AP Physics 2 Comprehensive Reference Guide and Equations List

Standard Reference Conventions

• The frame of reference of any problem is assumed to be inertial unless otherwise stated.
• Air resistance is assumed to be negligible unless otherwise stated.
• Springs and strings are assumed to be ideal unless otherwise stated.
• The electric potential is zero at an infinite distance from an isolated point charge.
• The direction of current is the direction in which positive charges would drift.
• All batteries, wires, and meters are assumed to be ideal unless otherwise stated.

Physical Constants and Conversion Factors

• Avogadro's number: $N_A = 6.02 \times 10^{23}\,\text{mol}^{-1}$
• Universal gas constant: $R = 8.31\,\text{J}/(\text{mol}\cdot\text{K})$
• Boltzmann's constant: $k_B = 1.38 \times 10^{-23}\,\text{J}/\text{K}$
• $1\,\text{atmosphere of pressure}$: $1\,\text{atm} = 1.0 \times 10^5\,\text{N}/\text{m}^2 = 1.0 \times 10^5\,\text{Pa}$
• Coulomb constant: $k_c = \frac{1}{4\pi\varepsilon_0} = 8.99 \times 10^9\,\text{N}\cdot\text{m}^2/\text{C}^2$
• Proton mass: $m_p = 1.67 \times 10^{-27}\,\text{kg}$
• Neutron mass: $m_n = 1.67 \times 10^{-27}\,\text{kg}$
• Electron mass: $m_e = 9.11 \times 10^{-31}\,\text{kg}$
• Elementary charge: $e = 1.60 \times 10^{-19}\,\text{C}$
• Vacuum permittivity: $\varepsilon_0 = 8.85 \times 10^{-12}\,\text{C}^2/(\text{N}\cdot\text{m}^2)$
• Vacuum permeability: $\mu_0 = 4\pi \times 10^{-7}\,\text{T}\cdot\text{m}/\text{A}$
• $1\,\text{electron volt}$: $1\,\text{eV} = 1.60 \times 10^{-19}\,\text{J}$
• Planck's constant: $h = 6.63 \times 10^{-34}\,\text{J}\cdot\text{s} = 4.14 \times 10^{-15}\,\text{eV}\cdot\text{s}$
• Speed of light: $c = 3.00 \times 10^8\,\text{m}/\text{s}$
• Wien's constant: $b = 2.90 \times 10^{-3}\,\text{m}\cdot\text{K}$
• Stefan-Boltzmann constant: $\sigma = 5.67 \times 10^{-8}\,\text{W}/(\text{m}^2\cdot\text{K}^4)$
• $1\,\text{unified atomic mass unit}$: $1\,\text{u} = 1.66 \times 10^{-27}\,\text{kg} = 931.5\,\text{MeV}/c^2$
• Universal gravitational constant: $G = 6.67 \times 10^{-11}\,\text{m}^3/(\text{kg}\cdot\text{s}^2)$
• Magnitude of the acceleration due to gravity at Earth's surface: $g = 9.8\,\text{m}/\text{s}^2$
• Magnitude of the gravitational field strength at Earth's surface: $g = 9.8\,\text{N}/\text{kg}$

Unit Symbols

• ampere: $\text{A}$
• coulomb: $\text{C}$
• degree Celsius: $^\circ\text{C}$
• electron volt: $\text{eV}$
• farad: $\text{F}$
• hertz: $\text{Hz}$
• joule: $\text{J}$
• kelvin: $\text{K}$
• kilogram: $\text{kg}$
• meter: $\text{m}$
• mole: $\text{mol}$
• newton: $\text{N}$
• ohm: $\Omega$
• pascal: $\text{Pa}$
• second: $\text{s}$
• tesla: $\text{T}$
• volt: $\text{V}$
• watt: $\text{W}$

Metric Prefixes

• Factor $10^{12}$: prefix tera, symbol $\text{T}$
• Factor $10^9$: prefix giga, symbol $\text{G}$
• Factor $10^6$: prefix mega, symbol $\text{M}$
• Factor $10^3$: prefix kilo, symbol $\text{k}$
• Factor $10^{-2}$: prefix centi, symbol $\text{c}$
• Factor $10^{-3}$: prefix milli, symbol $\text{m}$
• Factor $10^{-6}$: prefix micro, symbol $\mu$
• Factor $10^{-9}$: prefix nano, symbol $\text{n}$
• Factor $10^{-12}$: prefix pico, symbol $\text{p}$

Values of Trigonometric Functions for Common Angles

• For angle $\theta = 0^\circ$:
  • $\sin(0^\circ) = 0$
  • $\cos(0^\circ) = 1$
  • $\tan(0^\circ) = 0$
• For angle $\theta = 30^\circ$:
  • $\sin(30^\circ) = \frac{1}{2}$
  • $\cos(30^\circ) = \frac{\sqrt{3}}{2}$
  • $\tan(30^\circ) = \frac{\sqrt{3}}{3}$
• For angle $\theta = 37^\circ$:
  • $\sin(37^\circ) = \frac{3}{5}$
  • $\cos(37^\circ) = \frac{4}{5}$
  • $\tan(37^\circ) = \frac{3}{4}$
• For angle $\theta = 45^\circ$:
  • $\sin(45^\circ) = \frac{\sqrt{2}}{2}$
  • $\cos(45^\circ) = \frac{\sqrt{2}}{2}$
  • $\tan(45^\circ) = 1$
• For angle $\theta = 53^\circ$:
  • $\sin(53^\circ) = \frac{4}{5}$
  • $\cos(53^\circ) = \frac{3}{5}$
  • $\tan(53^\circ) = \frac{4}{3}$
• For angle $\theta = 60^\circ$:
  • $\sin(60^\circ) = \frac{\sqrt{3}}{2}$
  • $\cos(60^\circ) = \frac{1}{2}$
  • $\tan(60^\circ) = \sqrt{3}$
• For angle $\theta = 90^\circ$:
  • $\sin(90^\circ) = 1$
  • $\cos(90^\circ) = 0$
  • $\tan(90^\circ) = \text{undefined}$

Electricity Equations and Symbols

• Notation Key:
  • $A$ = area
  • $C$ = capacitance
  • $d$ = distance
  • $E$ = electric field
  • $F$ = force
  • $I$ = current
  • $\ell$ = length
  • $P$ = power
  • $q, Q$ = charge
  • $r$ = distance, radius, or position
  • $R$ = resistance
  • $t$ = time
  • $U$ = potential energy
  • $V$ = electric potential
  • $\kappa$ = dielectric constant
  • $\rho$ = resistivity
  • $\tau$ = time constant
• Governing Equations:
  • Coulomb's Law for electrostatic force:     $|\mathbf{F}_E| = \frac{1}{4\pi\varepsilon_0}\frac{|q_1 q_2|}{r^2}$
  • Electric field definition and point charge field:     $\mathbf{E} = \frac{\mathbf{F}_E}{q}$$|\mathbf{E}| = \frac{1}{4\pi\varepsilon_0}\frac{|q|}{r^2}$
  • Electric potential difference and field relation:     $\Delta V = \frac{\Delta U_E}{q}$$E = \left|\frac{\Delta V}{\Delta r}\right|$
  • Electric potential of a point charge:     $V = \frac{1}{4\pi\varepsilon_0}\frac{q}{r}$
  • Electric potential energy:     $U_E = qV = \frac{1}{4\pi\varepsilon_0}\frac{q_1 q_2}{r}$
  • Capacitance definition and parallel-plate capacitor equation:     $C = \frac{Q}{\Delta V}$$C = \frac{\kappa \varepsilon_0 A}{d}$
  • Energy stored in a capacitor:     $U_C = \frac{1}{2}Q\Delta V = \frac{1}{2}C(\Delta V)^2 = \frac{1}{2}\frac{Q^2}{C}$
  • Electric current definition:     $I = \frac{\Delta Q}{\Delta t}$
  • Resistance of a uniform conductor:     $R = \frac{\rho \ell}{A}$
  • Ohm's Law:     $I = \frac{\Delta V}{R}$
  • Electric power:     $P = I\Delta V = I^2 R = \frac{(\Delta V)^2}{R}$
  • Series and parallel resistor combinations:     $R_s = \sum_i R_i$$\frac{1}{R_p} = \sum_i \frac{1}{R_i}$
  • Series and parallel capacitor combinations:     $C_p = \sum_i C_i$$\frac{1}{C_s} = \sum_i \frac{1}{C_i}$
  • RC circuit time constant:     $\tau = RC$

Magnetism Equations and Symbols

• Notation Key:
  • $A$ = area
  • $B$ = magnetic field
  • $F$ = force
  • $I$ = current
  • $\ell$ = length
  • $q$ = charge
  • $r$ = distance, radius, or position
  • $t$ = time
  • $v$ = velocity or speed
  • $\mathcal{E}$ = electromotive force (emf)
  • $\theta$ = angle
  • $\Phi_B$ = magnetic flux
• Governing Equations:
  • Magnetic force on a moving charge:     $F_M = |q| v B \sin(\theta)$
  • Magnetic force on a current-carrying wire:     $F_M = I \ell B \sin(\theta)$
  • Magnetic field of a long straight current-carrying wire:     $B = \frac{\mu_0 I}{2\pi r}$
  • Magnetic flux:     $\Phi_B = B A \cos(\theta)$
  • Faraday's law of induction:     $\mathcal{E} = -\frac{\Delta \Phi_B}{\Delta t}$
  • Motional electromotive force:     $\mathcal{E} = B \ell v$

Thermal Physics Equations and Symbols

• Notation Key:
  • $A$ = area
  • $c$ = specific heat
  • $F$ = force
  • $k$ = thermal conductivity
  • $K$ = kinetic energy
  • $L$ = length
  • $m$ = mass
  • $n$ = number of moles
  • $N$ = number of atoms
  • $P$ = pressure
  • $Q$ = energy transferred to a system by heating
  • $t$ = time
  • $T$ = temperature
  • $U$ = internal energy
  • $v$ = velocity or speed
  • $V$ = volume
  • $W$ = work done on a system
• Governing Equations:
  • Rate of thermal conduction:     $\frac{Q}{\Delta t} = \frac{k A \Delta T}{L}$
  • Ideal gas law:     $PV = nRT = N k_B T$
  • Average translational kinetic energy per molecule:     $K = \frac{3}{2} k_B T$
  • Root-mean-square speed of gas molecules:     $v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}} = \sqrt{\frac{3 R T}{M}}$
  • First law of thermodynamics:     $\Delta U = Q + W$
  • Work done on a gas at constant pressure:     $W = -P \Delta V$
  • Internal energy of a monatomic ideal gas:     $U = \frac{3}{2} n R T = \frac{3}{2} N k_B T = \frac{3}{2} P V$
  • Specific heat capacity heat transfer:     $Q = m c \Delta T$

Waves, Sound, and Optics Equations and Symbols

• Notation Key:
  • $a$ = width
  • $A$ = amplitude
  • $d$ = separation
  • $D$ = path length
  • $f$ = frequency or focal length
  • $F$ = force
  • $h$ = height
  • $\ell$ = length
  • $L$ = distance
  • $m$ = order or mass
  • $M$ = magnification
  • $n$ = index of refraction
  • $s$ = position
  • $t$ = time
  • $T$ = period
  • $v$ = speed
  • $x$ = position
  • $y$ = position
  • $\lambda$ = wavelength
  • $\theta$ = angle
  • $\omega$ = angular frequency
• Governing Equations:
  • Wave speed relation:     $\lambda = \frac{v}{f}$
  • Index of refraction definition:     $n = \frac{c}{v}$
  • Snell's Law of refraction:     $n_1 \sin(\theta_1) = n_2 \sin(\theta_2)$
  • Thin lens and spherical mirror equation:     $\frac{1}{f} = \frac{1}{s_o} + \frac{1}{s_i}$
  • Magnification equation:     $M = \frac{h_i}{h_o} = -\frac{s_i}{s_o}$
  • Double-slit and diffraction grating interference maxima:     $d \sin(\theta) = m \lambda$
  • Single-slit diffraction minima:     $a \sin(\theta) = m \lambda$
  • Interference path length difference relations:     $\Delta D = m \lambda$$\Delta D = \left(m + \frac{1}{2}\right)\lambda$
  • Beat frequency:     $f_{\text{beat}} = |f_1 - f_2|$

Modern Physics Equations and Symbols

• Notation Key:
  • $A$ = area
  • $E$ = energy
  • $f$ = frequency
  • $K$ = kinetic energy
  • $m$ = mass
  • $N$ = number of particles
  • $p$ = momentum
  • $P$ = power
  • $t$ = time
  • $T$ = absolute temperature
  • $\phi$ = work function
• Governing Equations:
  • Photon energy:     $E = h f = \frac{h c}{\lambda}$
  • Photoelectric effect maximum kinetic energy:     $K_{\text{max}} = h f - \phi$
  • Photon momentum and de Broglie wavelength:     $p = \frac{h}{\lambda}$$\lambda = \frac{h}{p}$
  • Mass-energy equivalence:     $E = m c^2$
  • Radioactive decay equation:     $N = N_0 e^{-\lambda t}$
  • Wien's displacement law:     $\lambda_{\text{max}} T = b$
  • Stefan-Boltzmann law for radiant power:     $P = \sigma A T^4$

Mechanics and Fluids Equations and Symbols

• Notation Key:
  • $a$ = acceleration or angular acceleration
  • $A$ = amplitude or area
  • $d$ = distance
  • $E$ = energy
  • $f$ = frequency
  • $F$ = force
  • $h$ = height
  • $I$ = rotational inertia
  • $J$ = impulse
  • $k$ = spring constant
  • $K$ = kinetic energy
  • $\ell$ = length
  • $L$ = angular momentum
  • $m, M$ = mass
  • $p$ = momentum
  • $P$ = pressure or power
  • $r$ = radius or distance
  • $t$ = time
  • $T$ = period
  • $U$ = potential energy
  • $v$ = velocity or speed
  • $V$ = volume
  • $W$ = work
  • $x, y$ = position or vertical position
  • $\alpha$ = angular acceleration
  • $\theta$ = angle or angular position
  • $\mu$ = coefficient of friction
  • $\rho$ = density
  • $\tau$ = torque
  • $\omega$ = angular speed
• Governing Equations for Fluids:
  • Density definition:     $\rho = \frac{m}{V}$
  • Pressure definition:     $P = \frac{F}{A}$
  • Hydrostatic pressure at depth $h$:     $P = P_0 + \rho g h$
  • Buoyant force (Archimedes' principle):     $F_B = \rho V g$
  • Continuity equation for fluid flow:     $A_1 v_1 = A_2 v_2$
  • Bernoulli's equation:     $P_1 + \rho g y_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho g y_2 + \frac{1}{2}\rho v_2^2$

Geometry and Trigonometry Formulas

• Rectangle:
  • Area: $A = b h$
• Triangle:
  • Area: $A = \frac{1}{2} b h$
• Circle:
  • Area: $A = \pi r^2$
  • Circumference: $C = 2\pi r$
• Rectangular Solid:
  • Volume: $V = \ell w h$
• Cylinder:
  • Volume: $V = \pi r^2 \ell$
  • Surface Area: $S = 2\pi r \ell + 2\pi r^2$
• Sphere:
  • Volume: $V = \frac{4}{3}\pi r^3$
  • Surface Area: $S = 4\pi r^2$
• Right Triangle:
  • Pythagorean Theorem: $c^2 = a^2 + b^2$
  • Sine definition: $\sin(\theta) = \frac{a}{c}$
  • Cosine definition: $\cos(\theta) = \frac{b}{c}$
  • Tangent definition: $\tan(\theta) = \frac{a}{b}$