Constants and conversion factors
$G = 6.67 \times 10^{-11}\ \mathrm{m^{3}/(kg \cdot s^{2})} = 6.67 \times 10^{-11}\ \mathrm{N \cdot m^{2}/kg^{2}}$universal gravitational constant
$1\ \mathrm{atm} = 1.0 \times 10^{5}\ \mathrm{N/m^{2}} = 1.0 \times 10^{5}\ \mathrm{Pa}$one atmosphere of pressure
$g = 9.8\ \mathrm{m/s^{2}}$magnitude of the acceleration due to gravity at the surface of the Earth
$g = 9.8\ \mathrm{N/kg}$magnitude of the gravitational field strength at the surface of the Earth
Prefixes
| Factor | Prefix | Symbol |
|---|---|---|
| $10^{12}$ | tera | $T$ |
| $10^{9}$ | giga | $G$ |
| $10^{6}$ | mega | $M$ |
| $10^{3}$ | kilo | $k$ |
| $10^{-2}$ | centi | $c$ |
| $10^{-3}$ | milli | $m$ |
| $10^{-6}$ | micro | $\mu$ |
| $10^{-9}$ | nano | $n$ |
| $10^{-12}$ | pico | $p$ |
Unit symbols
| hertz | $\mathrm{Hz}$ |
| newton | $\mathrm{N}$ |
| joule | $\mathrm{J}$ |
| pascal | $\mathrm{Pa}$ |
| kilogram | $\mathrm{kg}$ |
| second | $\mathrm{s}$ |
| meter | $\mathrm{m}$ |
| watt | $\mathrm{W}$ |
Trigonometric functions at common angles
| $\theta$ | $0^{\circ}$ | $30^{\circ}$ | $37^{\circ}$ | $45^{\circ}$ | $53^{\circ}$ | $60^{\circ}$ | $90^{\circ}$ |
|---|---|---|---|---|---|---|---|
| $\sin\theta$ | $0$ | $1/2$ | $3/5$ | $\sqrt{2}/2$ | $4/5$ | $\sqrt{3}/2$ | $1$ |
| $\cos\theta$ | $1$ | $\sqrt{3}/2$ | $4/5$ | $\sqrt{2}/2$ | $3/5$ | $1/2$ | $0$ |
| $\tan\theta$ | $0$ | $\sqrt{3}/3$ | $3/4$ | $1$ | $4/3$ | $\sqrt{3}$ | $\infty$ |
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.
- Fluids are assumed to be ideal, and pipes are assumed to be completely filled by fluid, unless otherwise stated.
Geometry and trigonometry
Rectangle
$A = bh$
Triangle
$A = \tfrac{1}{2}bh$
Circle
$A = \pi r^{2}$
$C = 2\pi r$
$s = r\theta$
Rectangular solid
$V = \ell w h$
Cylinder
$V = \pi r^{2}\ell$
$S = 2\pi r\ell + 2\pi r^{2}$
Sphere
$V = \tfrac{4}{3}\pi r^{3}$
$S = 4\pi r^{2}$
Right triangle
$a^{2} + b^{2} = c^{2}$
$\sin\theta = a/c$
$\cos\theta = b/c$
$\tan\theta = a/b$
Kinematics
$v_{x} = v_{x0} + a_{x}t$
$x = x_{0} + v_{x0}t + \tfrac{1}{2}a_{x}t^{2}$
$v_{x}^{2} = v_{x0}^{2} + 2a_{x}\left(x – x_{0}\right)$
Center of mass
$\vec{x}_{\mathrm{cm}} = \frac{\sum m_{i}\vec{x}_{i}}{\sum m_{i}}$
$\vec{v}_{\mathrm{cm}} = \frac{\sum \vec{p}_{i}}{\sum m_{i}} = \frac{\sum m_{i}\vec{v}_{i}}{\sum m_{i}}$
Force and translational dynamics
$\vec{a}_{\mathrm{sys}} = \frac{\sum \vec{F}}{m_{\mathrm{sys}}} = \frac{\vec{F}_{\mathrm{net}}}{m_{\mathrm{sys}}}$
$\vec{F}_{\mathrm{net}} = \frac{\Delta\vec{p}}{\Delta t} = m\frac{\Delta\vec{v}}{\Delta t} = m\vec{a}$
$\left|\vec{F}_{g}\right| = G\frac{m_{1}m_{2}}{r^{2}}$
$\left|\vec{F}_{f}\right| \le \left|\mu \vec{F}_{N}\right|$
$\vec{F}_{s} = -k\Delta\vec{x}$
$a_{c} = \frac{v^{2}}{r}$
Work, energy and power
$K = \tfrac{1}{2}mv^{2}$
$W = F_{\parallel}d = Fd\cos\theta$
$\Delta K = \sum W_{i} = \sum F_{\parallel,i}\,d_{i}$
$U_{s} = \tfrac{1}{2}k\left(\Delta x\right)^{2}$
$U_{G} = -\frac{Gm_{1}m_{2}}{r}$
$\Delta U_{g} = mg\Delta y$
$P_{\mathrm{avg}} = \frac{W}{\Delta t} = \frac{\Delta E}{\Delta t}$
$P_{\mathrm{inst}} = F_{\parallel}v = Fv\cos\theta$
Linear momentum
$\vec{p} = m\vec{v}$
$\vec{J} = \vec{F}_{\mathrm{avg}}\Delta t = \Delta\vec{p}$
Torque and rotational dynamics
$\omega = \omega_{0} + \alpha t$
$\theta = \theta_{0} + \omega_{0}t + \tfrac{1}{2}\alpha t^{2}$
$\omega^{2} = \omega_{0}^{2} + 2\alpha\left(\theta – \theta_{0}\right)$
$v = r\omega$
$a_{T} = r\alpha$
$\tau = r_{\perp}F = rF\sin\theta$
$I = \sum m_{i}r_{i}^{2}$
$I’ = I_{\mathrm{cm}} + Md^{2}$
$\alpha_{\mathrm{sys}} = \frac{\Sigma\tau}{I_{\mathrm{sys}}} = \frac{\tau_{\mathrm{net}}}{I_{\mathrm{sys}}}$
Energy and momentum of rotating systems
$K = \tfrac{1}{2}I\omega^{2}$
$W = \tau\Delta\theta$
$L = I\omega$
$L = rmv\sin\theta$
$\Delta L = \tau\Delta t$
$\Delta x_{\mathrm{cm}} = r\Delta\theta$
Oscillations
$T = \frac{1}{f}$
$T_{s} = 2\pi\sqrt{\frac{m}{k}}$
$T_{p} = 2\pi\sqrt{\frac{\ell}{g}}$
$x = A\cos\left(2\pi ft\right)$
$x = A\sin\left(2\pi ft\right)$
Fluids
$\rho = \frac{m}{V}$
$P = \frac{F_{\perp}}{A}$
$P = P_{0} + \rho gh$
$P_{\mathrm{gauge}} = \rho gh$
$F_{b} = \rho Vg$
$A_{1}v_{1} = A_{2}v_{2}$
$P_{1} + \rho gy_{1} + \tfrac{1}{2}\rho v_{1}^{2} = P_{2} + \rho gy_{2} + \tfrac{1}{2}\rho v_{2}^{2}$
What the letters mean
$a$ 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
$m$ mass
$M$ mass
$p$ momentum
$P$ power or pressure
$r$ radius or distance
$t$ time
$T$ period
$U$ potential energy
$v$ velocity or speed
$V$ volume
$W$ work
$x$ position
$y$ vertical position
$\alpha$ angular acceleration
$\theta$ angle or angular position
$\mu$ coefficient of friction
$\rho$ density
$\tau$ torque
$\omega$ angular speed
The constants, conventions and equations on this sheet are those the College Board provides on the AP Physics exam reference information.