Formula Sheet

Official-style AP reference. Hover a formula to copy its LaTeX.

Constants and Conversion Factors

k=\frac{1}{4\pi\epsilon_0}=9.0\times10^9\,\mathrm{N\,m^2/C^2}

Coulomb constant

\epsilon_0=8.85\times10^{-12}\,\mathrm{C^2/(N\,m^2)}

Vacuum permittivity

\mu_0=4\pi\times10^{-7}\,\mathrm{T\,m/A}

Vacuum permeability

m_p=1.67\times10^{-27}\,\mathrm{kg}

Proton mass

m_n=1.67\times10^{-27}\,\mathrm{kg}

Neutron mass

m_e=9.11\times10^{-31}\,\mathrm{kg}

Electron mass

e=1.60\times10^{-19}\,\mathrm{C}

Elementary charge

1\,\mathrm{eV}=1.60\times10^{-19}\,\mathrm{J}

Electron volt

c=3.00\times10^8\,\mathrm{m/s}

Speed of light

1\,\mathrm{u}=1.66\times10^{-27}\,\mathrm{kg}=931\,\mathrm{MeV}/c^2

Unified atomic mass unit

G=6.67\times10^{-11}\,\mathrm{m^3/(kg\,s^2)}

Universal gravitational constant

g=9.8\,\mathrm{m/s^2}=9.8\,\mathrm{N/kg}

Gravity at Earth's surface

Unit Symbols and Prefixes

\begin{array}{c|ccccccccccccccc}\text{quantity}&\text{ampere}&\text{coulomb}&\text{electronvolt}&\text{farad}&\text{henry}&\text{hertz}&\text{joule}&\text{kilogram}&\text{meter}&\text{newton}&\text{ohm}&\text{second}&\text{tesla}&\text{volt}&\text{watt}\\\text{symbol}&A&C&eV&F&H&Hz&J&kg&m&N&\Omega&s&T&V&W\end{array}

Official unit symbols

\begin{array}{c|ccccccccc}\text{prefix}&\text{tera}&\text{giga}&\text{mega}&\text{kilo}&\text{centi}&\text{milli}&\text{micro}&\text{nano}&\text{pico}\\\text{symbol}&T&G&M&k&c&m&\mu&n&p\\\text{factor}&10^{12}&10^9&10^6&10^3&10^{-2}&10^{-3}&10^{-6}&10^{-9}&10^{-12}\end{array}

Metric prefixes

Common Trigonometric Values

\begin{array}{c|ccccccc}\theta&0^\circ&30^\circ&37^\circ&45^\circ&53^\circ&60^\circ&90^\circ\\\sin\theta&0&1/2&3/5&\sqrt2/2&4/5&\sqrt3/2&1\\\cos\theta&1&\sqrt3/2&4/5&\sqrt2/2&3/5&1/2&0\\\tan\theta&0&\sqrt3/3&3/4&1&4/3&\sqrt3&\infty\end{array}

Trigonometric values for common angles

Exam Conventions

\mathrm{Inertial\ reference\ frame}

Assume an inertial frame of reference unless otherwise stated.

F_{air}\approx0

Assume air resistance is negligible unless otherwise stated.

\mathrm{Ideal\ springs\ and\ strings}

Assume springs and strings are ideal unless otherwise stated.

V(\infty)=0

Electric potential is zero infinitely far from an isolated point charge.

I:\;\mathrm{positive\ charge\ flow}

Conventional current points in the direction positive charges would drift.

\mathrm{Ideal\ batteries,\ wires,\ and\ meters}

Assume batteries, wires, and meters are ideal unless otherwise stated.

Electricity and Magnetism — Electrostatics

|\vec F_E|=\frac{1}{4\pi\epsilon_0}\frac{|q_1q_2|}{r^2}=k\frac{|q_1q_2|}{r^2}

Coulomb's law

\vec E=\frac{\vec F_E}{q}

Electric field definition

\vec E=\frac{1}{4\pi\epsilon_0}\int\frac{dq}{r^2}\hat r

Electric field from a continuous charge distribution

\Phi_E=\int\vec E\cdot d\vec A

Electric flux

\oint\vec E\cdot d\vec A=\frac{q_{enc}}{\epsilon_0}

Gauss's law

Q_{total}=\int\rho(\vec r)\,dV

Total charge from volume charge density

Electricity and Magnetism — Potential and Capacitance

U_E=\frac{1}{4\pi\epsilon_0}\frac{q_1q_2}{r}

Electric potential energy of two point charges

V=\frac{1}{4\pi\epsilon_0}\int\frac{dq}{r}

Electric potential from a charge distribution

\Delta V=-\int_a^b\vec E\cdot d\vec r

Potential difference from electric field

E_x=-\frac{dV}{dx}

Electric field from potential

\Delta U_E=q\Delta V

Electric potential-energy change

C=\frac{Q}{\Delta V}

Capacitance definition

C=\frac{\kappa\epsilon_0A}{d}

Parallel-plate capacitance with dielectric

U_C=\frac12Q\Delta V

Energy stored in a capacitor

\kappa=\frac{\epsilon}{\epsilon_0}

Dielectric constant

Electricity and Magnetism — Current and Circuits

I=\frac{dq}{dt}

Electric current

I=\int\vec J\cdot d\vec A

Current from current density

\vec E=\rho\vec J

Microscopic form of Ohm's law

R=\frac{\rho\ell}{A}

Resistance of a uniform conductor

I=\frac{\Delta V}{R}

Ohm's law

P=I\Delta V

Electrical power

R_{eq}=\sum_iR_i

Equivalent resistance in series

\frac1{R_{eq}}=\sum_i\frac1{R_i}

Equivalent resistance in parallel

\frac1{C_{eq}}=\sum_i\frac1{C_i}

Equivalent capacitance in series

C_{eq}=\sum_iC_i

Equivalent capacitance in parallel

\tau=R_{eq}C_{eq}

RC time constant

Electricity and Magnetism — Magnetic Fields

\oint\vec B\cdot d\vec A=0

Gauss's law for magnetism

\vec F_B=q(\vec v\times\vec B)

Magnetic force on a moving charge

d\vec B=\frac{\mu_0}{4\pi}\frac{I(d\vec\ell\times\hat r)}{r^2}

Biot–Savart law

\vec F_B=\int I(d\vec\ell\times\vec B)

Magnetic force on a current-carrying wire

\oint\vec B\cdot d\vec\ell=\mu_0I_{enc}

Ampère's law

B_{sol}=\mu_0nI

Magnetic field inside an ideal solenoid

Electricity and Magnetism — Induction and Inductance

\Phi_B=\int\vec B\cdot d\vec A

Magnetic flux

\mathcal E=\oint\vec E\cdot d\vec\ell=-\frac{d\Phi_B}{dt}

Faraday's law

|\mathcal E_{sol}|=N\left|\frac{d\Phi_B}{dt}\right|

Induced emf in an N-turn coil

L_{sol}=\frac{\mu_0N^2A}{\ell}

Inductance of an ideal solenoid

U_L=\frac12LI^2

Energy stored in an inductor

\mathcal E=-L\frac{dI}{dt}

Self-induced emf

\tau=\frac{L}{R_{eq}}

LR time constant

\omega_{LC}=\frac1{\sqrt{LC}}

Angular frequency of an LC circuit

Mechanics — Translational Kinematics

v_x=v_{x0}+a_xt

Velocity under constant acceleration

x=x_0+v_{x0}t+\frac12a_xt^2

Position under constant acceleration

v_x^2=v_{x0}^2+2a_x(x-x_0)

Constant-acceleration relation without time

\Delta x=\int_{t_1}^{t_2}v_x(t)\,dt

Displacement from velocity

\Delta v_x=\int_{t_1}^{t_2}a_x(t)\,dt

Velocity change from acceleration

\vec x_{cm}=\frac{\sum_i m_i\vec x_i}{\sum_i m_i}

Center of mass for discrete particles

\vec r_{cm}=\frac{\int\vec r\,dm}{\int dm}

Center of mass for a continuous body

\lambda=\frac{d}{d\ell}m(\ell)

Linear mass density

Mechanics — Forces and Circular Motion

\vec a_{sys}=\frac{\sum_i\vec F_i}{m_{sys}}=\frac{\vec F_{net}}{m_{sys}}

Acceleration of a system

|\vec F_G|=G\frac{m_1m_2}{r^2}

Universal gravitation

|\vec F_f|\le\mu|\vec F_N|

Friction-force limit

\vec F_s=-k\Delta\vec x

Hooke's law

a_c=\frac{v^2}{r}=r\omega^2

Centripetal acceleration

T=\frac1f

Period and frequency

Mechanics — Work and Energy

K=\frac12mv^2

Translational kinetic energy

W=\int_a^b\vec F\cdot d\vec r

Work by a variable force

\Delta K=\sum_iW_i=\sum_iF_{\parallel,i}d_i

Work–kinetic energy theorem

\Delta U=-\int_a^b\vec F_{cf}(r)\cdot d\vec r

Potential-energy change for a conservative force

F_x=-\frac{dU(x)}{dx}

Force from potential energy

U_s=\frac12k(\Delta x)^2

Spring potential energy

U_G=-G\frac{m_1m_2}{r}

Gravitational potential energy

\Delta U_g=mg\Delta y

Near-Earth gravitational potential-energy change

P_{avg}=\frac{W}{\Delta t}=\frac{\Delta E}{\Delta t}

Average power

P_{inst}=\frac{dW}{dt}

Instantaneous power

Mechanics — Momentum

\vec p=m\vec v

Linear momentum

\vec F_{net}=\frac{d\vec p}{dt}

Newton's second law in momentum form

\vec J=\int_{t_1}^{t_2}\vec F_{net}(t)\,dt=\Delta\vec p

Impulse–momentum theorem

\vec v_{cm}=\frac{\sum_i\vec p_i}{\sum_i m_i}=\frac{\sum_i m_i\vec v_i}{\sum_i m_i}

Center-of-mass velocity

Mechanics — Rotation

\omega=\frac{d\theta}{dt}

Angular velocity

\alpha=\frac{d\omega}{dt}

Angular acceleration

\omega=\omega_0+\alpha t

Angular velocity under constant angular acceleration

\theta=\theta_0+\omega_0t+\frac12\alpha t^2

Angular position under constant angular acceleration

\omega^2=\omega_0^2+2\alpha(\theta-\theta_0)

Constant-angular-acceleration relation without time

v=r\omega

Tangential speed

a_T=r\alpha

Tangential acceleration

\vec\tau=\vec r\times\vec F

Torque

I_{tot}=\sum_iI_i=\sum_i m_i r_i^2

Rotational inertia for discrete masses

I=\int r^2\,dm

Rotational inertia for a continuous body

I'=I_{cm}+Md^2

Parallel-axis theorem

\alpha_{sys}=\frac{\sum\tau}{I_{sys}}=\frac{\tau_{net}}{I_{sys}}

Angular acceleration of a system

K_{rot}=\frac12I\omega^2

Rotational kinetic energy

W=\int\vec\tau\cdot d\vec\theta

Work by a torque

\vec L=\vec r\times\vec p=I\vec\omega

Angular momentum

\Delta\vec L=\int\vec\tau\,dt

Angular impulse

\Delta x=r\Delta\theta

Arc displacement

Mechanics — Oscillations

T=\frac{2\pi}{\omega}=\frac1f

Period, angular frequency, and frequency

T_s=2\pi\sqrt{\frac{m}{k}}

Period of a mass–spring oscillator

T_p=2\pi\sqrt{\frac{\ell}{g}}

Period of a simple pendulum

T_{phys}=2\pi\sqrt{\frac{I}{mgd}}

Period of a physical pendulum

x=x_{max}\cos(\omega t+\phi)

Simple harmonic motion

Geometry and Trigonometry

A=bh

Area of a rectangle

V=\ell wh

Volume of a rectangular solid

A=\frac12 bh

Area of a triangle

V=\pi r^2\ell

Volume of a cylinder

S=2\pi r\ell+2\pi r^2

Surface area of a cylinder

A=\pi r^2,\quad C=2\pi r,\quad s=r\theta

Circle area, circumference, and arc length

V=\frac43\pi r^3,\quad S=4\pi r^2

Sphere volume and surface area

a^2+b^2=c^2

Pythagorean theorem

\sin\theta=\frac ac,\quad\cos\theta=\frac bc,\quad\tan\theta=\frac ab

Right-triangle trigonometric ratios

Vectors

\vec A\cdot\vec B=AB\cos\theta

Vector dot product

|\vec A\times\vec B|=AB\sin\theta

Magnitude of a vector cross product

\vec r=A\hat i+B\hat j+C\hat k

Vector in component form

\vec C=\vec A+\vec B

Vector addition

\vec C=(A_x+B_x)\hat i+(A_y+B_y)\hat j

Vector addition by components

Calculus

\frac{df}{dx}=\frac{df}{du}\frac{du}{dx}

Chain rule

\frac{d}{dx}(x^n)=nx^{n-1}

Power rule for derivatives

\frac{d}{dx}(e^{ax})=ae^{ax}

Derivative of an exponential

\frac{d}{dx}(\ln ax)=\frac1x

Derivative of a logarithm

\frac{d}{dx}[\sin(ax)]=a\cos(ax)

Derivative of sine

\frac{d}{dx}[\cos(ax)]=-a\sin(ax)

Derivative of cosine

\int x^n\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\ne-1

Power rule for integrals

\int e^{ax}\,dx=\frac1a e^{ax}+C

Integral of an exponential

\int\frac{dx}{x+a}=\ln|x+a|+C

Integral yielding a logarithm

\int\cos(ax)\,dx=\frac1a\sin(ax)+C

Integral of cosine

\int\sin(ax)\,dx=-\frac1a\cos(ax)+C

Integral of sine

Identities

\log(a\cdot b^x)=\log a+x\log b

Logarithm product and power identity

\sin^2\theta+\cos^2\theta=1

Pythagorean trigonometric identity

\sin(2\theta)=2\sin\theta\cos\theta

Double-angle identity for sine

\frac{\sin\theta}{\cos\theta}=\tan\theta

Tangent identity