 0$

Elasticity and Hooke's Law

Elasticity describes how materials deform under force. Hooke's Law relates the force to the extension or compression of a spring.

• $F = -kx$

• Young's modulus: $Y = \frac{F/A}{\Delta L/L}$

Chapter 9: Momentum

Linear and Angular Momentum

• Linear momentum: $\vec{p} = m\vec{v}$

• Conservation of momentum: $\sum \vec{p}_{i} = \sum \vec{p}_{f}$ (in the absence of external forces)

• Angular momentum: $L = I\omega$

Chapter 10: Energy and Work

Work and Energy

• Kinetic energy: $K = \frac{1}{2}mv^2$

• Potential energy (gravitational): $U_g = mgy$

• Work: $W = Fd\cos\theta$

Chapter 11: Using Energy

Thermal Energy and Efficiency

• First Law of Thermodynamics: $\Delta E_{th} = W + Q$

• Efficiency: $e = \frac{\text{what you get}}{\text{what you had to pay}} = \text{COP}$

• Maximum efficiency (Carnot): $e_{max} = 1 - \frac{T_C}{T_H}$

Chapter 12: Thermal Properties of Matter

Ideal Gas Law and Heat Transfer

• $pV = Nk_B T$

• $pV = nRT$

• Heat transfer: $Q = nC_v \Delta T$ (at constant volume), $Q = nC_p \Delta T$ (at constant pressure)

Work Done by a Gas

• Work is the area under the p-V curve: $W = \int p dV$

Chapter 13: Fluids

Fluid Dynamics

• Continuity equation: $Q = \frac{\Delta V}{\Delta t} = v_1 A_1 = v_2 A_2$

Chapter 14: Equilibrium and Oscillations

Simple Harmonic Motion (SHM)

• Total energy in SHM: $E = \frac{1}{2}mv_x^2 + \frac{1}{2}kx^2 = \frac{1}{2}mv_{max}^2 = \frac{1}{2}kA^2$

• Frequency: $f = \frac{1}{T}$

• Position, velocity, acceleration:

Chapter 15: Traveling Waves and Sound

Wave Properties

• Wave equation: $y(x,t) = A \cos(2\pi f t)$

• Speed of sound: $v = 343$ m/s (in air at 20°C)

Doppler Effect

• $\Delta f = \pm 2f_s \frac{V_0}{V}$

Chapter 16: Superposition and Standing Waves

Interference Conditions

• Constructive and destructive interference depend on phase changes and path differences.

Chapter 17: Wave Optics

Reflection and Refraction

• Law of reflection: $\theta_r = \theta_i$

• Snell's Law: $n_1 \sin \theta_1 = n_2 \sin \theta_2$

Chapter 18: Ray Optics

Image Formation by Lenses and Mirrors

• Lens equation: $\frac{1}{s} + \frac{1}{s'} = \frac{1}{f}$

• Magnification: $m = -\frac{s'}{s}$

Chapter 19: Optical Instruments

Microscopes and Telescopes

• Total magnification: $M = m_o m_e = -\frac{L}{f_o} \frac{0.25\,\text{m}}{f_e}$

Chapter 20: Electric Fields and Forces

Coulomb's Law and Electric Field

• $F = k \frac{q_1 q_2}{r^2}$

• $\vec{E} = \frac{F}{q}$

Chapter 21: Electric Potential

Potential and Capacitance

• $V = \frac{kq}{r}$

• $C = \varepsilon_0 \frac{A}{d}$

Chapter 22: Current and Resistance

Ohm's Law

• $I = \frac{\Delta V}{R}$

• $R = \rho \frac{L}{A}$

Chapter 23: Circuits

Series and Parallel Circuits

• Series resistance: $R_{series} = R_1 + R_2 + \ldots$

• Parallel resistance: $\frac{1}{R_{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \ldots$

Chapter 24: Magnetic Fields and Forces

Magnetic Force and Right-Hand Rule

• $F = qvB \sin \alpha$

Chapter 25: Electromagnetic Induction and Waves

Faraday's Law and EM Waves

• $\varepsilon = -\frac{d\Phi}{dt}$

• $c = \lambda f$

Chapter 26: AC Electricity

AC Circuit Elements

• Resistor: $V_R = I_R R$

• Capacitor: $X_C = \frac{1}{2\pi f C}$

• Inductor: $X_L = 2\pi f L$

Chapter 27: Relativity

Special Relativity

• $\beta = \frac{v}{c} \leq 1$

• $\gamma = \frac{1}{\sqrt{1-\beta^2}}$

• $E = mc^2$

Chapter 28: Quantum Physics

Photon Energy and Uncertainty Principle

• $E = hf$

• $\Delta x \Delta p_x \geq \frac{h}{4\pi}$

Chapter 29: Atoms and Molecules

Atomic Structure and Spectra

• $E_n = \frac{13.60\,\text{eV}}{n^2}$

• $\lambda_{max} = 91.1\,\text{nm} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$

Chapter 30: Nuclear Physics

Radioactive Decay and Nuclear Reactions

• $N = N_0 e^{-\lambda t}$

• $t_{1/2} = \frac{0.693}{\lambda}$

Appendix: Physical Constants

Common physical constants used in physics calculations.

• Speed of light: $c = 3.00 \times 10^8$ m/s

• Gravitational constant: $G = 6.67 \times 10^{-11}$ N m2/kg2

• Planck's constant: $h = 6.63 \times 10^{-34}$ J·s

• Elementary charge: $e = 1.60 \times 10^{-19}$ C